## Mathematics And Computer Science Math Emory Edu-Free PDF

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Numerical methods for volume preserving,image registration. Eldad Haber Jan Modersitzki,February 17 2004, Image registration techniques are used routinely in a variety of. today s medical imaging diagnosis Since the problem is ill posed. one may like to add additional information about distortions This. applies for example to the registration of contrast enhanced images. where variations of substructures are not related to patient motion but. to contrast uptake Here one may only be interested in registrations. which do not alter the volume of any substructure, In this paper we discuss image registration techniques with a focus. on volume preserving constraints These constraints can reduce the. non uniqueness of the registration problem significantly Our imple. mentation is based on a constrained optimization formulation Upon. discretization we obtain a large discrete highly nonlinear optimiza. tion problem and the necessary conditions for the solution form a dis. cretized nonlinear partial differential equation To solve the problem. we use a variant of the Sequential Quadratic Programming method. Moreover we present results on synthetic as well as on real life data. 1 Introduction, Image registration is one of the fundamental tasks in today s image processing. and in particular in medical imaging see e g 13 and references therein. Dept of Mathematics and Computer Science Emory University Atlanta GA 30322. haber modersit mathcs emory edu, The objective of image registration is to make images which are taken at.
different times from different perspectives and or from different devices to. be more alike Loosely the goal of image registration is to find a reasonable. deformation such that the distance between a reference image R and a. deformed version of a template image T becomes small. An application of particular clinical interest is the registration of pairs of. images acquired before and after contrast administration see e g 30 and. references therein A typical example is depicted in Fig 1 In this appli. cation magnetic resonance images of a female breast are taken at different. times images from Bruce Daniel Lucas Center for Magnetic Resonance Spec. troscopy and Imaging Stanford University The first image shows an MRI. section taken during the so called wash in phase of a radiopaque marker and. the second image shows the analogous section during the so called wash out. phase A comparison of these two images indicates a suspicious region in the. upper part of the images This region can be detected easily if the images. have been registered tissue located at a certain position in the wash in image. is related to the tissue at the same position in the wash out phase Gener. ally however a quantitative analysis is a delicate matter since observable. differences are not only related to contrast uptake but also due to motion of. the patient like for example breathing or heart beat. Figure 1 MRI s of a female breast left during the wash in phase. middle during the wash out phase and right difference image. As pointed out by Rohlfing et al 30 there is a substantial difficulty. with the registration of pre and post contrast images Bright regions seem. to enlarge during the so called wash in phase This enhancement is due to. contrast uptake but not to movement of the patient Fig 2 illustrates an ideal. situation Without external information it is impossible to answer whether. Figure 2 Synthetic images simulating contrast uptake left wash in. right wash out, the white area has been enlarged or the grey area turned to white. The idea is to restrict the set of feasible transformations a priori in a. reasonable way For this situation we constrain the transformations to be. volume preserving VP In contrast to 30 we use a variational setting. Therefore we are not restricted to parametric and in particular B spline. based deformations Also we do not introduce an additional penalty term. but use a constrained approach, The VP approach is also connected to the so called mass preserving MP. registration which is related to the Monge Kantorovich transport problem. see 19 36 From a mathematical point of view the MP approach contains. the VP approach setting the density to 1 However in the MP approach the. images enter directly into the constraints whereas in the VP they do not In. contrast to the MP approach the VP approach aims to correct geometrical. distortions but do not alter gray values Another major difference between. the above work and ours is the numerical treatment For the MP approach a. Helmholtz decomposition is exploited and after eliminating the constraints. the necessary conditions for the optimization problem are solved using a. steepest descent method Here we discretize the transformation and the. constraints directly and use a Sequential Quadratic Programm to solve the. discrete optimality conditions, In this paper we present a flexible constrained image registration ap. proach It has three main ingredients a distance measure a regularizer and. the constraints, Our mathematical framework is general enough to handle a variety of. distance measures including the most popular ones like those based on the. sum of squared differences SSD mutual information MI or correlation. as long as a Ga teaux derivative exists see e g 29 22 20 For presentation. purposes we explicitly discuss the approach only for the SSD measure. Even with this additional constraints however image registration is an ill. posed problem cf e g 21 26 If one considers for example the registration. of an image of a disc to a copy of this image any rotation of the image gives. a solution with respect to any reasonable distance measure Note that a. pure rotation is a rigid transformation and rigid transformations are volume. preserving For this reason also the constrained image registration problem. has to be regularized The framework presented here is based on a general. regularizer too Any regularizing term with a Ga teaux derivative can be. used This includes well known choices like for example the elastic 4 6. 15 9 the diffusion 23 10 and the biharmonic 1 12 regularizer For. ease of presentation we emphasize on the most popular of these the elastic. registration, Also our approach to the constraints is very general However since the.
implementation of the volume preserving constraints is not straightforward. we restrict ourselves to these constraints It is very important to note that. volume preservation constraints are pointwise differential constraints which. after discretization apply to any pixel voxel, Finally we suggest the use of a staggered grid discretization This is a. well known established and often used technique in many fields like for. example fluid dynamics or electromagnetics However we are not aware of. any image registration algorithm based on this discretization. The paper is organized as follows In Section 2 we present the continuous. mathematical setup of the constrained registration problem and derive the. continuous Euler Lagrange equations Although we do not use these equa. tions in our numerical implementation they give insight into the problem s. structure and serve as a reference for the discrete analogs In Section 3 we. discuss the discretization of the problem For readers from image process. ing which might not be so familiar with staggered grids we give a brief and. formal introduction Staggered grid discretization is well known to be stable. when working with tightly coupled system of partial differential equations. Our constraints are discretized using a finite volume discretization of. each displaced pixel voxel and we show that the resulting formula mimics. the continuous one In Section 4 we discuss a numerical scheme for the. optimization of the discretized image registration problem In Section 5 we. present numerical results and finally in Section 6 we summarize the paper. and discuss future work,2 Mathematical setup, With d N we denote the spatial dimension of the given images R T Rd. R which are assumed to be sufficiently smooth Thus T x gives a gray. value at a spatial position x We assume that the supports of the images are. contained in a bounded domain 0 L d i e R x T x 0 for x. Our goal is to find a reasonable deformation u such that the distance. between the reference image R and the deformed template image T x u x. becomes small Note that u u1 ud T Rd Rd It is well known that. this problem is ill posed and therefore needs to be regularized In general it is. common to use a Tikhonov style regularization A mathematical formulation. of the regularized and constrained problem reads,D R T u S u min 1a. subject to C u x 0 for all x 1b, where D is some distance measure S is some regularization term and C are. some constraints Here 0 is a regularization parameter and compromises. between similarity and regularity, The three building blocks are discussed below The constraints can be.
used to supply additional information about the registration problem For. example in some applications it is of importance that particular points like. e g anatomical landmarks are in a precise one to one correspondence With. the setting C u T u T R one can guaranty the correspondence of the. landmarks R and T see e g 5 11 In this paper we focus on applications. demanding for volume preserving VP transformations However one may. set C u 0 to recover the unconstrained problem, For the following analysis and numerics any choices and combinations of. the building blocks D S and C are feasible as long as they do have a Ga teaux. derivative For the purpose of this presentation we restrict the discussion to. the following choices, Distance measures The distance measure used in this note is the so called. sum of squared differences or L2 norm,D R T u T x u x R x dx 2. with Ga teaux derivative cf e g 26,du v D R T u hf x u x v x iRd dx 3. f x u x T x u x R x T x u x x Rd 4, Other distance measures like e g those based on mutual information 7 34.
might be used as well, Remark 1 Note that due to the chain rule any differentiable distance func. tional based on R and T x u x has a factor T x u x Thus since. we assume R and T to be zero for x also f is zero for x There. fore the integration in Eqs 2 and 3 reduces to an integration over the. domain only, Regularizer In this work we consider the well known elastic regularizer 4. 6 9 However our formulation is flexible enough that we could use many. other regularizers like e g the fluid 6 3 diffusion 10 or curvature regu. larizers 12 or any combinations of these Each of the above regularizers is. based on a differential operator B Particularly,S u hB u B u iRd dx 5. and therefore its Ga teaux derivative is,du v S u hA u x v x iRd dx A B B 6. where for ease of presentation we assume natural boundary conditions on. u For the elastic regularizer with Lame constants and we have. B u u A u u u 7,with the divergence the curl and the Laplacian.
Constraints We require the transformation x x u x to be volume. preserving see also 30 The transformation is volume preserving if and. only if for any domain V,dx dx where V x x V 8, This constraint implies that not only the overall volume but also and most. importantly the volume of each arbitrarily small subdomain is conserved. For a smooth transformation we use the transformation rule to derive the. point wise constraints,1 det det Id u 9,or C u det Id u 1 0 10. Here Id Rd d denotes the d by d identity matrix and j k its j k th entry. The Ga teaux derivative of the volume preserving contraints is given by. the following lemma, Lemma 1 Let v be a suitable perturbation of u The Ga teaux derivative of. C cf Eq 10 is given by,du v C u x det Id u x Id u x v x Rd d. Proof A computation gives,du v det Id u,lim det Id u v det Id u.
det Id u lim det Id v Id u 1 1,det Id u trace v Id u 1. det Id u Id u v Rd d, Remark 2 Applying Cramer s rule it can be verified that the Ga teaux deriva. tive of C is a polynomial in j uk j k 1 d,Example 1 For d 2 we have. C u u 1 u1 2 u2 2 u1 1 u2,det I2 u I2 u u2 2 1 1 u2. We now investigate necessary conditions for a solution of the image regis. tration problem 1 For computational purposes we have to discretize either. the optimization problem 1 or the resulting necessary conditions In the. following section we choose to discretize the optimization problem directly. and therefore the continuous conditions are not used directly in our numer. ical scheme However the discrete conditions have to mimic the continuous. analogs and therefore we find it useful to study the latter. Introducing the Lagrange multiplier p Rd R the Lagrangian of 1 is. L u p D R T u S u C u x p x dx 12, and the continuous Euler Lagrange equations for 1 read.
0 du v L u p,du v D R T u du v S u du v C u x p x dx 13a. 0 dp q L u p C u x q x dx 13b, for any appropriate perturbations v and q Thus for any x we have. 0 f x u x A u x,det Id u x Id u x p x 14a,0 det Id u x 1 14b. where we imposed zero Dirichlet boundary conditions for the Lagrange mul. The system 14 presents a highly coupled system of nonlinear partial. differential equations PDE The quantity f in 14a which depends non. linearly on u and the images can be viewed as forces pushing the template. towards similarity The differential operator A in 14b is a linear elliptic. pure rotation is a rigid transformation and rigid transformations are volume preserving For this reason also the constrained image registration problem has to be regularized The framework presented here is based on a general regularizer too Any regularizing term with a Ga teaux derivative can be used This includes well known choices like