Registration of Structures in Arbitrary Dimensions ... - Semantic Scholar

Registration of Structures in Arbitrary Dimensions: Implicit Representations, Mutual Information & Free form Deformations Xiaolei Huang

Nikos Paragios

Dimitris Metaxas

Computer and Information Sciences Rutgers, The State Univ. of NJ Piscataway, NJ 08854, USA [email protected]

Real Time Vision & Modeling Dept. Siemens Corporate Research Princeton, NJ 08540, USA [email protected]

Computer and Information Sciences Rutgers, The State Univ. of NJ Piscataway, NJ 08854, USA [email protected]

Abstract Registration is a core component in various applications of imaging and vision. While simple cases refer to the registration of clouds of points, a strong need exists for shape, image and volume alignment. In this paper, we propose a novel global-to-local registration method that integrates statistical and variational techniques. Registration is considered in an implicit higher dimensional space. The powerful space of distance transforms of arbitrary metric is used as an embedding function. Mutual information can support various motion models and is considered to perform global registration. A B-Spline approximation of grid is used within a free-from deformation criterion to recover a (complementary to the global) dense registration field that is continuous and guarantees one-toone mapping. Such framework exhibits robustness and can cope in an efficient manner with important local deformations. 2D/3D shapes are used to demonstrate the potentials of the proposed technique.

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Figure 1. (1.a) Initial conditions, (1.b) Global alignment using mutual information and implicit representations, (2.a) Establishing correspondences using FFD, (2.b) Registering locally using FFD, (2.c) Grid deformation.

constraint. The assumption of constant intensity of a same point in 3D scene when projected onto the image space is though often unrealistic. Pose of the object, illumination conditions as well as reflection properties of the observed structure can violate this condition. Such limitation can be dealt with stochastic criteria such as mutual information [3, 18], a method that can account for global illumination changes. The estimation of a dense correspondence/registration field is an ill-posed problem. The number of unknown variables is usually higher than the number of constraints. Additional smoothness constraints are considered to cope with this limitation [16]. Such techniques have good performance when dealing with images with low SNR but are not efficient with shapes since they cannot guarantee an oneto-one correspondence between the source and the target shape/image. Global alignment methods are an alternative to the complete recovery of the registration flow. The use of global linear models (rigid, affine, homographic, etc.) is a compromise between low complexity, robustness and fairly good matching between the source and the target shape. To this end, one would like to recover a parametric motion model that when applied to the source, it transforms

1 Introduction Shape alignment and image registration are topics of increasing attention in imaging and vision. Recognition [1], segmentation, and motion analysis are some of the domains that can benefit from the outcome of such techniques. Quite often, shape alignment [17] is considered to be a different problem than image registration [9]. One can define the registration problem as follows: recover a transformation between a source and a target shape/image that results in meaningful correspondences between their basic elements (points, curvature, medial axes, intensity properties, etc.). Various criteria have been considered to perform shape alignment based either on the local shape structure or on global geometric properties (medial axes, etc.). The recovery of dense correspondences in such scenarios is nontrivial. Local structure is not always well defined while global structure cannot account for this level of detail. Most image registration methods are driven from the brightness 1

the source close to the target. However, these methods cannot deal with local deformations and have poor performance when the global model cannot capture the properties of the transformation between the target and source. Nonlinear models are an alternative to perform registration. Deformable models [10] can deal with both global and local deformations, though they often require the explicit parameterization of the structure of interest. One can conclude that the following terms are critical in the registration process: (i) the information space, (ii) the nature of transformation supported by the method (iii) the mathematical framework introduced to perform that task. In this paper we propose a technique that addresses in an efficient manner the above limitations. Our approach is depicted in [fig. (1)]. The structures of interest (shapes, images, etc.) are represented in an implicit form [11], the space of distance functions of arbitrary metric that is a powerful registration space [12]. A well known stochastic criterion that is robust to scale variations, illumination changes and can deal with arbitrary motion models (mutual information [3, 18]) is considered to perform global registration [fig. (1.1.b)]. Local deformations are captured while preserving an one-to-one correspondence using a B-Spline based free form transformation [15] [fig. (1.2.b)]. A smooth function is recovered that when applied to control lattices [fig. (1.2.c)], makes the points of the source to gradually approach their positions at the target. The most closely related work with our approach can be found in [12]. The remainder of this paper is organized as follows; In section 2 we derive an implicit representation of the structure of interest, while in section 3 we present the global registration criterion. Local deformations are dealt with in section 4. We conclude with experimental results and discussion in section 5.

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Figure 2. Implicit Representations of Shapes: (a) hand, (b) dude.

In order to facilitate the introduction of the method, we consider the 2D case and let Φ : Ω → R+ be a Lipschitz function that refers to a distance transform representation for a given shape S. By definition Ω is bounded since it refers to the image domain. The shape defines a bi-modal partition; the region that is enclosed by S, [RS ] and the background [Ω−RS ]. Given these definitions the following representation [fig. (2)] is considered:  0, (x, y) ∈ S  +D((x, y), S) > 0, (x, y) ∈ RS ΦS (x, y) =  −D((x, y), S) < 0,

(x, y) ∈ [Ω − RS ]

where D((x, y), S) refers to the min Euclidean distance between the grid location (x, y) and the shape S. Such selection is a convenient feature space when considered with objective functions that are optimized using a gradient descent method. One can prove that the gradient is a unit vector in the direction of the vector distance function. Sufficient conditions for convergence of gradient descent methods require continuous first derivatives. The considered representation satisfies these conditions in several ways. The use of implicit representations provides additional support to the registration process since one would like to align the original structures as well as their clones that are positioned coherently in the image/volume plane. Last, but not least one can refer to recent work [20] that demonstrates robustness and stability of such representation when considered in shape registration. Prior work related with the use of implicit representations in registration can be found in [12]. The exploration of such representations has led to rigid-invariant techniques for alignment. To this end, standard similarity techniques were considered like the SSD criterion [12] where one would like to recover a rigid transformation A(Θ) with parameters Θ = (s, θ, T ) such that:

2 On the selection of the feature space The definition of similarity/di-similarity term as well as the feature space are critical components of the registration process. Simplistic methods aim to minimize an Euclidean metric between the cloud of points of the source and the target. More advanced techniques are based on implicit geometric characteristics of these structures, like the curvature, medial axes, normals , etc. or combination of them. The estimation of such implicit properties is a difficult task that often requires the parameterization of the structure of interest. Within the proposed framework, an implicit parameterization for the source and the target in a higher dimension space is considered. Such representation refers to a level set where the structure of interest is represented using an embedding function. Euclidean distance transforms are invariant to translation/rotation and one can predict the effect of scale variations on such representation [12]. Consequently, it is a powerful selection to embed a structure of interest in a higher dimension.

∀x ∈ Ω : [s ΦD (x) = ΦS (A(Θ; x))] where ΦS , ΦD are the implicit representation of the source and the target shape. Such condition is applicable when dealing with rigid transformations and can lead to the following objective function ZZ ρ (sΦD (x) − ΦS (A(Θ; x)) dΩ E(Θ) = Ω

2

where ρ is an error norm. Such term can deal fairly well with rigid motion but is not applicable to more generic global registration models. One cannot predict the effect of such models on the space of distance. Therefore, criteria that are based on establishing intensity-driven correspondences in the pixel level are not plausible solutions. Scale variations can be considered as global intensity change in the space of distance transforms. Therefore, registration is equivalent to the matching of rather different modalities that refer to the same structure of interest. Such application is quite popular in the medical image analysis domain. Statistical methods, like the mutual information criterion can address such limitations.

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3 Mutual Information & Implicit Representations

Figure 3. Bunny: Global Registration Using Implicit Representations & Mutual Information: (a) Initial Conditions, (b) Rigid Registration. Each column corresponds to a different trial.

Mutual information is an invariant technique for registration according to a monotonic transformation of the two input random variables. Therefore it can cope in a natural manner with scale variations. Furthermore, such criterion is based on the global characteristics of the structures of interest. The inference problem is considered on the space of global density functions derived from the structures of interest. Therefore, the need for seeking visual correspondences in the pixel level does not exist. In order to facilitate the notation let us denote: (i) the source representation ΦD as f , and (ii) the target representation ΦS as g. In the most general case, registration is equivalent with recovering the parameters Θ = (θ1 , θ2 , ..., θN ) of a parametric transformation A such that¡ the mutual ¢ information A between fΩ = f (Ω) and gΩ = g A(Θ; Ω) is maximized for a given sample domain Ω; h Ai h i £ ¤ A A M I(X fΩ , X gΩ ) = H X fΩ + H X gΩ − H X fΩ ,gΩ

3.1 Optimization of Mutual Information The definition of such framework leads to a natural way of dealing with various global motion models. These distributions as suggested in the literature can be approximated/estimated using a non-parametric Gaussian Kernel density model leading to the following expression for the joint density: ZZ A 1 pfΩ ,gΩ (l1 , l2 ) = G(l1 −f (x), l2 −g(A(Θ; x)))dx {z } | | {z } V (Ω) Ω α

β

where G(l1 − α, l2 − β) represents a two dimensional zeromean differentiable Gaussian kernel. A similar approach A can be considered in defining pfΩ (l1 ) and pgΩ (l2 ) using a 1D Gaussian kernel. The calculus of variations with a gradient descent method can be used to minimize the cost function EGlobal and recover the registration parameters θi ;

where H represents the differential entropy. Such quantity represents a measure of uncertainty, variability or complexity and consists of three components; (i) the entropy of the model, (ii) the entropy of the projection of the model given the transformation, and (iii) the joint entropy between the model and the projection that encourages transformations where f explains g. One can use the above criterion and an arbitrary transformation to perform global registration that is equivalent with minimizing:

∂EGlobal =− ∂θi ·

ZZ

R2

1 V (Ω)

³

A

1 + log

ZZ

pfΩ ,gΩ (l1 , l2 ) ´ A pfΩ (l1 )pgΩ (l2 )

−Gβ (l1 − α, l2 − β) ¸ ¢ ¡ ∂ A(Θ; x) dx dl1 dl2 ∇g(A(Θ; x)) · ∂θi Z Z ·Z Z ³ A 1 pfΩ ,gΩ (l1 , l2 ) ´ = − 1 + log A V (Ω) pfΩ (l1 )pgΩ (l2 ) Ω R2 ¸ ³ ´ − Gβ (l1 − α, l2 − β) dl1 dl2

A

EGlobal (A(Θ)) = −M I(X fΩ , X gΩ ) ZZ A A pfΩ ,gΩ (l1 , l2 ) dl1 dl2 pfΩ ,gΩ (l1 , l2 )log =− A pfΩ (l1 )pgΩ (l2 ) R2

¡

where (i) pfΩ corresponds to the implicit representation A [ΦD (Ω)] probability distribution in fΩ , (ii) pgΩ corresponds to the implicit representation [ΦS (A(Θ; Ω))] distribution in A A gΩ , and (iii) pfΩ ,gΩ their joint distribution.

Ω

∇g(A(Θ; x)) ·

¢ ∂ A(Θ; x) dx ∂θi

where V (Ω) represents the volume of the sample domain Ω. These partial differential equations can be used to perform 3

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(b) Figure 5. Empirical Validation of Global Registration Using Implicit Representations & Mutual Information.

(c) • Translation in x and rotation are unknown [fig. (5.[2.b])],

Figure 4. Fish: Global Registration Using Implicit Representations & Mutual Information: Rigid versus Affine (a) Initial Conditions, (b) Rigid Registration, (c) Affine Registration; Each column corresponds to a different trial.

An empirical evaluation test is considered where we have quantize the search space using uniform sampling rule (100 elements) for all unknown parameters in each case. Translation in (x, y) were in the range£ of [−20, ¤ 20] × [−20, 20], scale in [0.5, 2.0] and rotation in − π3 , π3 . Then, one can estimate the cost function in the space of two unknown parameters, by considering all possible combinations derived from the sampling strategy (the other two parameters are fixed). The resulting functional as shown in [fig. (5)] has some nice properties; it is smooth and exhibits a single global minimum. The objective function has a convex form for all combinations that involve two unknown registration variables (some of them are shown in [fig. (5)]) and is a good indicator for a well-behaved optimization criterion with smooth properties. Model-based registration can be an acceptable solution to a large number of image processing and computer vision applications. Medical imaging is an area where non-rigid motion is a common problem. It is well known that solving the correspondence problem between shapes as well as estimating the optical flow between two images are topics of increasing interest. Dense motion/registration estimation is an ill-posed problem since the number of variables to be recovered is larger than the number of available constraints. Smoothness as well as other form of constraints [16] were employed to cope with this limitation. Such components are efficient when used within an image registration problem that refers to structures with limited discontinuities. On the other hand, in the case of shapes such constraints cannot be used in a straightforward manner. The lack of ”visual” support in this scenario imposes such limitation. Furthermore, optical flow driven local deformations can potentially

registration according to a given model between two shapes. The method can support any global linear model, like rigid, affine, etc. Examples of such approach for rigid registration are given in [fig. (3,4,6)] and for affine in [fig. (4)]. The registration protocol is the following: given a source and a target shape, an implicit representation in the space of distance transforms is recovered. Then, the mutual information criterion is used to recover the parameters of the optimal transformation between the source and the target implicit representations. Gradient descent optimization techniques suffer from being very sensitive to the initial conditions. The form of the cost function is a good indicator regarding the efficiency/stability of the framework.

3.2 Empirical Validation In order to perform a study on the performance of the global registration technique introduced earlier, one can constrain the unknown parameter space in two dimensions. An empirical assessment on the performance as the one suggested in [12] can be considered for two ”dude” shapes [fig. (6.1)]. To this end, for the examples shown in [fig. (5)] we have studied the following cases: • Translation is unknown [fig. (5.[1,a])], • Scale and rotation are unknown [fig. (5.[1,b])], • Translation in x and scale unknown [fig. (5.[2.a])], 4

tionally expensive task [7]. We consider a slight variance of such model, an Incremental B-spline FFD. Such selection decreases the computational complexity of enforcing smoothness constraints and inherently supports a multiresolution approach. The resulting model is suitable for modeling large and small non-rigid deformations leading to a dense registration paradigm that is C 2 continuous and guarantees an one-to-one mapping. Local deformations are a complementary component to the global registration model. The mutual information criterion is considered to recover a global motion model (τ ) in the space of implicit representations. One can use such model to transform the source shape D to a new shape ˆ = τ (D) that is the projection of D to S. Then, local D registration is equivalent with recovering a pixel-wise deformation field that creates visual correspondences between the implicit representation [ΦS ] of the target shape S and the implicit representation [ΦDˆ ] of the warped source shape ˆ An example of such registration process is shown in [fig. D. (1)] where all the stages of the proposed technique are depicted.

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4.1 B-spline FFD and Implicit Representations The Sum of Squared Differences (SSD) defined in the space of implicit representations can be considered [12] to recover such deformation field L(Θ; x)); ZZ ¡ ¢2 ΦDˆ (x) − ΦS (L(Θ; x)) dx Edissim (Θ) =

(b) Figure 6. (1) Dude, (2) Hand. Global Registration Using Implicit Representations & Mutual Information: (a) Initial Conditions, (b) Rigid Registration. Each column corresponds to a different trial.

Ω

We consider an Incremental Cubic B-spline Free Form Deformation (FFD) to model the local transformation L. To this end, object deformations are recovered by evolving a control lattice P that is overlaid on structure of interest. The inference problem is then solved with respect to the control lattice coordinates that are the parameters of FFD. Our approach aims at recovering a deformation improvement [δP ] on the current deformed lattice. Such formulation is supported by the linear precision property of FFD [8], can naturally account for an efficient multi-resolution approach to FFD, and simplifies the integration of smoothness constraints. While such approach can register structures of arbitrary dimension, the example of 2D shape registration is considered to facilitate the introduction of the method. Let us consider a regular lattice of control points

create inconsistencies where a closed structure can be transformed to an open structure, or vice versa.

4 Free Form Incremental B-Spline Local Registration An elegant way to overcome in some extend such limitation refers to the use of warping techniques and free form deformations that are quite popular in graphics, animation and rendering [4]. The essence of traditional FFD is to deform an object by manipulating a regular control lattice P overlaid on the volumetric embedding space. Thin plate splines [6, 19] is a popular non-rigid transformation technique that requires finding two sets of corresponding landmark points, a difficult and ill-posed problem. Cubic B-spline based Free Form Deformations (FFD) is an alternative to TPS. It can model local transformations in a computationally efficient manner. Such selection provides a better local control compared to the high order Bernstein Polynomials [15]. Smoothness is a natural condition in the registration process. Introducing such constraint within the traditional Bspline based FFD is a rather complicated and computa-

x y Pm,n = (Pm,n , Pm,n ); m = 1, ..., M, n = 1, ..., N

overlaid to a region Γc = {x} = {(x, y)|1 ≤ x ≤ X, 1 ≤ y ≤ Y } in the embedding space that encloses the source structure. Let us denote the initial configuration of the control lattice as P 0 , and the deforming control lattice as P = P 0 + δP . Under these assumptions, the incremental FFD parameters 5

Data-driven and smoothness constraints can now be integrated to recover the local deformation component of the registration and solving the correspondence problem;

are the deformations of the control points in both directions (x, y); x y Θ = {(δPm,n , δPm,n )}; (m, n) ∈ [1, M ] × [1, N ]

E(Θ) =

The motion of a pixel x = (x, y) given the deformation of the control lattice from P 0 to P , is defined in terms of a tensor product of Cubic B-spline: L(Θ; x) =x + δL(Θ; x) =

3 X 3 X

0 Bk (u)Bl (v)(Pi+k,j+l

+ δPi+k,j+l )

where y x · M ⌋ − 1, l = ⌊ · N ⌋ − 1 X Y

The novel notation/terms of deformation component refer to • δPi+l,j+l , (k, l) ∈ [0, 3] × [0, 3] consists of the deformations of pixel x’s (sixteen) adjacent control points, • δL(x) =

3 X 3 X

Bk (u)Bl (v)δPi+k,j+l

is the incremental deformation at pixel x, • Bk (u) is the k th basis function of the Cubic B-spline given by; B0 (u) = (1 − u)3 /6, B1 (u) = (3u3 − 6u2 + 4)/6 B2 (u) = (−3u3 + 3u2 + 3u + 1)/6, B3 (u) = u3 /6 x X

¡

This flow consists of a data-driven update component and a diffusion term that constraints the parameters of the free form deformation to be locally smooth. The partial derivatives of the FFD formulation are given in the appendix. The performance of the proposed framework is demonstrated for various examples in [fig. (1,7)]. One can consider a multi-resolution implementation of the proposed framework as shown in [fig. (8)]. To this end, adaption of the multi-level B-spline Free Form Deformation control lattices takes place according to a coarseto-fine strategy. Coarser control lattices account for global non-rigid deformations, while finer control lattices handle highly local deformations. The hierarchy of control lattices are computed efficiently using a progressive B-spline subdivision algorithm [5]. The proposed framework can naturally support such a multi-level formulation. In a hierarchy of r level, the total deformation δL(x) at a pixel x is:

k=0 l=0

with u =

¢2 ΦDˆ (x) − ΦS (L(Θ; x)) dx + ¯¯ ¯¯ ! ¯¯ Z Z à ¯¯ ¯¯ ∂δL(Θ; x) ¯¯2 ¯¯ ∂δL(Θ; x) ¯¯2 ¯ ¯ ¯ ¯ ¯ ¯¯ dx ¯ α ¯¯ ¯¯ + ¯¯ ¯¯ ∂x ∂y Ω Ω

where α is the constant balancing the contribution of the two terms. The calculus of variations and a gradient descent method can be used to optimize such objective function. Then, one can obtain the following evolution equation for the parameter θi of L; ZZ ¢ ∂ (ΦDˆ (x) − ΦS (L(Θ; x)) E(Θ) = −2 ∂θi Ω ¢ ∂ δL(Θ; x) dx ∇ΦS (L(Θ; x)) · ∂θi ¶ µ ZZ ∂ ∂ ∂ δL(Θ; x) · δL(Θ; x) + 2α ∂θi ∂x Ω ∂x ¶ µ ∂ ∂ ∂ δL(Θ; x) · δL(Θ; x) dx + ∂y ∂θi ∂y

k=0 l=0

k=⌊

ZZ

x ·M −⌊X · M ⌋. Bl (v) is similarly defined.

The use of such technique to model the local deformation registration component introduces in an implicit form some smoothness constraint that can deal with a limited level of deformation. In order to further preserve the regularity of the recovered registration flow, one can consider an additional smoothness term on the deformation field δL. We consider a computationally efficient regularization term in the form of an objective function component;

δL(x) =

r X

δLk (x)

k=1 k

where δL (x) refers to the deformation improvement original FFD defined at the kth level ¤ £due toPthe r δP = k=1 δP k . The proposed framework and the derivation can be naturally extended to 3D, using a 3D tensor product of B-spline polynomials [13].

¯¯ ¯¯ ¯¯ ! Z Z à ¯¯ ¯¯ ∂δL(Θ; x) ¯¯2 ¯¯ ∂δL(Θ; x) ¯¯2 ¯ ¯ ¯ ¯ ¯ ¯¯ dx ¯ Esmooth (Θ) = ¯¯ + ¯¯ ¯¯ ¯¯ ∂x ∂y Ω

Such regularization term is based on a classic error norm that has certain known limitations. One can replace this smoothness component with more elaborated norms. Within the proposed framework, an implicit smoothness constraint is also imposed by the Spline FFD. Therefore the need for introducing complex and computationally expensive regularization components is not strong.

5 Discussion In this paper we have proposed a variational framework for global-to-local registration. Such framework has been 6

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(ii) Figure 8. B-Spline Multi-resolution FFD Registration; (1.a) Initial Conditions, (1.b) Global Registration. (2) Coarse resolution, (3) Finer resolution, (a) Correspondences, (b) Matching, (c) Grid deformation.

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(c) rion cannot account for such complex scenario and will have to be replaced with more appropriate local matching technique. One can consider a local criterion driven by mutual information as proposed in [2] to perform local registration. Such approach will be able to account for local information of different nature defined in the powerful space of implicit representations. Integration of both components (global and local registration) within a common framework is also an interesting direction as well as a mathematical justification of the method in terms of existence and uniqueness of the solution within a limited parameter search space. Acknowledgments: The authors are thankful to Dr. Sebastian and Prof. Kimia from Brown University for providing their shape database [14] that was used to demonstrate the potentials of the proposed framework as well as to Dr. Mokhtarian and Prof. Kittler for providing the fish examples.

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Figure 7. B-Spline FFD Registration; (i) Bunny, (ii) Fish, (iii) Brain Structure. (1.a) Initial Conditions, (1.b) Rigid Registration, (2.a) Correspondences, (2.b) Matching, (2.c) Grid deformation.

derived by integrating a powerful representation (implicit surface) with (i) a robust/efficient global registration technique (mutual information) and (ii) an elegant local deformation approach. The resulting paradigm is relatively free from the initial conditions [fig. (5)], exhibits robustness to noise, can deal with arbitrary forms of global transformations [fig. (4)] and can efficiently solve the correspondence problem [fig. (8,7)]. Furthermore, it can deal with important local variations [fig. (6.1)], severe occlusions and missing parts [fig. (6.2)], open structures [fig. (9)] and can be extended in higher dimensions [fig. (9)]. The problem of registering geometric shapes in 2D/3D was considered to demonstrate the potentials of such technique.

A Appendix Partial derivatives for the incremental B-Spline Free Form Deformation parameters x y δPm,n = (δPm,n , δPm,n ); m = 1, ..., M, n = 1, ..., N

are straightforward to be determined. Without loss of generality one can consider the (m, n)th control point and its deformation in both directions. Then, for the data driven term the following relation holds;  · ¸ Bm−i (u) Bn−j (v)   ,0 ≤ m − i, n − j ≤ 3  ∂ δL(Θ; x) 0 = x  ∂δPm,n   0, otherwise  · ¸ 0   ,0 ≤ m − i, n − j ≤ 3  ∂ δL(Θ; x) Bm−i (u) Bn−j (v) = y  ∂δPm,n   0, otherwise

The use of such framework for image and volume registration is the main future direction of our work. To this end, a slight modification will be required when dealing with images of different modalities. The use of SSD crite7

[8] S. Lee, K.-Y. Chwa, and S.Y. Shin. Image Metamorphosis with Scattered Feature Constraints. IEEE Transactions on Visualization and Computer Graphics, 2:337–354, 1996.

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[9] J. Maintz and M. Viergever. A Survey for Medical Image Registration. Medical Image Analysis, 2:1–36, 1998.

(d)

[10] D. Metaxas. Physics-Based Deformable Models. Kluwer Academic Publishers, 1996.

Figure 9. Rigid Registration Using Implicit Representations & Mutual Information for open 3D structures (from face range scan data). (a) Source, (b) target, (c) Initial Conditions, (d) Rigid Registration.

[11] S. Osher and J. Sethian. Fronts propagating with curvaturedependent speed : Algorithms based on the Hamilton-Jacobi formulation. Journal of Computational Physics, 79:12–49, 1988.

while for the smoothness term one can obtain the following analytical expression of the partial derivatives;

[12] N. Paragios, M. Rousson, and V. Ramesh. Matching Distance Functions: A Shape-to-Area Variational Approach for Global-to-Local Registration. In European Conference on Computer Vision, pages II:775–790, 2002.

∂ δL(Θ; x) ∂x

∂

∂ (δL(Θ;x)) ∂x x ∂δPm,n

∂ (δL(Θ;x)) ∂ ∂y y

∂δPm,n

=

P3

k=0

P3

l=0

¡

∂ B (u) ∂u k

¢ · ux Bl (v)δPi+k,j+l

[13] D. Rueckert, L. Sonoda, C. Hayes, D. Hill, M. Leach, and D. Hawkes. Nonrigid Registration Using Free-Form Deformations: Application to Breast MR Images. IEEE Transactions on Medical Imaging, 8:712–721, 1999.

 · ¡ ∂ ¸ ¢ 0≤m−i B (u) · ux Bn−j (v)   ∂u m−i ,  n−j ≤3 0 =    0, otherwise =

 ·   ¡    

The derivation for tained.

0 ¢ · ux Bn−j (v)

∂ B (u) ∂u m−i

∂ ∂y δL(Θ; x),

0,

¸

,

[14] T. Sebastian, P. Klein, and B. Kimia. Recognition of Shapes by Editting Shock Graphs. In IEEE International Conference in Computer Vision, pages 755–762, 2001.

0≤m−i n−j ≤3

[15] T. Sederberg and S. Parry. Free-Form Deformation of Solid Geometric Models. In ACM SIGGRAPH, volume 4, pages 151–160, 1986.

otherwise

etc. can be similarly ob-

[16] A. Tikhonov. Ill-Posed Problems in Natural Sciences. Coronet, 1992.

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