Patent Application: US-48044504-A

Abstract:
a statistical shape model is built by automatically establishing correspondence between a set of two dimensional shapes or three dimensional shapes . a parameterization of each shape is determined and , a statistical shape model is built using the parameterization . an objective function is used to provide an output which indicates the quality of the statistical shape model . by performing these steps repeatedly for different parameterizations and comparing the quality of the resulting statistical shape models using output of the objective function to determine which parameterization provides the statistical shape model having the best quality , the output of the objective function is a measure of the quantity of information required to code the set of shapes using the statistical shape model .

Description:
the illustrated embodiment of the invention is based upon a two dimensional ( 2 - d ) statistical shape model . as previously described in the introduction , a 2 - d statistical shape model is built from a training set of example shapes / boundaries . each shape , s i , can ( without loss of generality ) be represented by a set of ( n / 2 ) points sampled along the boundary at equal intervals , as defined by some parameterisation φ i of the boundary path ( the term parameterisation refers to the separation of the boundary path into the set of distances along the boundary between the sampled points ). using procrustes analysis [ 12 ] the sets of points can be rigidly aligned to minimise the sum of squared differences between corresponding points . this allows each shape s i to be represented by a n - dimensional shape vector x i , formed by concatenating the coordinates of its sample points , measured in a standard frame of reference . using principal component analysis , each shape vector can be approximated by a linear model of the form . where x is the mean shape vector , the columns of p describe a set of orthogonal modes of shape variation and b is a vector of shape parameters . new examples of the class of shapes can be generated by choosing values of b within the range found in the training set . this approach can be extended easily to deal with continuous boundary functions [ 6 ], but for clarity this embodiment of the invention is limited to the discrete case . an example of how a shape is sampled according to its parameterisation is shown in fig2 . the shape is a circle , and the origin used for the parameterisation is the lowermost point on the circle . it has been decided to use ten landmarks to parameterise the circle ( including the origin ). referring to the graph shown in fig2 , the landmarks are equally spaced along the horizontal axis of the graph which is labelled t ( t is in effect a measure of the number of landmarks used to parameterise the circle ). the parameterisation of any given landmark is represented by the clockwise distance around the circle between that landmark and the preceding landmark . these distances are represented by the vertical axis of the graph . the utility of the linear model of the shape shown in ( 2 ) depends on the appropriateness of the set of boundary parameterisations { φ i } that are used to construct the statistical shape model from the set of training boundaries { s i }. the embodiment of the invention defines a criterion for choosing the set of parameterisations { φ i }. the aim is to choose { φ i } so as to obtain the ‘ best possible ’ model . since it is desired to obtain a compact model with good generalisation properties the ‘ best ’ model is defined as that which can account for all of the observations ( the training boundaries ) in as simple a way as possible . in other words , the inventors have based their parameterisation method upon the statement that the ‘ best ’ model is that which describes the whole training set as efficiently as possible . in order to determine which parameterisation is the most efficient a minimum description length criterion is used ( i . e . the optimisation method attempts to determine the set of parameterisations which uses the least amount of data to describe the whole training set ). this is formalised by stating that it is desired to find { φ i } that minimises the information required to code the whole training set to some accuracy δ on each of the elements of { x i }. note that to describe { x i } to arbitrary accuracy would require infinite information ; δ is chosen to reflect the measurement errors involved in acquiring the training boundaries . a set { s i } of s training shapes are to be parameterised using { φ i } and sampled to give a set of n - dimensional shape vectors { x i }. following ( 2 ), { x i } can be approximated to an accuracy of δ in each of its elements using a linear shape model of the form where x is the mean of { x i }, p has t columns which are the t eigenvectors of the covariance matrix of { x i } corresponding to the t largest eigenvalues λ j , b i is a vector of shape parameters , and r i is a vector of residuals . the elements of r i can be shown to have zero mean and a variance of λ r = 1 n ⁢ ∑ j = i + 1 s ⁢ λ j the total information required to code the complete training set of images using this encoding is given by where i model is the information required to code the model ( the mean vector x and the covariance matrix p ), i n is the average information required to code each parameter vector b i , and i r the average information required to code each residual vector , r i . for simplicity , it is assumed that the elements of the mean x and the matrix p are uniformly distributed in the range [− 1 , 1 ], and that k m bits are used per element for the mean , and k j bits per element for the j th column of p giving quantisation errors δ m = 2 − k m and δ j = 2 − k j respectively . thus the elements of b , are assumed to be normally distributed over the training set with zero mean and variance λ j . to code them to an accuracy λ m it can be shown [ 14 ] that the information required to code each parameter vector b i is on average similarly , to code the n elements of r i to an accuracy of δ r = 2 − k r the information required on average is i total = nk m + n ⁢ ∑ j = 1 t ⁢ k j + ⁢ s ⁢ ∑ j = 1 t ⁢ [ k b + 0 . 5 ⁢ ⁢ log ⁡ ( 2 ⁢ π ⁢ ⁢ e ⁢ ⁢ λ j ) ] + ⁢ s ⁢ ⁢ n ⁡ [ k r + 0 . 5 ⁢ ⁢ log ⁡ ( 2 ⁢ π ⁢ ⁢ e ⁢ ⁢ λ r ) ] ( 8 ) this is the total amount of information required to code the complete training set of images using a linear shape model of the form given in ( 3 ). it can be seen from ( 8 ) that i total is a function of the quantisation parameters k m , k j , k b , and k r , which are related to δ , the overall approximation error . since it is desired ultimately to minimise i total with respect to { φ i } it is first required to find the minimum with respect to the quantisation parameters . this can be found analytically , leading to an expression in terms of s , n , t , { λ j } and λ r . i total = ⁢ - 0 . 5 ⁢ ( n + n ⁢ ⁢ t + st ) ⁢ log ⁡ ( 12 ⁢ ⁢ αλ r / s ) + snk + ⁢ 0 . 5 ⁢ ( n + s ) ⁢ ∑ j = 1 t ⁢ log ⁡ ( λ j ) + 0 . 5 ⁢ ⁢ n ⁢ ⁢ s ⁢ ⁢ log ⁢ ( αλ r ) + ⁢ 0 . 5 ⁢ s ⁢ ( n + t ) ⁢ log ⁡ ( 2 ⁢ π ⁢ ⁢ e ) - 0 . 5 ⁢ st ⁢ ⁢ log ⁡ ( s ) ( 9 ) where s is the number of training shapes , n is double the number of points sampled along the boundary of each shape ( each point being represented as a pair of coordinates ), t is the number of columns of the covariance matrix ( i . e . the number of eigenvalues used to code the training set of images , or equivalently the number of modes of the model ), λ j is the value of each eigenvalue . λ r is the variance over the training set of images of the vector of residuals , and α = ( n ⁢ ⁢ s n ⁡ ( s - 1 ) - t ⁡ ( n - s ) ) . thus , for a fixed number of modes , t , to optimise i total it is required to minimise the number of modes to use , t , is chosen so as to be able to represent the training set to a given accuracy . in this instance the root mean square difference between points on the training set and the model reconstructions of the set are less than a suitable threshold . the threshold is typically chosen to be related to the estimated uncertainty in the original data . the motivation behind the generation of the function f is to allow an optimal dense correspondence to be established automatically between a set of training shapes . as set out above , corresponding points are described in terms of parameterisation φ i of the boundary path of the shape . it is desired to find the global optimum of f in ( 10 ) with respect to the set of shape parameterisations { φ i }. the first described method of finding the global optimum of f is to use an explicit representation of { φ i } coupled with a stochastic search . a representation of { φ i } is required that ensures a diffeomorphic mapping between each pair of training shapes . in 2 - d this can be achieved by enforcing the ordering of corresponding points around the training shapes . in 3 - d , however , no such ordering exists . the invention provides a new method of representation that guarantees diffeomorphic mapping without using an explicit ordering constraint . here the method is described for 2 - d shapes ; appendix b explains how it can be extended to 3 - d . a piecewise linear parameterisation is defined for each training shape by recursively subdividing boundary intervals by inserting nodes ( landmarks ). the position of each new landmark is coded as fraction of the boundary path length between neighbouring landmarks . thus by constraining the subdivision parameters to the range [ 0 , 1 ] a hierarchical ordering is enforced where , at each level of the hierarchy , landmarks are positioned between those already present . this is illustrated by the example in fig3 which demonstrates the parameterisation of a circle . referring to fig3 a - d , the shape to be parameterized is a circle . an origin is marked onto the circle as shown in fig3 a ( the origin is the first landmark ). moving in a clockwise direction the circumference around the circle is constrained to have a length of 1 . a second landmark is positioned at 0 . 65 , measured from the origin , as shown in fig3 b . this is the first level of the hierarchy . starting from the origin the distance around the circle to the second landmark is constrained to be 1 . a third landmark is positioned at 0 . 65 , measured from the origin , as shown in fig3 c . starting from the second landmark the distance around the circle to the origin is constrained to be 1 . a fourth landmark is positioned at 0 . 8 , measured from the second landmark , as shown in fig3 c . this the second level of the hierarchy . further landmarks are added in the same manner , as shown in fig3 d , thereby providing the third level in the hierarchy . recursive subdivision can be continued until an arbitrarily exact parameterisation is achieved . correspondence is assumed across the whole training set between equivalent nodes in the subdivision tree ( i . e . equivalent levels of hierarchy ). a set of parameterisations { φ i } can be manipulated in order to optimise the objective function f . in practice the search space is high - dimensional with many local minima , and consequently it is preferred to use a stochastic optimisation method such as simulated annealing [ 15 ] or genetic algorithm search [ 16 ]. both the simulated annealing [ 15 ] and genetic algorithm search are well known to those skilled in the art . a genetic algorithm was used to perform the experiments reported below . the results of applying the invention to several sets of outlines of 2 - d biomedical objects are now described . qualitative results are shown by displaying the variation captured by the first three modes of each model ( first three elements of b varied by ± 2 [ standard deviations over training set ]). quantitative results are also given , tabulating the value of f , the total variance and variance explained by each mode for each of the models , comparing the automatic result with those for models built using manual landmarking and equally spaced points ( boundary length parameterisation ). the first test used 17 hand outlines . the qualitative results in fig4 show the ability to generalize plausibly to examples not in the training set . the results in tables 1 a - c ( fig1 a - c ) show that the automatic method produces a better model than both the equally spaced and manual model . the second test used 38 outlines of left ventricles of the heart , segmented from echocardiograms . again , fig5 shows good generalization . the quantitative results are presented in tables 2 a - d ( fig1 a - d ). once again , the automatic algorithm produces significantly better results than the other two methods . the final test used the outlines of 49 hip prostheses . the qualitative results in fig6 show that there is little variation in the three most significant modes of the automatic model . this is because the only variation in shape is caused by the rotation of the prosthesis in the plane . it is also interesting to note in tables 3 a - 3 ( fig1 a - d ) that the model produced by equally spacing landmarks is better than the manual model . this is because equally - spaced points suffice as there is little variation , but the manual annotation adds noise to the model . as before , the automatically constructed model is best . the second described method of finding the global optimum of f involves selecting corresponding points by uniformly sampling a continuous parameterisation φ ( m ), of each shape , and then manipulating the set of parameterisations { φ i }, in a way that minimises the value of f min . the method described in this section is applicable to both open and closed curves ; for clarity , the discussion is limited to the closed case . a legal reparameterisation function φ ( m ) is a monotomically increasing function of m , with a range ( 0 ≦ φ ( m )≦ 1 ). an example of such a function is shown in fig8 . referring to fig8 , rather than determining a set of landmark points which together provide a parameterisation function ( as shown in fig2 ) it is desired instead to use two - dimensional curves to generate a parameterisation function φ ( m ). such a function can be represented as the cumulative distribution function of some normalised , positive definite density function we choose , for example , to represent ρ ( x ) as a sum of gaussian kernels : ρ ⁡ ( x ) = c ⁡ [ 1 + ∑ i ⁢ a i σ ⁢ ( 2 ⁢ π ) ⁢ exp ⁡ ( - 1 2 ⁢ σ i 2 ⁢ ( x - o i ) 2 ) ] ( 11 ) where the coefficients a i control the height of each kernel ; σ i specifies the width and a i the position of the centre and c is the normalisation constant . the constant term is included to ensure that when all a i &# 39 ; s are zero , φ ( m ) is an arc - length parameterisation . φ ⁡ ( m ) = ⁢ ∫ 0 m ⁢ ρ ⁡ ( x ) ⁢ ⁢ ⅆ x = ⁢ c ⁡ [ m + ∑ i ⁢ a i 2 ⁢ erf ⁢ ⁢ ( m - a i σ i ⁢ 2 ) + ∑ i ⁢ a i 2 ⁢ erf ⁢ ⁢ ( o i σ i ⁢ 2 ) ] ( 12 ) c - 1 = 1 + ∑ l ⁢ a i 2 ⁢ erf ⁢ ⁢ ( m - a i σ i ⁢ 2 ) + ∑ l ⁢ a i 2 ⁢ erf ⁢ ⁢ ( a i σ i ⁢ 2 ) ( 13 ) the parameterisation is manipulated by varying { a l }, the heights of the kernels ; the widths { σ t } and the positions { a i } are fixed . in other words , a set of gaussian curves is combined together to generate a function whose cumulative value is the parameterisation of the shape ( i . e . the set of curves together provide a curve of the type shown in fig8 ). if n k kernels are used to represent the parameterisation , the configuration space becomes ( n k s )- dimensional . one possibility is to use a stochastic optimisation method as described above , but a more effective approach would be desirable . this search space is generally too large for a direct optimisation scheme to converge rapidly and reliably . this problem is overcome by using the following multiresolution approach : begin with a single kernel of width σ l = ¼ , center at a l = ½ on each shape . the height , a l of the kernel on each shape is initialised to zero — equivalent to an arc - length parameterisation . employ an optimisation algorithm to find the magnitude a l , of the kernel on each shape that minimises f . once these values are found , they are fixed and recorded . at each subsequent iteration k , add an additional 2 k − 1 kernels of width ¼ ( ½ ) k . the new kernels are positioned at intervals of ( ½ ) k between m = 0 and m = 1 so that they lie halfway between the kernels added on previous iterations . the optimisation algorithm is used to find the best height for each kernel . continue recursively adding additional kernels until the parameterisation is suitably defined . an advantage of this approach is that it avoids becoming trapped in a local minima of f since it starts at very low resolution and gradually steps up to higher resolutions . the best alignment of the training set depends on the parameterisation of each shape . this is addressed by performing a procrustes analysis [ 6 ] on the reparameterised shapes before evaluating f min . it will appreciated that other suitable forms of kernels may be used in place of the gaussian kernels , for example cauchy functions may be used . in the experiments reported below , it has been assumed that a correspondence exists between the origins of each shape in the training set . if the correspondence does not exist , ( 12 ) must be modified so that φ ( t )→( ε + φ ( t )) mod1 , where s specifies the offset of the origin . qualitative and quantitative results of applying the second method to several sets of outlines of 2d biomedical objects have been obtained . an investigation of how the objective function behaves around the minimum and how it selects the correct number of modes to use has also been carried out . the method was tested on a set of 17 hand outlines , 38 left ventricles of the heart , 24 hip prostheses and 15 cross - sections of the femoral articular cartilage . the algorithm was run for four iterations , giving 16 kernels per shape , a matlab implementation of the algorithm using a standard iterative optimisation algorithm takes between 20 and 120 minutes , depending on the size of training set . the results are compared with models built by equally - spacing points along the boundary and hand - built models , produced by identifying a set of ‘ natural ’ landmarks on each shape . fig9 shows qualitative results by displaying the variation captured by the first three modes of each model ( the first three elements of b varied by ± 2σ ). quantitative results are given in table 2 , tabulating the value of f min , the total variance , and variance explained by each mode for each of the models . the qualitative results in fig9 show that the shapes generated within the allowed range of b are all plausible . the qualitative results in table 2 show that the method produces models that are significantly more compact than either the models built by hand of those obtained using equally - spaced points . it is interesting to note that the model produced by equally - spacing points on the hip prosthesis is more compact than the manual model . this is because equally - spaced points suffice as there is little variation , but errors in the manual annotation adds additional noise which is captured as a statistical variation . to demonstrate the behavior of the objective function some corresponding points were taken from the automatically generated hand model , and random noise was added to each one . fig1 shows a plot of f min against the standard deviation of the noise . the plot shows that as the points are moved further away from their corresponding positions , the value of f min increases — the desired behavior . this method of describing parameterisation can also be applied in 3d . the second method of parameterisation , described above , uses two dimensional kernels to determine the parameterisation . the third method , described below , is similar to the second method but uses landmark points rather than gaussian kernels to represent the parameterisation . the training data used by the third method are a set { ψ i } of shapes that are represented as curves in two dimensions : ψ i ( t )=[ ψ i x ( t ), ψ i y ( t )], 0 ≦ t ≦ 1 ( 15 ) the shape outlines can be open [ ψ i ≠ ψ i ( 1 )] or closed [ ψ i ( 0 )≠ ψ i ( 1 )] it is required to find the optimal set of parameterisations { φ i } of these curves , where φ i is a diffeomorphic mapping : a piecewise - linear approximation to the parameterisation φ i is used by specifying a set of n c control points , { p k } on each shape and equally spacing n / k points along the boundary between each interval . the configuration space is therefore ( n c n s )- dimensional . this search space is generally too large for a direct optimisation scheme to converge rapidly and reliably . this is overcome by using the following multiresolution approach . begin with one control point , p 1i , on each shape . add n / 2 equally - spaced points along the contours between ψ i ( 0 ) and ψ i ( p 1i ) and between ψ i ( p 1i ) and ψ i ( 0 ). search for the best set of values for { p 1i } in the range [ 0 , 1 ]. a circle to be parameterised is shown in fig1 a . the ‘ x ’ represents the origin , the circles represent the current ( flexible ) control points and the squares represent the fixed control points . at each iteration , the current control points are allowed to move between the endpoints of the arrow . for the first iteration , the best control point p 1i is found to be at 0 . 35 . once the best values are found , fix and record them . place two additional control points p 2i and p 3i between 0 and p 1i and between p 1i and 1 respectively . equally spaced n / 4 points between [ 0 , p 2i ], [ p 3i , p 1i ], [ p 1i , p 31 ] and [ p 3i , 1 ]. referring to fig1 b the best control points p 2i and p 3i are 0 . 12 and 0 . 85 respectively . fix and record the optimal positions of { p 2i } and { p 3i }. continue adding additional control points in a similar fashion between the fixed control points { p 1 } until the parameterisation is suitably defined . referring to fig1 c the best control points p 4i , p 5i , p 6i and p 7i are 0 . 6 , 0 . 25 , 0 . 7 and 0 . 95 . fig1 d shows the full set of best control points . at each iteration , the position of each control points is initialised as halfway along its allowed range — the equivalent of an arc - length parameterisation . since an explicit ordering constraint has not been used , the method may be used on shaped in 3d ( see [ 19 ] for details ). the pose of each shape affects the value of f . it is therefore required to optimise the four parameters that allow a rigid transformation of each shape : translations d x , d y , scaling s and rotation θ . it has been found that adding an additional 4 n s dimensions to each iteration significantly slows the optimisation and introduces many additional false minima . better results can be achieved by performing a procrustes analysis [ 12 ] of the reparameterised shapes after each iteration . the third method was tested on a set of 17 hand outlines , 38 left ventricles of the heart . 24 hip prostheses and 15 outlines of the femoral articular cartilage . the algorithm was run for four iterations , giving 16 control points per shape . the results are compared to models built by equally - spacing points along the boundary and hand - built models , produced by identifying a set of natural landmarks on each shape . in fig1 qualitative results are shown by displaying the variation captured by the first three modes of each model ( the first three elements of b varied by ± 2σ ). quantitative results are given in table 3 , tabulating the value of f , the total variance , and variance explained by each mode for each of the models . the automatic results are compared with those obtained for models built using manual landmarking and equally spaced points . the qualitative results in fig1 show that the shapes generated within the allowed range of b are all plausible . the quantitative results in table 3 show that the method produces models that are significantly more compact than either the models built by hand or those obtained using equally - spaced points . it is interesting to note that , although the total variance of the hand - built model of the heart ventricle is larger than the equally - spaced model , the value of the objective function is lower . this is because much of the variation of the hand - built model is captured within the first few modes whereas the equally - spaced model requires more modes to describe the same amount of variation . this is the behavior that is desired from the objective function — the ability to represent much variation with few modes . the invention provides an objective function that can be used to evaluate the quality of a statistical shape model . the expression used has a theoretical grounding in information theory , is independent of quantisation error and unlike other approaches [ 4 , 6 ], does not involve any arbitrary parameters . as well as providing good results when used as an objective function for automatically building statistical shape models , the function may also be used to calculate the correct number of modes to use for a given model . the objective function includes a σlog ( λ i ) term which is equivalent to the determinant of the covariance matrix , as used by kotcheff and taylor [ 6 ], but the more complete treatment here shows that other terms are also important . three novel representations of correspondence have been described that enforce diffeomorphic mapping and are applicable in 2 - d and 3 - d . it has been found that these new representations improve the performance of genetic algorithm search in comparison with the representation described by kotcheff and taylor [ 6 ] and allow more direct methods of optimisation to be applied . the results described in relation to 2 - d objects offer a significant improvement over those from a hand - built model . the test has been run with different random seeds and achieved almost identical results each time . the various approaches described here can also be extended in modelling full appearance ( grey - levels , not just shape ) as described in paper [ 20 ]. since the mathematical description of the appearance model is identical to the shape model this is straight forward . i total in ( 7 ) is a function of the quantisation parameters δ m , { δ j }, δ b and δ r . since we wish ultimately to minimise i total with respect to { φ i } we need first to find the minimum with respect to these parameters . first , we need to determine what quantisations δ m , { δ j }, δ b and δ r are required to achieve a quantisation error δ = 2 − k in the final reconstruction . we assume that by quantising a parameter , we effectively add noise to that parameter . we have used error propagation to estimate the effects of noise on the final reconstruction . in our linear model ( 2 ), noise of variance σ 2 on the elements of x induces noise of variance σ 2 on x i . similarly , noise of variance of σ 2 on the elements of b t can be shown to induce an average of noise of variance σ 2 / 2 on the elements of x i . noise of variance σ 2 on the elements of the j th column of p induces an average noise of variance λ j σ 2 on each element of x i . quantising a value to δ induces noise with a flat distribution in [− δ / 2 , δ / 2 ] and thus a variance of δ 2 / 12 . thus quantising x , p and b i , causes an additional error that must be corrected by the residual term , r i . in effect , the variance of the residual is increased from the original λ r . taking this into account , the variance on the elements of the residual is given by using the central limit theorem we assume that the residuals are normally distributed . λ r 1 is substituted for λ r in ( 7 ) giving we can now find the minimum of i total with respect to δ m , { δ i }, δ b and δ r . by equating the differentials to zero , we can show that at the optimum δ m 3 = 2 − 2k n = 12λ r 1 / s ( 13 ) δ j 2 = 2 − 2k j = 12λ r 1 /( s λ j )= δ m 2 / λ j ( 14 ) δ n 2 = 2 − 2k r = 12λ r 1 = sδ m 2 ( 15 ) λ r ′ = αλ r ⁢ ⁢ where ⁢ ⁢ α = ( ns n ⁡ ( s - 1 ) - t ⁡ ( n - s ) ) ( 16 ) this appendix contains a description of how our parameterisation can be extended to 3 - d . our ultimate goal is to build 3 - d statistical models of biomedical objects . to ensure that all shape instances are legal , we must constrain each landmark to lie on the surface . a seemingly plausible solution is to use spherical co - ordinates whose centre is at the centre of gravity of the shape . this representation , however , is not unique , in that two different points on the surface may have the same co - ordinates . consequently , triangles can flip over between examples and as this is captured as a statistical variation , the specificity and compactness of the model is affected . we can overcome this by flattening the surface to that of a unit sphere using the conformal mapping technique of angenent et al [ 18 ]. this method solves a second order partial differential equation to provide a function that maps any point on the original surface to that of a unit sphere . as the mapping is a diffeomorphism , each point on the original surface has a unique , corresponding point on the sphere . this allows us to navigate the shapes surface using two spherical co - ordinates . we may now use hierarchical parameterisation to position landmarks on the sphere . we guarantee that the landmarks stay in order by constraining each point to lie inside the spherical triangle formed by the triple of points from the hierarchy tier above the parameterisation is now the position of the point in relation to the three vertices of the triangle . this allows us to constrain each landmark to lie between its neighbouring points , thus defining an explicit correspondence across the entire set of shapes . when a new landmark is added , it forms three new triangles , as demonstrated in fig7 . once all the landmark points have been positioned on the sphere , they can be projected onto the shapes surface using the inverse of the conformal mapping and evaluate them using our objective function . an alternative approach is the extension of the continuous parameterisation of a line ( given in equation 16 )— the parameterisation of homeomorphisms on the sphere ( i . e . mappings which do not induce folding or tearing of the sphere ). if we define on parameterisable transformation which does not tear or fold , we can generate a general transformation by applying a sequence of such individual transformation . below we describe one such transformation , which essentially stretches the area around a particular point and compresses the area at the opposite side of the sphere . by applying a sequence of such stretches , centred on different points , we can generate a wide variety of homeomorphisms . to automatically build a model we optimise the parameters of these individual transformations so as to minimise the objective function defined above . we wish to construct a parameterised set of exact homeomorphisms on the sphere , that is , a set of mappings from the sphere to itself which are continuous , one - to - one , and onto ( i . e . no gaps or wrinkles ). we would also like the mapping to be differentiable everywhere other than at a finite number of isolated points . we would like the members of this set to be localised in terms of their action , and continuous with the identity . from a computational point of view , we would like the set to use as a few parameters as possible , and be computationally efficient ( i . e . involve only the evaluation of elementary functions ). we can construct such a set as follows . consider an arbitrary point p on the unit sphere . we construct spherical polar co - ordinates ( θ , φ ) on the sphere such that p corresponds to the point θ = 0 . let us now consider a set of homeomorphisms that re - parameterises the θ co - ordinate : we first take the rotationally symmetric case where ƒ ( θ , φ )= ƒ ( θ ) for the mapping to be differentiable over the range 0 & lt ; θ & lt ; π and continuous with the identity , ƒ must be a differentiable non - decreasing monotonic function over the range 0 & lt ; θ & lt ; π , with ƒ ( 0 )= 0 , ƒ ( π )= π . any such monotonic function ƒ can be re - written in terms of the cumulative distribution function of some density function p ( θ ), defined over the range 0 & lt ; θ & lt ; π . we take as our normalised density function a constant term plus a wrapped cauchy distribution . the wrapped cauchy [ 1 ] is a normalisable , uni - modal distribution for circular data , or variable width , which has an analytic indefinite integral : f ⁡ ( θ ) = π ⁢ ∫ 0 0 ⁢ ⁢ ⅆ φρ ⁡ ( φ ) ⁢ ⁢ ⁢ = 1 1 + a ⁡ [ θ + a ⁢ ⁢ arccos ⁡ ( ( 1 + α 2 ) ⁢ cos ⁢ ⁢ θ - 2 ⁢ α 1 + α 2 - 2 ⁢ α ⁢ ⁢ cos ⁢ ⁢ θ ) ] where ( α ≡ e − a , αε ) is the width - parameter is the width - parameter of the cauchy , with amplitude a ≧ 0 . in this case , the mapping is also differentiable everywhere . we can extend this to the non - symmetric case if we make the amplitude a a smooth function of the co - ordinate φ . one such way to do this is to again use the wrapped cauchy distribution to obtain : a -& gt ; a ⁡ ( φ ) = a c ⁡ [ 1 - β 2 1 + β 2 - 2 ⁢ β ⁢ ⁢ cos ⁢ ⁢ φ - 1 - β 2 ( 1 + β ) 2 ] ( 3 ) where β ≡ e − b is the width of the subsidiary cauchy , and we have chosen the formulation such that a ( φ ) has a minimum value of zero . note that we have also introduced a further parameter in terms of the definition of the zero of the φ co - ordinate . this mapping is differentiable everywhere except at the point p and the point diametrically opposite to it . we can also consider transformations of the φ co - ordinate , which are equivalent to shearing and twisting the sphere about the axis defined by our point p . so , for example , we could consider a re - parameterisation of the form : φ -& gt ; φ + g ⁡ ( θ ) ⁢ ⁢ where ⁢ ⁢ g ⁡ ( θ ) = ⁢ ( b 2 ⁢ π ) ⁢ 1 - τ 2 1 + τ 2 - 2 ⁢ ⁢ τ ⁢ ⁢ cos ⁡ ( θ - θ 0 ) , τ = ⁢ ⅇ - 1 , t ∈ ℜ ( 4 ) where b is the amplitude , τ the width and θ 0 the center . this transformation is continuous with the identity at b = 0 . it can also be localised about θ = θ 0 in the limit of zero width . we take a set of transformations as defined above , each about a different point , and apply them in succession to generate a combined transformation . in general , the final transformation will depend on the order on which we compose the individual transforms . 1 . k . v . maridia statistics of directional data . academic press , london , 1972 1 . cootes , t ., a . hill , and c . taylor , the use of active shape models for locating structures in medical images . image and vision computing , 1994 . 12 : p . 353 - 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