WO2001013332A2 - Method for generating and animating a three-dimensional human body model - Google Patents

Abstract

The present invention concerns methods of representing a limbed creature in a three-dimensional model, comprising defining no more than about 3000 adjacent polygons (in particular quadrilaterals), at least 90 % of which are quadrilaterals, each tubular section of the limbed creature having a cross-section having a multiple of two sides, and the model describing substantially all of the salient points on the body of the limbed creature. It also concerns methods of representing a head in a three-dimensional model, comprising defining no more than 500 polygons describing substantially all of the salient points on the head. Also provided is a method for reconstructing a surface representation of an individual using the models of the invention, together with methods for using a computer for reconstructing a three-dimensional surface of a limbed creature and head, systems, computer programs and computer program products for same.

Description

TITLE OF THE INVENTION Surface Representation

BACKGROUND OF THE INVENTION FIELD OF THE INVENTION

It has been well known for many years to represent three-dimensional objects using a wire frame model, and this has been widely used in the computer industry. In wire frame modelling techniques, a number of points on the surface of an object to be represented are identified and a calculation or measurement made of the position of each of those points to identify their location in some 3-dimensional coordinate system. Typically, each identified point is joined with its nearest neighbours by lines or "wires". An object may be represented with whatever degree of fidelity desired by changing the spacing of the points; in other words, by adapting the resolution with which points on the surface are stored. This is analogous to e.g. pressing chicken wire against a 3-dimensional surface and allowing it to deform such that the intersections of the wires represent points on the surface of the 3-dimensional object. The areas enclosed by wire boundaries correspond to elemental surface elements often described as being "geometric primitives". Depending on how the chicken wire is constructed, these elements may for example be triangular, rectangular, hexagonal or any other polygonal shape.

Once the coordinates of each of the vertices have been recorded, the wire frame model can be displayed and viewed from any desired angle, or the vertices moved to change the shape of the model.

The wire frame can be made to appear more realistic, for example by eliminating the transparent nature of a wire frame so that a viewer sees only the portion of the wire frame which would be visible when viewing the wire frame from a particular perspective and eliminating those lines and vertices which would be hidden by the surface in front. Textures can also be applied to the surfaces of each of the adjacent polygons and in that way a wire model can provide a single surface representing the object. Lighting effects can be applied to achieve a more realistic look. Additionally, the textures applied to the surfaces of the polygons can actually be images of sections of the surface of the object being modelled, i.e. a polygon representing a section of the surface of an arm can be given a texture comprising an image of the surface of the corresponding section of the arm being modelled. This polygonal surface can then be manipulated as the model is displayed or animated to give a realistic look to the model. This process is well known in the field of 3D computer graphics and is referred to as UV mapping. In UV mapping, each vertex (a three-dimensional X,Y,Z coordinate) of a polygon is considered have a (two-dimensional U,V) coordinate in the computer file containing the image of the surface (UV space). Thus, a triangle in three-dimensional space has coordinates (X₁,Y₁,Z₁),(X₂,Y₂,Z₂),(X₃,Y₃,Z₃) and is displayed using pixels from the equivalent triangle in UV space with the coordinates (U₁,V₁),(U₂,V₂),(U₃,V₃).

Scanning techniques have also been employed in which a grid of rectangles is projected onto an object (for example a person). The projected rectangles deform in shape depending on the shape of the object. Pairs of cameras are then used to identify the relative positions of each of the vertices of the grid, allowing subsequent data processing to calculate the three-dimensional position of each vertex, in turn allowing the construction of a three dimensional model of the object. However, such scanning techniques are limited by their ability to only operate at a fixed resolution - either they must scan everything at a very high resolution in order to define the desired features of shape of the object, in turn resulting in a very large set of data defining the object, or they must use a lower resolution and accept a loss of resolution of the desired features. For example, the human face requires a higher resolution to define its features than does the human leg. Prior art modelling and animation techniques include e.g. those of US 5870220, US 5741211, US 5732204, US 5747822, US 5793372, US 5793392 and US 5742291.

It has also been known to provide a template model for an object (e.g. a heart - see US 5889524) having specifically defined salient (i.e. anatomical or morphological) features. The defined points, edges or planes on the template model can then be mapped (realigned) to specific points, edges or planes on the object being modelled. This can be useful in allowing comparison of different members of an object class, and can help optimise modelling by providing a basic template for the model. However, these models use triangles and are not constructed to minimise the number of polygons used. Other publications of interest include US 5883631, US 5815401, US 5796400 and US 5726896.

However, the techniques used to date have been computer-driven, using "optimised" triangular template meshes and suchlike which fail to recognise that the object being modelled has a template morphology which when animated will change in specific ways. They have also failed to ensure that a minimum amount of data required to represent the object is used. Similarly, they fail to ensure that changes in resolution of the model are accompanied by a minimum possible increase in calculations required for modelling.

BRIEF SUMMARY OF THE INVENTION

The present invention overcomes the prior art disadvantages by comprising a three- dimensional model of a limbed creature, particularly quadrupeds (including quadrupeds which use feet as hands), more particularly of a human, which uses a significantly reduced number of polygons to provide a high resolution representation. The simple structure of the model means that its rendering requires fewer calculations, and is therefore faster. It is also designed so that animation of the model, particularly movements of limbs and of the face require a simplified manipulation of vertices when compared to existing models. This in turn reduces the calculations required for animation meaning that animation can be achieved more quickly. The polygons used in the model also provide the significant advantage that their resolution can be readily increased without a disproportionate increase in calculations. The increased resolution models also retain all of the original vertices (a feature which is not common to prior art techniques). This means that preliminary modelling and animation can be achieved using a low resolution model, with all of the advantages that it entails (for example fast animation). Once that has been satisfactorily completed, a higher resolution model can replace the low resolution model and the output animation generated. Since the original vertices are also present in the high resolution model, the series of transformations used to animate the low resolution model can also be used to animate the high resolution model.

Also provided by the invention are methods for reconstructing a surface representation of an individual, methods for using a computer for reconstructing a three-dimensional surface of a limbed creature, systems for reconstructing a surface of a limbed creature, computer programs for reconstructing the surface of a limbed creature, and computer program products for same.

Another aspect of the invention also concerns surface representations of just the head portion of a three dimensional model, and having the same advantages as previously described due in particular to the use of quadrilaterals, including the simplicity of the model, its ease of animation and the ease of changing its resolution. The invention also provides the above mentioned methods, systems, computer programs and computer program products for the surface representation of the head as for the surface representation of the individual.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS Figure 1 shows a back view of a first human model;

Figure 2 shows a bottom view of a first human model; Figure 3 shows a front view of a first human model;

Figure 4 shows a perspective view of a first human model;

Figure 5 shows a side view of a first human model;

Figure 6 shows a top view of a first human model;

Figure 7 shows a back view of a first human model foot;

Figure 8 shows a bottom view of a first human model foot;

Figure 9 shows a front view of a first human model foot;

Figure 10 shows a perspective view of a first human model foot;

Figure 11 shows a side view of a first human model foot;

Figure 12 shows a top view of a first human model foot;

Figure 13 shows a back view of a first human model hand;

Figure 14 shows a bottom view of a first human model hand;

Figure 15 shows a front view of a first human model hand;

Figure 16 shows a perspective view of a first human model hand;

Figure 17 shows a side view of a first human model hand;

Figure 18 shows a top view of a first human model hand;

Figure 19 shows a back view of a first human model head;

Figure 20 shows a bottom view of a first human model head;

Figure 21 shows a front view of a first human model head;

Figure 22 shows a perspective view of a first human model head;

Figure 23 shows a side view of a first human model head;

Figure 24 shows a top view of a first human model head;

Figure 25 shows a close-up view of the front of the first human model chest of Figure 3;

Figure 26 shows a back view of a second human model;

Figure 27 shows a bottom view of a second human model;

Figure 28 shows a front view of a second human model;

Figure 29 shows a perspective view of a second human model;

Figure 30 shows a side view of a second human model; Figure 31 shows a top view of a second human model; Figure 32 shows a back view of a second human model foot; Figure 33 shows a bottom view of a second human model foot; Figure 34 shows a front view of a second human model foot; Figure 35 shows a perspective view of a second human model foot; Figure 36 shows a side view of a second human model foot; Figure 37 shows a top view of a second human model foot; Figure 38 shows a bottom view of a second human model hand; Figure 39 shows a front view of a second human model hand; Figure 40 shows a perspective view of a second human model hand; Figure 41 shows a side view of a second human model hand (from the front of the model); Figure 42 shows a side view of a second human model hand (from the rear of the model); Figure 43 shows a top view of a second human model hand; Figure 44 shows a back view of a second human model head; Figure 45 shows a bottom view of a second human model head; Figure 46 shows a front view of a second human model head; Figure 47 shows a perspective view of a second human model head; Figure 48 shows a side view of a second human model head; Figure 49 shows a top view of a second human model head; Figure 50 is a flowchart representing steps of an embodiment of the present invention for using a computer for reconstructing a three-dimensional surface of a limbed creature; Figure 51 illustrates a representation of a system for reconstructing a surface of a limbed creature in accordance with one embodiment of the present invention; Figure 52 shows a perspective view of a third human model head; Figure 53 shows a standard UV space map; and Figure 54 shows a space map guide.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a method of representing a limbed creature in a three dimensional model, comprising defining no more than about 3000 adjacent polygons, at least 90% of which are quadrilaterals, each tubular section of the limbed creature having a cross-section having a multiple of two sides, and the model describing substantially all of the salient points on the body of the limbed creature.

The present invention also provides for a method of representing a limbed creature in a three-dimensional model, comprising defining no more than about 2900, 2800, 2700, 2600, 2500, 2400, 2300, 2200, 2100 or in particular 2000 adjacent polygons.

In particular the tubular sections may be cylindrical and may have at least four sides. As described in the model below, it is particularly useful for cylindrical structures such as limbs, fingers etc. to have eight sides.

It has not been previously suggested that the whole of a limbed creature, particularly a quadruped may successfully be modelled using no more than about 2000 polygons, and as demonstrated below it can be readily achieved using no more than 1820 polygons. The reason that such a small number of polygons can be used is due to the fact that they primarily comprise quadrilaterals, and also because the edges of the quadrilaterals define axes which 'flow' through the model. The use of quadrilaterals makes the joining of adjacent quadrilaterals simple, and provides the distinct advantage of allowing a mathematically simple sub-division of the polygons themselves. For example the subdivision of triangles is not as simple as sub-division of quadrilaterals and typically results in the production of a "bumpy" surface. Since the sub-division of quadrilaterals is mathematically simple it means that a single sub-division or multiple sub-divisions of the whole of the model or just parts of it may be made whilst not causing the same magnitude of increase in calculations and other disadvantages usually encountered with sub-division of e.g. triangular polygons.

The use of quadrilaterals also provides the advantage that when sub-division occurs the vertices of the original quadrilateral are retained. Thus two models of a quadruped may be constructed - the first being a low-resolution model, and the second being a high- resolution model. The (second) high resolution model is arrived at by the sub-division of the quadrilaterals of the (first) low resolution model, together with alignment of the newly defined quadrilaterals with image data from the quadruped to be modelled. The first model may then be rapidly manipulated, for exampled animated, and when the exact nature of the set of manipulations is decided, the manipulations may then be applied to the high resolution model. In this way a whole series of possibilities may be easily explored before the relatively processor-intensive manipulation of the high resolution model is undertaken.

Similarly, the simultaneous animation of a plurality of models for example at least 10, 20, 50 or 100 models, may be performed quickly using low resolution models. Once a series of steps defining the animation of the model has been performed at low resolution, the high resolution models may then be animated using the same manipulations (i.e. transformations) of the vertices shared by the low- and high-resolution models.

Additionally, where a model is displayed so that it is perceived to be at a distance from a viewer and so quite small and having indistinguishable details, a subset of vertices and thus a smaller amount of data can be used. In particular, regions that would not be easily distinguishable at a distance, for example fingers, can be modelled with fewer vertices than are needed when the model appears to be nearby. This means that the same manipulations (i.e. transformations) of the vertices that would animate the model if perceived to be nearby can be used to animate the perceived distant model whilst requiring fewer calculations due to the reduced number of vertices, the distant model behaving in the same way that a nearby model would.

As mentioned above, the substantial advantages of the present invention derive in part from the use of quadrilaterals. As demonstrated below, a model comprising 100% quadrilaterals may be readily created. Generally, models comprising at least 90%, for example at least 95, 96, 97, 98 or 99% quadrilaterals may be created. Such models will also benefit from the general advantages provided by the use of quadrilaterals, although not to as great an extent as the model comprising solely quadrilaterals.

Due to the anatomical and morphological similarities between limbed creatures such as quadrupeds, and also due to the nature of the model which is generated, the model may be of for example a primate, ape, monkey, chimpanzee, canine, feline, equestrian or bovine. In particular, quadrupeds may be a quadrumane, for example an anthropoid, for example a human. Thus the term quadruped includes animals which use feet as hands and thus includes animals with four limbs but which display a bipedal stance.

An important feature of the present invention is the fact that by using no more than about 3000 polygons, for example by using 1820 quadrilaterals, it is possible to describe substantially all of the salient points on the body of the limbed creature (for example quadruped), and achieve a photorealistic representation (reconstruction) of the surface of the limbed creature. It may be that in some models there is no desire to model particular salient points (anatomical or morphological features) to a high resolution. However, the salient points to be described may include substantially (for example at least eight or nine of) the group comprising (i) the top of the head; (ii) the tip of the chin; (iii) the centre of each pupil; (iv) the north, south, east and west extremities of each eye; (v) the top, bottom and widest points of each ear; (vi) the top of the nose, centre of the bridge of the nose, the centre of each nostril, and the widest points of the nose; (vii) the centre, centre-left and centre-right of the septum; (viii) the centre of the mouth, the end- points of the mouth, the two apexes of the upper lip, and the bottom most point of the lower lip; (ix) the centre of each nipple; and (x) the front-most, back-most and widest points of each foot.

Other anatomical features not necessarily identified by a single point but by an edge or a plane may also be described. In particular these include the lengths, widths and circumferences of each skeletal section, for example the jointed sections of each limb, together with the length and width of the neck and trunk.

As mentioned above, it may not be desirable to describe (i.e. define as part of the model) all of the morphological features of a limbed creature (quadruped) - for example the toes of the feet might not be described, nor may the fingers of the hand. If the features of the hand are described in detail, they may include each jointed section of each digit, the tip of each digit (finger and thumb), and may describe surfaces defining the fingernails. The digits (toes) of the feet may be described in the same way as the fingers. Similarly the Adam's apple may be described. The genitalia may also be described. The model detailed below described the hands in detail, but does not provide detailed descriptions of the feet or genitalia. However, it is possible to describe these features in a model having no more than about 3000 polygons in total, and achieve a photorealistic model of a limbed creature such as a quadruped.

In order to simplify the model which is constructed, the left and right sides may be symmetrical. However, the re-alignment of the vertices of the model in order to effect a reconstruction of a surface representation may result in a loss of symmetry. Starting with an initially symmetrical model whose vertices are subsequently realigned to reconstruct a surface representation of a limbed creature, the data set describing the symmetrical polygons and vertices may be reduced in size by only describing about one half of them (if the model has a central axis about which the left and right sides are symmetrical and on which rests vertices then the appropriate vertices and polygons will need describing).

The models constructed according to the present invention are particularly useful for animation purposes due to their simplicity. Their usefulness for animation purposes (as well as general non-animation purposes) may be enhanced by the provision of a plurality of "loft levels" which describe jointed sections of the body and which when animated will deform in a manner similar to the modelled body. In this way, stretching of surfaces of the model which are not stretched in the limbed creature being modelled is avoided, helping to make the models more lifelike. In particular, each long-bone section (upper- and lower-limbs) may have two loft levels, providing what has been found to be the minimum number of vertices necessary to represent the section properly at multiple levels of sub-division. Each simple joint (knuckle, wrist/ankle and knee/elbow) may have two loft levels, positioned above and below the joint itself. It has been found that this is the minimum configuration which can represent the joints through their full range of movement at multiple levels of sub-division. Each complex joint (shoulder, hip and base of the thumb) may have three loft levels. This has been found to be the minimum configuration which can represent the complex joints through their full range of movement at multiple levels of sub-division.

The model may include polygons at the front upper section of the trunk describing breasts and nipples.

Also useful in ensuring mathematical simplicity of the model and easy animation is the feature of having flowing axial lines. These axial lines may in particular include lines which rest on the Y=0, Z=0 and X=0 axes. In particular, the model may have, defined by vertices of the polygons, in outstretched stance (as shown by the Figures), front and back medial axial lines on the forelimbs forming a continuous loop connecting each forelimb via the chest and back and continuing around every digit, said medial lines resting on the Z=0 axis. The model may have, defined by vertices of the polygons, in outstretched stance side medial axial lines on the head, neck, forelimbs, trunk and hind limbs forming a continuous loop connecting the top of the head via the centre of each forelimb, the centre of the side of the trunk, and the centre of the lower limbs, said medial lines resting on the Y=0 axis. The model may have, defined by edges of the polygons, in outstretched stance front and back medial axial lines on the head, neck and trunk forming a continuous loop connecting the top of the head via the centre of the head, centre of the chest and centre of the back, said medial lines resting on the X=0 axis.

In particular, the method of the present invention may define exactly or substantially the model of Figures 1-25 or the model of Figures 26-49, optionally further defined by Tables 1 and 2 respectively. Individual polygons or sets of polygons may be subdivided to for example provide a smoother surface or allow for further detail, whilst still falling within the scope of the invention. For example, sub-dividing the region of the mesh that represents the hair may allow a better representation of long hair with a smooth silhouette, or the extra detail may allow better representation of headgear such as hats. An example of such a subdivision is shown in the model of Figure 52 which shows a human head and can be compared to Figure 22 where subdivision has not been used. Subdivision can be applied wherever it is necessary to show greater detail.

The method of the present invention may be a method for reconstructing a surface representation of individual member of the modelled class of limbed creature, comprising the additional step of determining the coordinates of the salient points of the individual which are described by the model limbed creature, and aligning the salient points of the model limbed creature to the salient points of the individual limbed creature to reconstruct the surface of the individual limbed creature. The interpolated points may also be aligned. Figure 50 is a flow chart representing steps in a method provided in accordance with an embodiment of the present invention for using a computer for reconstructing a three- dimensional surface of a limbed creature. In a first step 502, instructions executed by the computer define a three-dimensional model of a limbed creature according to the method of the present invention. In one embodiment of the invention, the model comprises no more than about 3000 adjacent polygons, at least ninety percent of which are quadrilaterals, wherein each tubular section of the limbed creature has a cross-section having a multiple of two sides, and the model describes salient points on the body of the limbed creature. In another step 504, instructions executed by the computer generate data from imaging the limbed creature, the data including the coordinates of the salient points described by the model limbed creature. In a step 506, instructions executed by the computer align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature. Figure 51 illustrates a representation of a system for reconstructing a surface of a limbed creature in accordance with one embodiment of the present invention. The system 510 includes an imaging system 512 for producing images of a limbed creature. The system 510 also includes a memory 514. The memory 514 stores a three-dimensional model of a limbed creature 516, which model, in one embodiment of the invention, comprises no more than about 3000 adjacent polygons, at least ninety percent of which are quadrilaterals, each tubular section of the limbed creature having a cross-section having a multiple of two sides, said model describing salient points on the body of the limbed creature. The memory 514 also stores data 518 defining the coordinates of the salient points of the imaged limbed creature as described by the model limbed creature. The memory 514 also stores machine instructions 520 that define steps for processing the data derived from the images using the three-dimensional model. The system 510 further includes a processor 522 that is coupled to the memory 514, said processor 522 executing the machine instructions 520, causing the processor 522 to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature 524. Also provided according to the present invention is a method for using a computer for reconstructing a three-dimensional surface of a limbed creature comprising the steps of: a) defining a three-dimensional model of a limbed creature according to the method of the present invention, said model describing salient points on the body of the limbed creature; b) providing data from imaging the limbed creature, said data including the coordinates of the salient points described by the model limbed creature; and c) aligning the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature.

Also provided according to the present invention is a system for reconstructing a surface of a limbed creature, comprising: a) an imaging system for producing images of the limbed creature; b) a memory for storing: i) a three-dimensional model of a limbed creature according to the present invention, said model describing salient points on the body of the limbed creature; ii) data defining the coordinates of the salient points of the imaged limbed creature as described by the model limbed creature; and iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and c) a processor that is coupled to the memory, said processor executing the machine instructions, causing the processor to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature.

Also provided according to the present invention is a system for displaying a reconstruction of a limbed creature, comprising: a) a memory for storing: i) a three-dimensional model of a limbed creature according to the present invention, said model describing salient points on the body of the limbed creature; ii) data defining the coordinates of the salient points of an imaged limbed creature as described by the model limbed creature; iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and iv) machine instructions that define steps for displaying the images obtained; b) display means for displaying images; and c) a processor that is coupled to the memory and display means , said processor executing the machine instructions, causing the processor to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature and display said reconstruction on said display means.

Also provided according to the present invention is a computer program for reconstructing the surface of a limbed creature, comprising: i) program code comprising data defining a three-dimensional model of a limbed creature according to the present invention, said model describing salient points on the body of the limbed creature; ii) program code for effecting the input of data defining the coordinates of the salient points of imaged limbed creature as described by the model limbed creature; and iii) program code defining steps for processing the inputted data using the three-dimensional model to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature. Also provided according to the present invention is a computer program product for reconstructing the surface of a limbed creature, comprising a computer usable medium having computer readable program code means according to the present invention embodied in said medium.

The present invention can be put to a wide range of uses, for example the photorealistic representation of a human head or whole body by re-alignment of vertices against two photographs (i.e. forward-facing and side-profile). Similarly vertices can be re-aligned against salient points taken from a 45 ° side elevation (i.e. an INS-style photograph, midway between a front and side view), although this technique does require approximations of some features to be made by e.g. assuming symmetry. This technique may in particular be used to reconstruct face surfaces. Vertices can also be re-aligned against a front-view. This method requires the approximation of depth information by eg. assuming average values. Again this can be applied to the reconstruction ef faces.

The invention can also be used to measure the bodily dimensions of people from e.g. video or still images by aligning the salient points of the model to those in the images and then using the aligned model as a "ruler". Once a surface image of a person has been created, the vertices may be altered to represent the results of surgical procedures, for example cosmetic surgical procedures. Alternatively, models of humans can be used to provide "virtual tailoring" where the wearing of items of clothing can be modelled, or where a model of individual can be used to enable the manufacture of custom-fitted clothing.

The model can be used to reconstruct a surface representation of a person in a photo-fit style fashion in order to produce a three-dimensional model of a suspect.

Video games are also an important application of the present invention - it is always desirable to produce the most accurate three-dimensional models in video games whilst ensuring that their rendering and animation requires the least possible computer processing power. The models of the present invention are, as described above, mathematically simple and so their use in video games is particularly desirable.

When a reconstruction of the surface of a limbed creature as a quadrilateral mesh is to be displayed on a computer system, it is usually converted into a mesh of triangles for rendering by the hardware of the computer system due to limitations of the display hardware. This triangulation process is a simple well-known procedure built into the hardware or device-level software. It is dependent upon the order that the vertices are sent to the hardware rendering system, but this order is not usually considered when developing a model. When an image is displayed without vertex order being taken into account during development, there can be errors of the smoothness of the surface particularly in areas with complex curvature. In the model of Example 1 (described in more detail later) the order of the vertices has been selected so that the vertices are presented to the hardware rendering system in an order that forces each quadrilateral to be triangulated so that no errors in smoothness occur. Further information on triangulation in general can be found at the following website written by Jonathan Shewchuk, Assistant Professor in Computer Science University of California at Berkeley - http://www.cs.berkeley.edu/~jrs/meshf99/ .

The models and surface reconstructions of the present invention may also be used to construct a database, for example to enable computer-recognition of people by the location of their salient points as described by the model.

The models may also be used in virtual-reality environments or to provide e.g. video conferencing and virtual-meetings at low-bandwidth between computers which have stored on them the same basic limbed creature model - once an individual has been imaged and a surface representation of the individual constructed on a first computer by re-alignment of vertices of a model limbed creature according to the salient points defined by the imaging, the coordinates of the salient points and any surface textures (or the differences between them and the same points/textures of the basic limbed creature stored on the second computer) can be transmitted between first and second computers as a relatively small set of data, and used to achieve the same reconstruction on the second computer by re-aligning vertices of the model limbed creature which is already stored on the second computer. Alternatively the set of re-alignments of vertices may be sent between the two computers. New limbed creatures may be defined on a computer by simply providing it with a new set of coordinates for vertices of the model, the model using the same polygons defined by the same sets of vertices to describe the different limbed creatures. Animation of models can also be achieved using relatively small sets of data which define only the re-alignments of the vertices which occur from one frame to the next. Similarly, using a basic model limbed creature stored on first and second computers, a streaming animation of a reconstructed surface representation of an individual may be sent between the two computers in the form of a stream of data first defining the vertices of the reconstructed surface representation and the textures to be applied to the surface, and then a series of frames which need only define the realignments of vertices from frame to frame. Assuming that from frame to frame the realignments of the vertices are limited in scope (for example primarily re-alignments of vertices around the mouth during an animation of a conversation) then the set of data required from frame to frame will be limited, allowing for a low-bandwidth transmission. Key-frames may be sent, for example every few seconds, which define the absolute coordinates of the vertices, rather than the re-alignments necessary, and/or which define the surface textures. In this way, loss of data packets over the connection between two computers would not necessitate the re-sending of the whole of the transmission, instead being corrected by the next key frame. Thus the present invention is particularly useful for providing models, reconstructions of surface representations and animations of same across connections such as internet connections, even relatively low bandwidth connections, for example as streaming broadcasts. Using the quadrilateral meshes (i.e. the models) of the present invention it is possible to define a photorealistic image of an individual for transmission as just the 1822 vertices plus a suitable (mainly facial) image map. With the polygonal relationships between vertices being predefined, there is no need for them to be transmitted. The vertices when stored as real numbers occupy 3 (x, y, z) x 1822 x 2 bytes, ie 10932 bytes. If symmetry is assumed, this can be reduced to approximately 6000 bytes (all axial x=0 vertices are included). A suitable image map for the polygons of the face in a compressed format such as jpeg can be provided in under 40k, which means that it is possible to provide an animatable, photorealistic 3D individual in less than 50k of data.

Data compression may be enhanced in a number of ways, in particular in a client-server situation where the client is only capable of displaying or animating a certain number of polygons, or where the client only wishes to display a certain part of a model. Examples of data compression include basing specific models upon a generic model as mentioned above. For example, a standard "human character" could be pre-installed on a client machine. Data for vertices of a specific model could then be sent identifying the generic model to be used and then the deviations from the selected generic model, thus minimising the amount of data to be sent. A catalogue of standard "characters"could be pre-installed on a client machine, for example representing males and females, and different boy morphologies such as endomorph, mesomorph and ectomorph. Similarly, non-human characters could form part of a pre-installed catalogue. Standard image maps or textures may be provided for the polygons of the model, again possibly in the format of a catalogue which can simply be referenced in a data set representing a model. Thus data sets required to describe a specific model and its animation can be significantly reduced in size. In a client-server situation it would also be possible for the client to indicate to the server that it only desired to be sent a reduced resolution model. This could for example be because of limited bandwidth in the client-server connection or because the client is only capable of displaying or animating a limited number of polygons. The server, whilst maintaining the resolution of its model could then selectively filter the data sets sent to the client describing the model and/or its animation in order to reduce the size of data set. For example a limited bandwidth connection could be optimised by sending at full resolution the representation of a model referencing a character in a catalogue. Although taking a relatively long period to send, once received by the client, the animation of the model could then be achieved at full speed despite the low bandwidth of the connection by only sending animation data for key vertices (for example the salient points defined by the model), the client machine then estimating the movement of other vertices. In order to aid such estimates by client machines, models (for example characters in a catalogue) may also be provided with reference data which defines the specific or general movement of non-key vertices in relation to key vertices. Similarly, if the client machine is only capable of displaying or animating a limited number or polygons then the initial data set defining the model may be filtered to reduce the number of polygons in the model. Data sets can also be minimised if a client machine is only displaying part of a model, for example the head. In such a case it would not be necessary for the client machine to be sent data representing the movement of other parts of the body. Thus a client could indicate to a server that it required only data for the head of a model and the server could filter data sets accordingly. When the client again wanted to display the full model then the server could for example send a "key frame" (above), or a data set defining movement of the model since the last full set of data was sent, or for example a data set defining movement of the head since the last filtered data set for head-only movements, and the movement of the rest of the body since the last full data set.

The models may also be used to provide "virtual celebrities" or "virtual characters" comprising heads or entire bodies which may be incorporated into for example games or video productions. Such virtual characters may additionally comprise a digital signature as part of the data set defining them to verify the content of the data set. Such digital signatures could be used to ensure that software including such characters was appropriately licensed or authorised or that data defining a character was from a verified source. In particular the signature could be produced using standard public/private key cryptographic techniques such as PGP. It could for example include a signature part from the licensor of the character and a signature part from the licensee. In that way, a signature for a particular data set would verify both the licensor and licensee, meaning that a non-licensee could not simply copy a signed character data set from a licensee's product.

The data sets defining the models, be they virtual celebrities, characters defined for specific individuals or otherwise, may be readily stored on any appropriate medium. In particular they may be stored on portable media for example magnetic cards, so-called "smart cards", mobile phones (for example on SIM cards used with mobile phones) or any other suitable media. This provides the opportunity for applications such as security (identification of people), medical applications such as imaging, shopping in which a user could provide their stored "character" and have it used as a template for modelling clothes and suchlike, video games and other entertainment. In order to guarantee the authenticity of such stored data sets, digital signatures and other cryptographic techniques may be used as discussed above. In the case of storage on portable media carried by e.g. a mobile phone, where the storage device also having a data transmission capability such as infra-red (e.g. using the IrDA protocol) or Bluetooth, data may be transmitted to a receiver without needing to physically contact the receiver with the storage means.

The model may also be used in e.g. morphing techniques - models of different limbed creatures may be constructed, for example of a gorilla and human, having a common set of quadrilaterals, the quadrilaterals having different vertices. Using the set of vertices for the model human, a surface representation of a human could then be reconstructed from applying image data and coordinates of salient points to re-align the vertices. The realignments could then be applied to the vertices of the model gorilla, in effect morphing the human into a gorilla. A set of intermediate steps between the coordinates of the vertices of the human and gorilla surface representations could then be generated to animate the morphing process. Heads of different creatures may be similarly morphed.

As discussed above, a surface representation of an individual may be reconstructed from scanned images by aligning the salient points of a model to the salient points of the individual identified in the scanned images. This can be readily achieved using standard scanning apparatus which have been configured to identify the salient points of the models of the present invention, and is well within the abilities of one skilled in the art, given the information disclosed herein. Similarly computer chips, graphics cards and software may be optimised for the modelling and reconstruction of the present invention.

Other uses of the invention lie in modelling in engineering simulations, design and ergonomic applications.

The heads modelled in Figures 19-24 and 44-49 and described by Tables 1 and 2 also forms a separate part of the invention. The head can be described by a set of 495 vertices defining 486 adjacent quadrilaterals as shown in Figures 19-24 and Table 1, the head model including the neck. Alternatively the head model may exclude the neck and may comprise 461 vertices defining 453 adjacent quadrilaterals as shown in Figures 44-49 and Table 2. This model of the head has been created to enable photorealistic representation of an individuals head, particularly their face. In particular it is simple to animate due to the fact that the polygons defining the head are arranged such that facial movements such as movements of the mouth, cheeks and eyes cause a "natural" seeming distortion of the polygons. It has not previously been suggested that it is possible to define a model of a face, particularly a face suited to animation or allowing the photorealistic representation or reconstruction of an individual's face, using so few polygons, in particular not using quadrilaterals with all of the advantages that their use entails. Thus according to the present invention there is also provided a method of representing a head in a three dimensional model, comprising defining no more than 500 polygons describing substantially all of the salient points on the head.

At least 95, 96, 97, 98 or 99%) of said polygons may be quadrilaterals, and all of said polygons may be quadrilateral.

The model may comprise for example 453 or 486 quadrilaterals.

The model may describe salient points comprising substantially (for example at least six, seven or eight of) the group of: i) the top of the head; ii) the tip of the chin; iii) the centre of each pupil, iv) the north, south, east and west extremities of each eye; v) the top, bottom and widest points of each ear; vi) the top of the nose, centre of the bridge of the nose, the centre of each nostril, and the widest point of the nose; vii) the centre of the mouth, the end point of the mouth, the two apexes of the upper lip, and the bottom most point of the lower lip; viii) the centre, centre-left and centre-right of the septum; and ix) the tip of the Adam's apple.

As with the limbed creature model, the left and right sides of the model may be symmetrical, although of course this symmetry may be lost upon realignment of vertices to reconstruct a surface representation of an individual face. The model may in particular be used for animation, and as discussed above it provides a particular advantage in being simple (i.e. mathematically simple) to animate in contrast with existing prior art models.

The method may be a method for reconstructing a surface representation of the head of an individual, comprising the additional step of determining the coordinates of the salient points of the individual which are described by the model head, and aligning the salient points of the model head to the salient points of the individual's head to reconstruct the surface of the individual's head.

Also provided is a system for displaying a reconstruction of a head, comprising: a) a memory for storing: i) a three-dimensional model of a head, said model describing salient points on the head; ii) data defining the coordinates of the salient points of an imaged head as described by the model head; iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and iv) machine instructions that define steps for displaying the images obtained; b) display means for displaying images; and c) a processor that is coupled to the memory and display means , said processor executing the machine instructions, causing the processor to align the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head and display said reconstruction on said display means.

Also provided is a method for using a computer for reconstructing a three-dimensional surface of a head comprising the steps of: a) defining a three-dimensional model of a head according to the method of the present invention, said model describing salient points of the head; b) providing data from imaging the head, said data including the coordinates of the salient points described by the model head; and c) aligning the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

Also provided is a system for reconstructing a surface of a head, comprising: a) an imaging system for producing images of the head; b) a memory for storing: i) a three-dimensional model of a head according to the present invention, said model describing salient points of the head; ii) data defining the coordinates of the salient points of the imaged head as described by the model head; and iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and c) a processor that is coupled to the memory, said processor executing the machine instructions, causing the processor to aligning the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

Also provided is a computer program for reconstructing the surface of a head, comprising: i) program code comprising data defining a three-dimensional model of a head according to the present invention, said model describing salient points of the head; ii) program code for effecting the input of data defining the coordinates of the salient points of the imaged head as described by the model head; and iii) program code defining steps for processing the inputted data using the three-dimensional model to align the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

Also provided is a computer program product for reconstructing the surface of a head, comprising a computer usable medium having computer readable program code means according to the present invention embodied in said medium.

As an alternative to representation of shapes by a series of polygons, it is possible to use objects well known in the art such as splines, nurbs, metaballs and voxels. These objects may be linked to form a mesh using models having control points defined by the vertices of the models of the present invention, and the present invention extends to such models, their uses, systems, computer programs and computer program products embodying same.

An alternative to the mapping of salient points described previously is the use of UV mapping, using a standardised UV space map for the image to be applied to the mesh. In such a standardised UV space map, the source image data is modified to fit the standard UV space. An example of a standardised UV space map is shown in Figure 53 where the block of polygons labelled 310 represents the torso of a model, 315 represents the head, 320 represents right arm, 325 represents the left arm, 330 represents the back of the hand, 335 represents the palm of the hand, 340 represents the upper foot, 345 represents the lower foot, 350 represents the right leg, 355 represents the left leg, 360 represents the tongue, 365 represents the upper teeth, 370 represents the lower teeth, 375 represents the left eye and 380 represents the right eye. Data from such UV space maps can be fully interchanged in whole or in part with data from other UV space maps constructed in the same way. As all the points in any mesh derived from the original mesh, such as a low resolution mesh or a subdivided mesh, are fixed relative to the original mesh a single UV space map can be applied to any model derived from the original mesh. This has the advantages of being able to mix and match details such as clothing or hairstyles between models, or alternatively the standard UV space coordinates can be stored with an application rather than stored with the models used in the application thus reducing the size of the models. Mapping the data from the UV space map onto a mesh is a well-known procedure, further information about the subject can be found, for example, at the website of Primitive Itch of Richmond, California - http://www.primitiveitch.com/truetexture/tutorial2.html .

To facilitate manual UV space mapping, a guide system allows a user to match image features to the UV space map using a standard computer paint package. Figure 54 shows such a guide system derived from the standardised mesh of Figure 53. A user can apply features directly onto the guide and they will appear correctly placed on the three- dimensional model. Alternatively, the salient points of the guide map may be moved so that they align with the salient points of the image so that salient points are correctly located on the three-dimensional model. An example of a space map guide system is shown in Figure 54 where area labelled 310 represents the torso of a model, 315 represents the head, 320 represents right arm, 325 represents the left arm, 330 represents the back of the hand, 335 represents the palm of the hand, 340 represents the upper foot, 345 represents the lower foot, 350 represents the right leg, 355 represents the left leg, 360 represents the tongue, 365 represents the upper teeth, 370 represents the lower teeth, 375 represents the left eye and 380 represents the right eye. Further information about mapping a two-dimensional image onto a three-dimensional model can be found, for example,at the website of Uni graphics Solutions Inc of St Louis, Missouri http://www.ugsolutions.com/products/unigraphics/cad/photo/user/uvtext.html .

EXAMPLE 1

As can be seen from Figures 1-25 and Table 1, a first quadruped is modelled in a neutral stance, with the legs apart, 12% off vertical, the arms outstretched horizontally to their maximum span and the fingers straight and apart. The model comprises a total 1822 vertices used to define 1820 quadrilaterals, and is symmetrical about the X=0 axis (i.e. the left and right sides of the body are symmetrical). The following salient points are described: the top of the head 10, the tip of the chin 20, the centre of each pupil 30, the north, south, east and west extremities of each eye 41-44, the top, bottom and widest points of each ear 51-53, the top of the nose 60, centre of the bridge of the nose 61, the centre of each nostril 62, and the widest point of the nose 63, the centre of the mouth 70, the end-point of the mouth 71, the two apexes of the upper lip 72, and the bottom most point of the lower lip 73, the centre 80, centre-left 81 and centre-right 82 of the septum, the centre of each nipple 90, the front-most 100, widest 101,102 and back 103 points of each foot, the tips of each finger 110-113 and thumb 114, and the Adam's apple 120.

Each of the four limbs 130-133 and each of the fingers and thumb 140-144 is octagonal, allowing for simple subdivision. The two legs 130,131 join to the trunk 150 at the pelvis 160 and all of the axial lines from the legs 130,131 flow through to the trunk 150, which is thus primarily a 16-sided cylinder.

The two arms 132, 133 join to the trunk at the shoulders 170. Three of the axial lines flow through to the neck 180 and on through the head 190. Three of the remaining axial lines flow through to the trunk 150 and legs 130,131. The two remaining axial lines (front and back medial axial lines) on the arms 132,133 form a continuous loop 200 (marked as section I-I) resting on the Z=0 axis, connecting each arm 132,133 via the upper region of the trunk 150 (i.e. chest) and back 210 and continue around every finger 140-143.

Where each body part joins the next (e.g. digits 140-144 to hands 220, hands 220 to arms 132,133, arms 132,133 to trunk 150 etc.) all connecting polygons are quadrilateral, optimised for smooth subdivision. Each long bone section and simple joint has two loft-levels, with each complex joint having three loft-levels. For example, shoulder 230 comprises three loft levels 231,232,233 connecting it to upper portion 240 of arm 132 having two loft levels 241,242 which in turn connect to elbow 250 having two loft levels 251,252.

Breasts 260 are formed from eight quadrilaterals 261-268 arranged around irregular octagon which itself is split into four quadrilaterals 271-274.

The model has, defined by edges of the polygons side medial axial lines on head 10, neck 180, arms 132,133, trunk 150 and legs 130,131 forming a continuous loop 280 (marked as section II-II) connecting the top of head 10 via the centre of each arm 132,133, the centre of the side of trunk 150, and the centre of each leg 130,131, said medial lines resting on the Y=0 axis.

The model has, defined by edges of the polygons front and back medial axial lines on head 10, neck 180 and trunk 150 forming a continuous loop 290 (marked as section III- III) connecting the top of head 10 via the centre of head 10, centre of trunk 150 and centre of back 210, said medial lines resting on the X=0 axis.

In use, the person to be modelled is scanned to produce a series of images and the X, Y and Z coordinates of the salient points 10, 20, 30, 41-44, 51-53, 60-63, 70-73, 80-81, 90, 100-102, 100-114 and 120 determined. The salient points 10, 20, 30, 41-44, 51-53, 60- 63, 70-73, 80-81, 90, 100-102, 100-114 and 120 of the model of Figures 1-25 are then aligned to the coordinates derived from the scanned images to give the model the correct basic shape and allow the interpolation of the alignment of the other vertices of the model. The surface of the person being modelled can then be reconstructed, The quadrilaterals joining to provide a single surface. Smoothing algorithms are then be applied to the reconstructed surface to make it more realistic. Finally, sections of the scanned images correlating to the polygons of the model are then used to provide texture (i.e. surface images) for the smoothed surface to provide a finished model. The smoothing stage is optional one and may be performed using a wide range of algorithms which are available in commercially available software packages such as the World Toolkit produced by Engineering Animation Inc. (www.sense8.com). Other particularly useful software packages include TrueSpace and 3D Studio.

EXAMPLE 2

In a second model (Figures 26-49; Table 2) comprises 1820 adjacent quadrilaterals defined by 1822 vertices. It has the same basic structure as the first model and the same reference numerals are used for the first and second models.

As can be seen from the first model head of Figures 19-24, the quadrilaterals defining the top of the head are arranged such that their outer edges define a series of 16-sided concentric polygons centred on the top of the head 10. The second model head of Figures 44-49 has a different arrangement at the top of the head - a more rectangular arrangement of the quadrilaterals. Other differences between the first and second models include the quadrilaterals defining the shoulders (when viewed from the back).

Thus it is clear that a range of models according to the present invention can be readily created and modified by a person skilled in the art, particularly by the modification of the first and second models described herein.

It will be appreciated that it is not intended to limit the invention to the above example only, many variations, such as might readily occur to one skilled in the art, being possible, without departing from the scope thereof as defined by the appended claims.

Table 1 defines the vertices and polygons of the first human model of Figures 1-25. Table 2 defines the vertices and polygons of the second human model of Figures 26-49. The vertices and polygons of Table 1 are presented in a commonly recognised format. The Table is split into two sections - the first section defines the number of vertices then lists them in numerical order. Values given for coordinates are the distance in metres from the origin (i.e. the point in the Figures at which the planes I-I, II-II and III-III intersect) and are given in the order X,Y,Z. The second section of the Table comprises a list of polygons sorted in numerical order. The first column gives the number of the individual polygon, the subsequent columns identify the vertices that define the polygon.

Table 2 is presented in a similar format, again split into two sections. In the first section the vertices are listed in numerical order and the coordinates are again the distance in metres from the origin and given in the order Y,Z and X. The second section of the Table comprises a list of polygons sorted according to their numeric order. The first column defines the number of vertices in the polygon, and the subsequent columns identify the vertices which define the polygon. All polygons defined in the Tables face outwards in the resultant models.

Table 1

[bioVirtual Standard Mesh Report]

Mesh Contents:

Vertices : 1822

Polygons : 1820

Vertex Positions: (X, Y, Z)

Vertex 0) (-0 0005 1 2334 0 0002)

Vertex 1) (-0 0004 1 2279 0 1096)

Vertex 2) (-0 0005 1 2152 -0 1122)

Vertex 3) ( o 1236 1 2116 0 0001)

Vertex 4) (-0 1245 1 2114 0 0002)

Vertex 5) ( o 1237 1 2079 0 1037)

Vertex 6) (-0 1245 1 2078 0 1039)

Vertex 7) ( o 1236 1 1952 -0 1057)

Vertex 8) (-0 1246 1 1950 -0 1056)

Vertex 9) (-0 0003 1 1867 0 2096)

Vertex 10) ( o 1238 1 1696 0 1975)

Vertex 11) (-0 1244 1 1694 0 1976)

Vertex 12) (-0 0005 1 1441 -0 2181)

Vertex 13) ( o 2065 1 1434 0 0004)

Vertex 14) (-0 2102 1 1428 0 0003)

Vertex 15) ( o 2018 1 1390 0 0876)

Vertex 16) (-0 2073 1 1384 0 0878)

Vertex 17) (-0 0003 1 1349 0 2603)

Vertex 18) ( o 1236 1 1314 -0 2043)

Vertex 19) (-0 1246 1 1312 r -0 2041)

Vertex 20) ( o 1977 1 1277 -0 0875)

Vertex 21) (-0 2019 1 1275 -0 0873)

Vertex 22) ( o 1238 1 1240 0 2479)

Vertex 23) (-0 1243 1 1238 0 2480)

Vertex 24) ( o 1968 1 1131 0 1748)

Vertex 25) (-0 1974 1 1128 0 1750)

Vertex 26) ( o 1893 1 0876 -0 1532)

Vertex 27) (-0 1902 1 0873 -0 1530)

Vertex 28) ( o 1895 1 0839 0 2117)

Vertex 29) (-0 1900 1 0836 0 2119)

Vertex 30) (-o 0002 1 0600 0 3030)

Vertex 31) ( o 1239 1 0554 0 2854)

Vertex 32) (-0 1243 1 0552 0 2856)

Vertex 33) ( o 1939 1 0408 0 2281)

Vertex 34) (-0 1958 1 0406 0 2283)

Vertex 35) (-0 0005 1 0382 -0 2783)

Vertex 36) ( o 1236 1 0328 -0 2627)

Vertex 37) (-0 1245 1 0327 -0 2626)

Vertex 38) ( o 2308 1 0219 0 1606)

Vertex 39) (-0 2312 1 0216 0 1608)

Vertex 40) ( o 2515 1 0055 0 0876)

Vertex 41) (-0 2521 1 0052 0 0878)

Vertex 42) ( o 2551 1 0052 -0 0000)

Vertex 43) ( o 2441 1 0052 -0 0876)

Vertex 44) ( o 2039 1 0052 -0 1901)

Vertex 45) (-0 2048 1 0049 -0 1899)

Vertex 46) (-0 2449 1 0048 -0 0873)

Vertex 47) (-0 2558 1 0048 0 0003)

Vertex 48) ( o 1240 0 9485 0 3179)

Vertex 49) (-0 1242 0 9483 0 3180)

Vertex 50) ( o 2009 0 9376 0 2547)

Vertex 51) (-0 2012 0 9373 0 2549)

Vertex 52) (-0 0001 0 9246 0 3457) Vertex 53) ( o.2625, 0.9183, -0.0000)

Vertex 54) ( o. 2607, 0. 9183, 0. 0875)

Vertex 55) ( o. 2551, 0. 9183, -0. 0876)

Vertex 56) ( o. 2443, 0. 9183, 0. 1532)

Vertex 57) ( o. 2113, 0. 9183, -0. 2091)

Vertex 58) ( o. 1237 0 9183, -0 2809)

Vertex 59) (-0. 0004 0 9182, -0 2977)

Vertex 60) (-0. 1245 0. 9181, -0 2808)

Vertex 61) (-0. 2120, 0. 9180, -0 2089)

Vertex 62) (-0 2447 0 9179, 0 1535)

Vertex 63) (-0 2557 0 9180, -0 0873)

Vertex 64) (-0 2611 0 9179, 0 0878)

Vertex 65) (-0 2630 0 9180, 0 0002)

Vertex 66) ( o 1240 0 9043, 0 3310)

Vertex 67) (-0 1241 0 9041 0 3311)

Vertex 68) ( o 1565 0 8945, 0 3171)

Vertex 69) (-0 1566 0 8943, 0 3173)

Vertex 70) ( o 0813 0 8937, 0 3478)

Vertex 71) (-0 0814 0 8936 0 3479)

Vertex 72) ( o 1240 0 8802 0 3365)

Vertex 73) (-0 1241 0 8800 0 3366)

Vertex 74) ( o 0671 0 8765 0 3507)

Vertex 75) (-0 0672 0 8764 0 3508)

Vertex 76) ( o 1715 0 8719 0 3054)

Vertex 77) (-0 1716 0 8716 0 3056)

Vertex 78) ( o 1970 0 8595 0 2762)

Vertex 79) (-0 0000 0 8593 0 3544)

Vertex 80) (-o 1971 0 8592 0 2764)

Vertex 81) ( o 1240 0 8576 0 3259)

Vertex 82) (-0 1241 0 8574 0 3260)

Vertex 83) ( o 0748 0 8528 0 3310)

Vertex 84) (-0 0748 0 8527 0 3311)

Vertex 85) ( o 1634 0 8507 0 3043)

Vertex 86) (-0 1635 r 0 8505 0 3045)

Vertex 87) ( o 1813 , o 8368 0 2817)

Vertex 88) (-0 1814 0 8366 0 2819)

Vertex 89) ( o 0372 0 8356 0 3270)

Vertex 90) (-0 0372 0 8356 0 3271)

Vertex 91) ( o 1240 , o 8339 0 2996)

Vertex 92) (-0 1241 , o 8337 0 2998)

Vertex 93) ( o 2160 0 8336 0 2324)

Vertex 94) (-0 2161 0 8333 0 2327)

Vertex 95) ( o 1416 r 0 8306 0 .2971)

Vertex 96) ( o 1007 , o 8306 0 .2971)

Vertex 97) (-0 1008 0 8304 0 .2972)

Vertex 98) (-0 1416 0 8304 0 2972)

Vertex 99) ( o 2786 0 8231 -0 0431)

Vertex 100) (-o 2790 r 0 8227 -0 .0428)

Vertex 101) (-0 0000 , o 8225 0 .3409)

Vertex 102) ( o 1656 , o 8222 0 .2843)

Vertex 103) (-0 1657 , o 8220 0 .2844)

Vertex 104) ( o 1240 0 8211 0 3025)

Vertex 105) (-0 .1241 , o 8209 , o .3027)

Vertex 106) ( o 0255 r 0 8207 , o .3241)

Vertex 107) (-0 0256 , o 8206 0 .3241)

Vertex 108) ( o 0708 , o 8181 0 .2894)

Vertex 109) (-0 .0708 , o 8180 , o .2895)

Vertex 110) ( o .1003 r 0 8178 , o .2971)

Vertex 111) (-0 .1004 r 0 8177 , o .2972)

Vertex 112) ( o .1478 r 0 8175 0 .2970)

Vertex 113) (-0 1478 , o 8173 , o .2972)

Vertex 114) ( o .1937 r 0 .8171 r 0 .2525) Vertex 115) (-0 1938, 0 8169 0.2527)

Vertex 116) ( o 2928, 0 8111 -0. 0446)

Vertex 117) (-0 2932, 0 8107, -0. 0443)

Vertex 118) ( o 2786, 0 8070, -0. 1001)

Vertex 119) (-0 2790, 0 8067, -0. 0998)

Vertex 120) ( o 2552, 0 8067 -0 0876)

Vertex 121) ( o 1795, 0 8065 0 2671)

Vertex 122) ( o 1496, 0 8065 0 2970)

Vertex 123) ( o 1241, 0 8065 0 3047)

Vertex 124) ( o 1238 0 8066 -0 2759)

Vertex 125) ( o 0985 0 8065 0 2971)

Vertex 126) ( o 0591, 0 8065 0 2843)

Vertex 127) ( o 0223, 0 8064 0 3278)

Vertex 128) ( o 0000, 0 8064 0 3431)

Vertex 129) (-0 0003 0 8065 -0 2940)

Vertex 130) (-0 0223 0 8064 0 3278)

Vertex 131) (-0 0591 0 8064 0 2844)

Vertex 132) (-0 0985 0 8063 0 2972)

Vertex 133) (-0 1241 0 8063 0 3049)

Vertex 134) (-0 1244 0 8064 -0 2757)

Vertex 135) (-0 1496 0 8063 0 2972)

Vertex 136) (-0 1796 0 8063 0 2673)

Vertex 137) (-0 2557 0 8063 -0 0874)

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Table 2 nff version 2 .1 null

1822

0.0000 0 .3380 0.0000

0.0300 0 .3365 0.0000

-0.0308 0 .3330 0.0000

0.0000 0 .3320 0.0340

0.0000 0 .3320 -0.0340

0.0284 0 .3310 0.0340

0.0284 0 .3310 -0.0340

-0.0290 0 .3275 0.0340

-0.0290 0 .3275 -0.0340

0.0574 0 .3252 0.0000

0.0541 0 .3205 0.0340

0.0541 0 .3205 -0.0340

-0.0598 0. .3135 0.0000

0.0001 0 .3133 0.0567

0.0000 0, .3132 -0.0575

0.0240 0 .3121 0.0554

0.0240 0, .3120 -0.0567

0.0713 0, .3110 0.0000

-0.0560 0 .3100 0.0340

-0.0560 0. .3100 -0.0340

-0.0240 0 .3090 0.0543

-0.0240 0, .3090 -0.0552

0.0679 0, .3080 0.0340

0.0679 0 .3080 -0.0340

0.0479 0. .3050 0.0540

0.0479 0, .3050 -0.0540

-0.0420 0. .2980 0.0520

-0.0420 0. .2980 -0.0520

0.0580 0, .2970 0.0520

0.0580 0. .2970 -0.0520

0.0830 0. .2905 0.0000

0.0782 0, .2892 0.0340

0.0782 0. .2892 -0.0340

0.0625 0, .2852 0.0532

0.0625 0. .2852 -0.0536

-0.0763 0. .2845 0.0000

-0.0720 0. .2830 0.0340

-0.0720 0. .2830 -0.0340

0.0440 0. .2800 0.0633

0.0440 0. .2800 -0.0633

0.0240 0. .2755 0.0690

0.0240 0. .2755 -0.0690

0.0000 0. .2754 0.0700

-0.0240 0. .2754 0.0670

-0.0521 0. .2754 0.0560

-0.0521 0. .2754 -0.0560

-0.0240 0. .2754 -0.0670

0.0000 0. .2754 -0.0700

0.0871 0. .2599 0.0340 0.0871 0.2599 -0.0340

0.0698 0 .2569 0.0551

0.0698 0 .2569 -0.0551

0.0947 0 .2534 0.0000

0.0000 0 .2516 0.0720

0.0240 0 .2516 0.0715

-0.0240 0 .2516 0.0700

0.0420 0 .2516 0.0670

-0.0573 0 .2516 0.0580

-0.0770 0 .2516 0.0340

-0.0816 0. .2516 0.0000

-0.0770 0 .2516 -0.0340

-0.0573 0. .2516 -0.0580

0.0420 0 .2516 -0.0670

-0.0240 0. .2516 -0.0700

0.0240 0, .2516 -0.0715

0.0000 0. .2516 -0.0720

0.0907 0, .2478 0.0340

0.0907 0. .2478 -0.0340

0.0869 0, .2451 0.0429

0.0869 0 .2451 -0.0429

0.0953 0, .2449 0.0223

0.0953 0. .2449 -0.0223

0.0922 0, .2412 0.0340

0.0922 0, .2412 -0.0340

0.0961 0, .2402 0.0184

0.0961 0. .2402 -0.0184

0.0837 0. .2389 0.0470

0.0837 0, .2389 -0.0470

0.0757 0. .2355 0.0540

0.0971 0, .2355 0.0000

0.0757 0. .2355 -0.0540

0.0893 0. .2350 0.0340

0.0893 0. .2350 -0.0340

0.0907 0. .2337 0.0205

0.0907 0, .2337 -0.0205

0.0834 0. .2331 0.0448

0.0834 0, .2331 -0.0448

0.0772 0. .2293 0.0497

0.0772 0. .2293 -0.0497

0.0896 0. ,2290 0.0102

0.0896 0. .2290 -0.0102

0.0821 0. .2285 0.0340

0.0821 0. .2285 -0.0340

0.0637 0. .2284 0.0592

0.0637 0. .2284 -0.0592

0.0814 0. .2276 0.0388

0.0814 0. .2276 0.0276

0.0814 0. .2276 -0.0276

0.0814 0. .2276 -0.0388

-0.0118 0. .2255 0.0764

-0.0118 0. .2255 -0.0764

0.0934 0. ,2254 0.0000

0.0779 0. ,2253 0.0454

0.0779 0. .2253 -0.0454

0.0829 0. ,2250 0.0340 0.0829 0.2250 -0.0340

0.0888 0.2249 0.0070

0.0888 0.2249 -0.0070

0.0793 0.2242 0.0194

0.0793 0.2242 -0.0194

0.0814 0.2241 0.0275

0.0814 0.2241 -0.0275

0.0814 0.2240 0.0405

0.0814 0.2240 -0.0405

0.0692 0.2239 0.0531

0.0692 0.2239 -0.0531

-0.0122 0.2222 0.0803

-0.0122 0.2222 -0.0803

-0.0274 0.2211 0.0764

-0.0274 0.2211 -0.0764

-0.0240 0.2210 0.0700

0.0732 0.2210 0.0492

0.0814 0.2210 0.0410

0.0835 0.2210 0.0340

-0.0756 0.2210 0.0340

0.0814 0.2210 0.0270

0.0779 0.2210 0.0162

0.0898 0.2210 0.0061

0.0940 0.2210 0.0000

-0.0806 0.2210 0.0000

0.0898 0.2210 -0.0061

0.0779 0.2210 -0.0162

0.0814 0.2210 -0.0270

0.0835 0.2210 -0.0340

-0.0756 0.2210 -0.0340

0.0814 0.2210 -0.0410

0.0732 0.2210 -0.0492

-0.0240 0.2210 -0.0700

0.0000 0.2209 0.0720

0.0240 0.2209 0.0715

0.0400 0.2209 0.0670

0.0604 0.2209 0.0601

-0.0569 0.2209 0.0580

0.0684 0.2209 0.0537

0.0684 0.2209 -0.0537

-0.0569 0.2209 -0.0580

0.0604 0.2209 -0.0601

0.0400 0.2209 -0.0670

0.0240 0.2209 -0.0715

0.0000 0.2209 -0.0720

-0.0249 0.2192 0.0815

-0.0249 0.2192 -0.0815

0.0689 0.2186 0.0534

0.0689 0.2186 -0.0534

0.0801 0.2179 0.0143

0.0801 0.2179 -0.0143

-0.0043 0.2177 0.0748

-0.0043 0.2177 -0.0748

0.0774 0.2167 0.0466

0.0774 0.2167 -0.0466

0.0814 0.2163 0.0400

ON

o o C 00 o σ. LD VO CN o <tf _"tf 00 o o o CO o CN in CO vo r~ σ. cτι o σ. ro in CN [ o vo ro CN CN o 00 <tf ^ VD ^ O rH rO -Φ CN O VO OO O OO σ. o o t-- σ

-# ω ( <tf LO r- ID o σ. in σι ι CN ro r- CN rH o ^t< ro in oo ro CN CN 00 o ^f ro ro cN H Vo _n o _- σ. H o r- o σ. >

O CN o ro r- o O o in o ro C O O H O in ro o o O CN O t~- vo o < o o o in O l O H O o o o •* o

• o • o o O o • o o o O • o o • O • o o o • o • o • O • o • o o o • o • o o o o • • O o • o o o • o • o o o o • o • o ■ σ • o o i o 1 o o oo 1 O 1 i o o 1 1 o O I o o I I O I o o 1 o I I I o I O I O I O I o o I

ro oo ω o o ro in in r n r^ c^ cD oo oo oo r^ r^ m ^ '^ ^ '^ o o σi σi o o ^ ^ i ^ r^ r^ in in i in in m vo i i in i ^ ^ ^ ^ 'tf M oj cN CN N CN r^ cN CN H H H H O o oo ∞ ro oo ∞ co r^ t^ D o o Ln i i^

H H H H H H H H H H H H H H H H H H H H H H H H H H O O O O O O O O O O O O O O O O O O O O O O O O θ σ. σ. σ. σ. σ. C CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN N CN CN CN CN CN CN CN C CN CN CN CN CN CN CN CN CN CN

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

C5 <tf CN CN

CN 00 00

O l r-

H O O

o O O

-0.0199 0..1955 0.0731

-0.0199 0. .1955 -0.0731

-0.0502 0. .1952 0.0520

0.0931 0. .1952 0.0119

0.0931 0. .1952 -0.0119

-0.0502 0. ,1952 -0.0520

0.0000 0. .1950 0.0699

0.0240 0. .1950 0.0685

0.0400 0. .1950 0.0670

0.0400 0, .1950 -0.0670

0.0240 0. .1950 -0.0685

0.0000 0. .1950 -0.0699

-0.0221 0, .1946 0.0657

-0.0221 0, .1946 -0.0657

-0.0249 0. .1938 0.0753

-0.0249 0. .1938 -0.0753

0.1106 0, .1935 0.0000

0.0909 0, .1932 0.0155

0.0909 0. .1932 -0.0155

0.0870 0, .1931 0.0307

0.0870 0, .1931 -0.0307

-0.0265 0, .1928 0.0684

-0.0265 0. .1928 -0.0684

-0.0687 0, .1921 0.0000

-0.0638 0, .1918 0.0300

-0.0638 0. .1918 -0.0300

-0.0095 0, .1913 0.0690

-0.0095 0. .1913 -0.0690

0.0948 0. .1905 0.0125

0.0948 0, .1905 -0.0125

0.1063 0, .1900 0.0085

0.1063 0, .1900 -0.0085

0.0580 0, .1896 0.0624

0.0580 0, .1896 -0.0624

0.0919 0, .1895 0.0146

0.0919 0, .1895 -0.0146

0.1138 0, .1885 0.0000

-0.0139 0, .1879 0.0704

-0.0139 0, .1879 -0.0704

0.0890 0, .1878 0.0201

0.0890 0, .1878 -0.0201

0.0935 0, .1875 0.0161

0.0935 0, .1875 -0.0161

0.1133 0 .1870 0.0038

0.1133 0, .1870 -0.0038

0.1006 0, .1867 0.0122

0.1006 0, .1867 -0.0122

0.0901 0, .1865 0.0185

0.0901 0, .1865 -0.0185

0.1166 0, .1842 0.0000

0.1158 0, .1840 0.0049

0.1158 0, .1840 -0.0049

0.1112 0, .1832 0.0103

0.1112 0 .1832 -0.0103

-0.0131 0, .1830 0.0642

-0.0131 0, .1830 -0.0642 0.1034 0.1827 0.0128

0.1034 0. 1827 -0.0128

0.0707 0. 1824 0.0546

0.0707 0. 1824 -0.0546

0.0926 0. ,1819 0.0184

0.0926 0. ,1819 -0.0184

0.0889 0. ,1807 0.0206

0.0889 0. ,1807 -0.0206

-0.0157 0. ,1805 0.0732

-0.0157 0. ,1805 -0.0732

-0.0044 0. ,1800 0.0660

-0.0121 0. ,1800 0.0637

-0.0121 0. ,1800 -0.0637

-0.0044 0. .1800 -0.0660

0.1079 0. .1798 0.0068

0.1079 0. .1798 -0.0068

0.1152 0. ,1797 0.0036

0.1152 0. .1797 -0.0036

0.1111 0. .1796 0.0091

0.1111 0. .1796 -0.0091

0.1094 0. .1795 0.0045

0.1158 0. .1795 0.0000

0.1094 0. .1795 -0.0045

-0.0171 0. .1793 0.0683

0.1032 0. .1793 0.0087

0.1032 0. .1793 -0.0087

-0.0171 0. .1793 -0.0683

0.1106 0. .1791 0.0084

0.1106 0. .1791 -0.0084

0.0500 0, .1790 0.0630

0.1041 0. .1790 0.0117

0.1041 0, .1790 -0.0117

0.0500 0. .1790 -0.0630

0.0830 0. .1789 0.0376

0.0830 0, .1789 -0.0376

0.1035 0, .1785 0.0101

0.1035 0. .1785 -0.0101

0.1048 0, .1784 0.0042

0.1048 0, .1784 -0.0042

0.0866 0, .1783 0.0243

0.1130 0. .1783 0.0025

0.1130 0, .1783 -0.0025

0.0866 0, .1783 -0.0243

0.1144 0, .1781 0.0000

-0.0156 0, .1775 0.0599

-0.0156 0, .1775 -0.0599

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Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rr rr α rr r. rr rr rτ rr r. rr rr rτ rr rr rτ r- α σ rr rr rr r. rτ rr cr r rr α ;τ :r ^ !__^j !_r ^ :_r ^ !_r :_r !_r ^ !_r ∑y ;_r .^j' ^ ^

it lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt lt

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H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H si si O si si si si si si si si si si si si si si si si si O si O si si si O si si si si si si si si si si O si si si si si i si si cn o si si σi si σ σ si si

W Ul Ul it Ul Ul it it W Ul it W W W M OJ W CTi tO Cn H it it -O it tO H W W H W tO tO H tO H H O O tO O O H O H VO O H H VD H VD VO O O si cD it vo o ui si cn M H it ui o ω si to ∞ cπ it H M vo tO H vo o it si oo si w cn σi ω H tO it cn si o vo w co it H ca o c^

H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H HH HH HH HH H H ^J H H H sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl σi cn rø o o ui it ui it oo oo ui ui ω ω ui si si σi cn ui si it it si it it ω ω ω w to to to to to to to to H to tO H O O H H H H H O O O

W t0 it Ul lt O V0 0. CB H O it H H O O W t0 Ul lt W tO W Ul it t0 W t0 U1 VD 00 lt V0 s] ai C» Ul it Ul lt Ul H O lt V0 00 Vv0o oo H si σi O si σ si

H H H H H H H H H H H H H H H H H H H H H H H H H H H H s] si i s] si si si -J si sl sl sl sl sl sl sl sl sl sl sl s] sl sl sl sl sl sl sl sl sl sl sl sl sl si si si si si si si si si si σi σi si σ. si si σi σi σ OJ ui ui it OJ Ul it lt Ul OJ OJ tO OJ tO OJ σi tO Ul tO it H tO it tO it OO H tO OO H OJ H tO H tO O H tO O O H O O O H H VD VD H VO H O VO VD s] 0J ιt CJl Ul O H ιt V0 0J σ. t0 Ul ιt 0J O 0J 00 Ul H oo o vo σi si si to σi cπ oo o to si ιt H O. Ul VO DO CO H O sl VO -0 00 it DO 00 O H

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O ooooo

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X O

Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

o rrorroσoc oαorfoσorτor orrorrorτorτorro<-roσorrorrorrorrorrorroctorrorτorτocrorrortorrorrorroooooooooooooooooooooooo

l lt _t _ . _t lt _ . . _t _t _ l . . _t lt _ l . _t _ lt . . l _t _t lt _t . . . . _t _t _ _t l . _ _t lt _ .t _ l lt . . l _t l _ lt

H H H H H H H H H H H H H H ' ^{1 1} ' ^ ' H H H H H H H H H sl sl sl sl sl sl sl sl sl sl sl sl ssil ssil sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl ssil ssil ssil ssil ss]l ss]l sl sl ssil ssil ssil ssil ss]l ss]l ss]l sl sl sl sl sl sl sl sl sl sl s]

VO OO OO OO OO OO OO sl sl sl sl s] si σi σ σi cri cn cn σi cn σi cri cn σi σi ui ui ui ui ui ui ui ui ui ui ui it it it it it it it it it it t it it it OJ OJ OJ OJ OJ O VO OO Ul it H O si σi Ul it H o vo vo oo oo si σi ui it ω to H H O vo oo oo si σ ui it OJ tO H O VD OO si si σ σ ui it OJ to H H O O VO OO sl Ol Ul

H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H sl OO sl OO sl sl CO sl OO sl sl CD sl OO sl sl sl sl sl sl OO sl sl sl sl sl sl sl sl sl sl sl sl sl OO sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl V0 O VO O V0 V0 D0 00 O 00 V0 O 00 H sl s] s] 00 00 sl H V0 σi si σi s] Cn C» s] V0 O 00 Ui σ\ O s] 00 0^ sl (n Ul Ul ιt 00 CD Ul VD sl ιt s] s^

^ ιn _o ι o (j o ^ ι. w _i o α) o a ι» c. ^ w w sj αι ^ μ oi ιi> (iι μ θ -i c. ifl a uι θ vi ι. o ^ -θ μ oι u oo ιo ι_ oo M (Λ (o ι μ

H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H

∞ C» C_0 C0 00 CD 00 J 00 s] s] CD 00 00 s] 00 sl s] s] 00 00 00 00 00 s] s] s] 00 O 00 O sl s] 00 00 sl sl s] s] O s] sl s] CD 00 σ_ 00 00 sl s] 00 00

O H H O O H H DO DO VO VO O O H OO H OO VD VO H H H O O σi OO OO O σi O VD VO VD H H VO VO OO OO σi sl si σi DO DO H O H OO OO H O VO VO VO OO ω tO si to vo oo H O W M Ui ω si si σ σi ω tO si σi o oo ^ vo vD OO it αJ o vo H O it w ui it ω M si H O cn H O H VO si o σi σi H OO si oi ui

H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H OO sl OO sl OO OO sl OO sl sl sl sl sl sl sl OO sl sI sl OO sl sl sl sl-sl sl si si s] s] s] si si si si si si si si si si s] s] s] 00 si si si si si si s] s] s] si O VO O VO O tO VD O OO VD CD OO VD sl sl O sl OO OO H sl σi VD OO sl σi si si σ. si D ui oo σ oo si s] σ it σ ui ui oo oo o ui si si it si VO it 0. 0J Ul tO OJ OJ H lt H lt σi σi Ul tO VO VO VD sl OO OO OJ OO it tO VD σi Ul sl s] O 01 tO H O s] it H litt H lt Ul Ol VO O Cn VO sl O H tO OJ VD it OO sl tO OJ OO Ul

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 0

Ω ^" Ω^" Ω^" Ω^' Ω^~ Ω Ω^" Ω^" Ω^"" Ω^" Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω O Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω o* Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω σtr σ r ff σtrσσσσσσtr trσσσ σσ trtrσσ tr r t ^ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rr rt <τ rr rr rτ σ rr rr rt rr rr rr rr <-r α rr π- rr rr <-r α rr π rr rr r. ' ' l- i_ ι_ ^ ^ ^ !j' i_ ι_ ^ i3' ff i_ ι_ ^ i3' ^ ^ ι_r ι_r ι_ t ^ ^ ι_ ^

1791 1807 1805 1800 Oxcccccc both

1792 1794 1818 1812 Oxcccccc both

1793 1815 1819 1795 Oxcccccc both

1796 1808 1809 1798 Oxcccccc both

1797 1801 1811 1810 Oxcccccc both

1802 1803 1812 1818 Oxcccccc both

1802 1818 1820 1804 Oxcccccc both

1805 1807 1819 1815 Oxcccccc both

1806 1821 1819 1807 Oxcccccc both

1808 1816 1813 1809 Oxcccccc both

1810 1811 1814 1817 Oxcccccc both

Claims

Claims

1. A method of representing a limbed creature in a three dimensional model, comprising defining no more than about 3000 adjacent polygons, at least 90% of which are quadrilaterals, each tubular section of the limbed creature having a cross-section having a multiple of two sides, and the model describing substantially all of the salient points on the body of the limbed creature.

2. A method according to claim 1, comprising defining no more than about 2900, 2800, 2700, 2600, 2500, 2400, 2300, 2200, 2100 or 2000 adjacent polygons.

3. A method according to claim 1 or 2, each tubular section of said limbed creature having a multiple of eight sides.

4. A method according to claim 1 or 2, at least 95, 96, 97, 98 or 99% of said polygons being quadrilaterals.

5. A method according to claim 4, all of said polygons being quadrilateral.

6. A method according to claim 5, said method comprising defining no more than 1820 adjacent quadrilaterals.

7. A method according to any one of the preceding claims, said limbed creature comprising a quadruped.

8. A method according to claim 7, said quadruped being selected from the group consisting primate, ape, monkey, chimpanzee, feline, canine, equestrian and bovine.

9. A method according to claim 8, said quadruped being anthropoid.

10. A method according to claim 8, said quadruped being a human.

11. A method according to any one of the preceding claims, said model describing salient points comprising substantially the group of: i) the top of the head; ii) the tip of the chin; iii) the centre of each pupil, iv) the north, south, east and west extremities of each eye; v) the top, bottom and widest points of each ear; vi) the top of the nose, centre of the bridge of the nose, the centre of each nostril, and the widest point of the nose; vii) the centre of the mouth, the end point of the mouth, the two apexes of the upper lip, and the bottom most point of the lower lip; viii) the centre, centre-left and centre-right of the septum; ix) the centre of each nipple; and x) the front-most, back-most and widest points of each foot.

12. A method according to claim 11 , said model additionally describing at least one salient point selected from the group of: i) the tip of each finger and thumb; ii) the tip of the Adam's apple; iii) the tip of each toe; and iv) the genitals.

13. A method according to any one of the preceding claims, the left and right sides of said model being symmetrical.

14. A method according to any one of the preceding claims, the model having, defined by vertices of the polygons, in outstretched stance front and back medial axial lines on the forelimbs forming a continuous loop connecting each forelimb via the chest and back and continuing around every finger, said medial lines resting on the Z=0 axis.

15. A method according to any one of the preceding claims, the model having, defined by vertices of the polygons, in outstretched stance side medial axial lines on the head, neck, forelimbs, trunk and hindlimbs forming a continuous loop connecting the top of the head via the centre of each forelimb, the centre of the side of the trunk, and the centre of the lower limbs, said medial lines resting on the Y=0 axis.

16. A method according to any one of the preceding claims, the model having, defined by edges of the polygons, in outstretched stance front and back medial axial lines on the head, neck and trunk forming a continuous loop connecting the top of the head via the centre of the head, centre of the chest and centre of the back, said medial lines resting on the X=0 axis.

17. A method according to any one of the preceding claims, the model comprising substantially the model of either one of the group consisting Figures 1-25 and Figures 26-49.

18. A method according to any one of the preceding claims, the model comprising substantially the model of either one of the group consisting Table 1 and Table 2.

19. A method according to any one of the preceding claims, said model being used for animation.

20. A method according to any one of the preceding claims, being a method for reconstructing a surface representation of an individual member of the modelled class of limbed creature, comprising the additional step of determining the coordinates of the salient points of the individual which are described by the model limbed creature, and aligning the salient points of the model limbed creature to the salient points of the individual limbed creature to reconstruct the surface of the individual limbed creature.

21. A method according to claim 20, the determination and alignment steps comprising the steps of: i) transforming said three-dimensional model into a two-dimensional representation of said model which includes said surface of each of the polygons; ii) providing a two-dimensional image of the surface of the individual limbed creature; iii) aligning the salient points of the transformed three-dimensional model and the two-dimensional image of the limbed creature; and iv) performing an inverse transformation of the aligned model to transform it into a three-dimensional model.

22. A method for using a computer for reconstructing a three-dimensional surface of a limbed creature comprising the steps of: a) defining a three-dimensional model of a limbed creature according to the method of any one of claims 1-18, said model describing salient points of the body of the limbed creature; b) providing data from imaging the limbed creature, said data including the coordinates of the salient points described by the model limbed creature; and c) aligning the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature.

23. A system for reconstructing a surface of a limbed creature, comprising: a) an imaging system for producing images of the limbed creature; b) a memory for storing: i) a three-dimensional model of a limbed creature according to the method of any one of claims 1-18, said model describing salient points on the body of the limbed creature; ii) data defining the coordinates of the salient points of the imaged limbed creature as described by the model limbed creature; and iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and c) a processor that is coupled to the memory, said processor executing the machine instructions, causing the processor to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature.

24. A system for displaying a reconstruction of a limbed creature, comprising: a) a memory for storing: i) a three-dimensional model of a limbed creature according to the method of any one of claims 1-18, said model describing salient points on the body of the limbed creature; ii) data defining the coordinates of the salient points of an imaged limbed creature as described by the model limbed creature; iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and iv) machine instructions that define steps for displaying the images obtained; b) display means for displaying images; and c) a processor that is coupled to the memory and display means , said processor executing the machine instructions, causing the processor to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature and display said reconstruction on said display means.

25. A computer program for reconstructing the surface of a limbed creature, comprising: i) program code comprising data defining a three-dimensional model of a limbed creature according to the method of any one of claims 1-18, said model describing salient points on the body of the limbed creature; ii) program code for effecting the input of data defining the coordinates of the salient points of imaged limbed creature as described by the model limbed creature; and iii) program code defining steps for processing the inputted data using the three-dimensional model to align the salient points of the model limbed creature to the salient points of the imaged limbed creature to reconstruct the surface of the imaged limbed creature.

26. A computer program product for reconstructing the surface of a limbed creature, comprising a computer usable medium having computer readable program code means according to claim 25 embodied in said medium.

27. A method of representing a head in a three dimensional model, comprising defining no more than 500 polygons describing substantially all of the salient points on the head.

28. A method according to claim 27, at least 95, 96, 97, 98 or 99% of said polygons being quadrilaterals.

29. A method according to claim 28, all of said polygons being quadrilateral.

30. A method according to claim 29, said model comprising 486 quadrilaterals.

31. A method according to any one of claims 27-30, said model describing salient points comprising substantially the group of: i) the top of the head; ii) the tip of the chin; iii) the centre of each pupil, iv) the north, south, east and west extremities of each eye; v) the top, bottom and widest points of each ear; vi) the top of the nose, centre of the bridge of the nose, the centre of each nostril, and the widest point of the nose; vii) the centre of the mouth, the end point of the mouth, the two apexes of the upper lip, and the bottom most point of the lower lip; viii) the centre, centre-left and centre-right of the septum; and ix) the tip of the Adam's apple.

32. A method according to any one of claims 27-31 , the left and right sides of said model being symmetrical.

33. A method according to any one of claims 27-32, the model comprising substantially the model of either one of the group consisting Figures 19-24 and Figures 44-49.

34. A method according to any one of claims 27-32, the model comprising substantially the head described by the model of either one of the group consisting Table 1 and Table 2.

35. A method according to any one of claims 27-34, said model being used for animation.

36. A method according to any one of claims 27-35, being a method for reconstructing a surface representation of the head of an individual, comprising the additional step of determining the coordinates of the salient points of the individual which are described by the model head, and aligning the salient points of the model head to the salient points of the individual's head to reconstruct the surface of the individual's head.

37. A method for using a computer for reconstructing a three-dimensional surface of a head comprising the steps of: a) defining a three-dimensional model of a head according to the method of any one of claims 27-34, said model describing salient points of the head; b) providing data from imaging the head, said data including the coordinates of the salient points described by the model head; and c) aligning the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

38. A system for reconstructing a surface of a head, comprising: a) an imaging system for producing images of the head; b) a memory for storing: i) a three-dimensional model of a head according to the method of any one of claims 27-34, said model describing salient points of the head; ii) data defining the coordinates of the salient points of the imaged head as described by the model head; and iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and c) a processor that is coupled to the memory, said processor executing the machine instructions, causing the processor to aligning the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

39. A system for displaying a reconstruction of a head, comprising: a) a memory for storing: i) a three-dimensional model of a head according to the method of any one of claims 27-34, said model describing salient points on the head; ii) data defining the coordinates of the salient points of an imaged head as described by the model head; iii) machine instructions that define steps for processing the data derived from the images using the three-dimensional model; and iv) machine instructions that define steps for displaying the images obtained; b) display means for displaying images; and c) a processor that is coupled to the memory and display means , said processor executing the machine instructions, causing the processor to align the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head and display said reconstruction on said display means.

40. A computer program for reconstructing the surface of a head, comprising: i) program code comprising data defining a three-dimensional model of a head according to the method of any one of claims 27-34, said model describing salient points of the head; ii) program code for effecting the input of data defining the coordinates of the salient points of the imaged head as described by the model head; and iii) program code defining steps for processing the inputted data using the three-dimensional model to align the salient points of the model head to the salient points of the imaged head to reconstruct the surface of the imaged head.

41. A computer program product for reconstructing the surface of a head, comprising a computer usable medium having computer readable program code means according to claim 40 embodied in said medium.