A variational framework for joint image registration, denoising and edge detection

A variational framework for joint image registration, denoising and edge detection
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  A Variational Framework for Joint ImageRegistration, Denoising and Edge Detection Jingfeng Han 1 , Benjamin Berkels 2 , Martin Rumpf  2 ,Joachim Hornegger 1 , Marc Droske 3 , Michael Fried 1 ,Jasmin Scorzin 2 and Carlo Schaller 2 1 Universit¨at Erlangen-N¨urnberg,  2 Universit¨at Bonn,  3 University of California Abstract.  In this paper we propose a new symmetrical framework thatsolves image denoising, edge detection and non–rigid image registrationsimultaneously. This framework is based on the Ambrosio–Tortorelli ap-proximation of the Mumford–Shah model. The optimization of a globalfunctional leads to decomposing the image into a piecewise–smooth rep-resentative, which is the denoised intensity function, and a phase field,which is the approximation of the edge-set. At the same time, the methodseeks to register two images based on the segmentation results. The keyidea is that the edge set of one image should be transformed to matchthe edge set of the other. The symmetric non–rigid transformations areestimated simultaneously in two directions. One consistency functionalis designed to constrain each transformation to be the inverse of theother. The optimization process is guided by a generalized gradient flowto guarantee smooth relaxation. A multi–scale implementation schemeis applied to ensure the efficiency of the algorithm. We have performedpreliminary medical evaluation on T1 and T2 MRI data, where the ex-periments show encouraging results. 1 Introduction Image registration, image denoising and edge detection are three important andstill challenging image processing problems in the field of medical image analy-sis. Traditionally, solutions are developed for each of these three problems mu-tually independent. However, in the various applications, the solutions of theseproblems are depend on each other. Indeed, tackling each task would benefitsignificantly from prior knowledge of the solution of the other tasks. Here, wetreat these different image processing problems in an uniform mathematicallysound approach.There already have been some attempts in the literature to develop meth-ods aligning the images and detecting the features simultaneously [1,2,3,4]. Dueto our knowledge, most of the existing approaches are restricted to lower di-mensional parametric transformations for image registration. Recently, in [5,6] anovel approach for non–rigid registration by edge alignment has been presented.The key idea of this work is to modify the Ambrosio–Tortorelli approximation of the Mumford–Shah model, which is traditionally used for image segmentation,  so that the new functional can also estimate the spatial transformation betweenimages, but in contrast to the method proposed by Droske et al. our method isfully “symmetric” with respect to the treatment of the edge sets in both imagesand the transformations in both directions. 2 Method Assume two gray images  R  and  T   are given, whose intensity values are describedby the function  u 0 R  and  u 0 T  , respectively. The goal of the joint framework is tofind piecewise smooth representatives  u R  and  u T   (denoising), phase field edgefunctions  v R  and  v T   (edge detection) and symmetric non–rigid spatial transfor-mations  φ  and  ψ  such that  u R  ◦ ψ  matches  u T   and  u T   ◦ φ  matches  u R  (reg-istration). For simplification of presentation, we denote all the unknowns with Φ  = [ u R ,u T  ,v R ,v T  ,φ,ψ ]. The associated functional is defined as E  G [ Φ ] =  E  u 0 R AT  [ u R ,v R ] + E  u 0 T  AT  [ u T  ,v T  ] + E  REG [ Φ ]  →  min ,  (1) In what follows, we define and discuss the variational formulation. 2.1 Denoising and Edge Detection The  E  u 0 AT [ u,v ] denotes the Ambrosio–Tortorelli (AT) approximation functionalproposed in [7,8]. This functional is originally designed to approximate theMumford–Shah model [9] for image segmentation. The functional is defined as E  u 0 ,ǫ AT  [ u,v ] = Z  Ω α 2( u − u 0 ) 2  | {z }  E 1 +  β  2 v 2 ∇ u  2  | {z }  E 2 + ν  2 “ ǫ ∇ v  2  | {z }  E 3 + 14 ǫ ( v  −  1) 2  | {z }  E 4 ” dx, with parameters  α,β,ν   ≥ 0. In the Ambrosio–Tortorelli approximation, the edgeset is depicted by a phase field function  v  with  v ( x )  ≈  0 if   x  is an edge pointand  v ( x ) ≈ 1 otherwise. The term  E  1  favors  u  to be as similar to  u 0 as possible.The term  E  2  allows  u  to be singular (large  ∇ u  2 ) where  v  ≈ 0 and favors  u  tobe smooth (small  ∇ u  2 ) where  v  ≈ 1. The term  E  3  constrains  v  to be smooth.The last term  E  4  prevents the degeneration of   v , i.e. without  E  4  the functionalwould be minimized by  v  ≡ 0 ,u ≡ u 0 . For the details of the Ambrosio–Tortorelliapproximation, we refer to [7]. 2.2 Edge Alignment The main goal of the registration functional  E  REG  is to find the transformationsthat match the edge sets of image  R  and image  T   to each other. In order toexplicitly enforce the bijectivity and invertibility of spatial mapping, we estimatethe two transformations in two directions simultaneously:  φ  :  Ω   →  Ω   is thetransformation from image  T   to image  R  and  ψ  :  Ω   →  Ω   is the one from  R  to T  . The functional  E  REG  is a linear combination of an external functional  E  ext ,an internal functional  E  int  and a consistent functional  E  con : E  REG [ Φ ] =  µE  ext [ Φ ] + λE  int [ φ,ψ ] + κE  con [ φ,ψ ] ,  (2)  where µ,λ and κ are just scaling parameters. The three functionals E  ext ,E  int ,E  con are defined as follows: E  ext [ Φ ] = Z  Ω 12( v T   ◦ φ ) 2 ∇ u R  2 + 12( v R  ◦ ψ ) 2 ∇ u T   2 dx ,  (3) E  int [ φ,ψ ] = Z  Ω 12   Dφ −  11  2 + 12   Dψ  −  11  2 dx ,  (4) E  con [ φ,ψ ] = Z  Ω 12  φ ◦ ψ ( x )  − x  2 + 12  ψ  ◦ φ ( x )  − x  2 dx .  (5) Here, 11 is the identity matrix. The external functional  E  ext  favors transforma-tions that align zero–regions of the phase field of one image to regions of highgradient in the other image. The internal functional  E  int  imposes a commonsmoothness prior on the transformations. The consistency functional  E  con  con-strains the transformations to be inverse to each other, since it is minimizedwhen  φ  =  ψ − 1 and  ψ  =  φ − 1 . 2.3 Variational Formulation The definition of the global functional  E  G [ Φ ] is mathematically symmetricalwith respect to the two groups of unknown [ u R ,v R ,φ ] and [ u T  ,v T  ,ψ ]. Thus, werestrict here to variations with respect to [ u R ,v R ,φ ]. The other formulas can bededuced in a complementary way.For testfunctions  ϑ ∈ C  ∞ 0  ( Ω  ) ,ζ   ∈ C  ∞ 0  ( Ω, R d ), we obtain  ∂  u R E  G ,ϑ   = Z  Ω α ( u R  − u 0 R ) ϑ  + βv 2 R ∇ u R  · ∇ ϑ  + µ ( v T   ◦ φ ) 2 ∇ u R  · ∇ ϑdx ,  ∂  v R E  G ,ϑ   = Z  Ω β  ∇ u R  2 v R ϑ  +  ν  4 ǫ ( v R  −  1) ϑdx + Z  Ω νǫ ∇ v R  · ∇ ϑ  + µ ∇ u T   ◦ ψ − 1  2 v R ϑ | det Dψ | − 1 dx ,  ∂  φ E  G ,ζ    = Z  Ω µ ∇ u R  2 ( v T   ◦ φ ) ∇ ( v T   ◦ φ )  · ζ   + λDφ  :  Dζ  + κ ([ φ ◦ ψ ]( x )  − x )  ·  [ ζ   ◦ ψ ]( x ) + ([ ψ  ◦ φ ]( x )  − x ) Dψ ( h ( x ))  · ζ  ( x ) dx . where  A  :  B  =  ij  A ij B ij .At first, a finite element approximation in space is applied [10]. Then, weminimize the corresponding discrete functional by finding a zero crossing of thevariation. Because of the high dimensionality of the minimization problem (sixunknown functions, two of them vector valued), we employ an EM type algo-rithm, i.e. we iteratively solve for zero crossings of the variations given before.Since the variations with respect to the images and the phase fields are linearin the given variable, we can solve these equations directly with a CG method.The nonlinear equations for the transformation are solved with a time discrete,regularized gradient flow, which is closely related to iterative Tikhonov regular-ization, see [11].  3 Results The first experiment was performed on a pair of T1/T2 MRI slices (See Fig.1 a ,1 b ),which have the same resolution (257 × 257) and come from the same patient.The experiment results in Fig.1 show that the proposed method successfully re-moves the noise (Fig.1 c ,1 d ) and detects the edge features (Fig.1 e ,1 f  ) of T1/T2slices. Moreover, the method computes the transformations such that the twotransformed slices (Fig.1 g ,1 h ) optimally align to the srcinal images accordingto the edge features, see (Fig.1 i ,1  j ). The second experiment was designed todemonstrate the effect of the proposed method in 3D. We deformed one MRIvolume (129 × 129 × 129) with Gaussian radial basis function (GRBF) and seekto recover the artificially introduced transformation via symmetric registrationmethod. See the registration results in Fig.2. a c e g ib d f h jFig.1.  Results of registration of T1/T2 slices with parameters:  α  = 2550 ,β   = 1 ,ν   =1 ,µ  = 0 . 1 ,λ  = 20 ,κ  = 1 ,ǫ  = 0 . 5 h . ( a ,  b ): The srcinal images  u 0 T  1  and  u 0 T  2 . ( c ,  d ):Piecewise smooth functions  u T  1  and  u T  2 . ( e ,  f  ): Phase field functions  v T  1  and  u T  2 . ( g , h ): The registered T1 and T2 slices. ( i ): Blending of transformed T1 slice and phasefield function of T2 slice. (  j ): Blending of transformed T2 slice and phase field functionof T1 slice. Acknowledgement.  The authors gratefully acknowledge the support of Deutsche Forschungsgemeinschaft (DFG) under the grant SFB 603, TP C10.The authors also thank HipGraphic Inc. for providing the software for volumerendering (InSpace). References 1. Z¨ollei, L., Yezzi, A., Kapur, T.: A variational framework for joint segmentation andregistration. In: MMBIA’01: Proceedings of the IEEE Workshop on MathematicalMethods in Biomedical Image Analysis, Washington, DC, USA, IEEE ComputerSociety (2001) 44–51  a b cFig.2.  Results of 3D registration. We denote the srcinal MRI volume as  R  and theartificially deformed volume as  T  . After symmetric registration, the resampled volumeare denoted as  R ′ and  T  ′ respectively. ( a ) The check board volume of   R  and  T  . ( b )The check board volume of   R  and  T  ′ .( c ) The check board volume of   T   and  R ′ . Theparameter setting:  α  = 2550 ,β   = 1 ,ν   = 1 ,µ  = 0 . 1 ,λ  = 20 ,κ  = 1 ,ǫ  = 0 . 5 h .2. Chen, Y., Thiruvenkadam, S., Huang, F., Gopinath, K.S., Brigg, R.W.: Simultane-ous segmentation and registration for functional mr images. In: Proceedings. 16thInternational Conference on Pattern Recognition. Volume 1. (2002) 747 – 7503. Young, Y., Levy, D.: Registration-based morphing of active contours for segmen-tation of ct scans. Mathematical Biosciences and Engineering  2  (2005) 79–964. Pohl, K.M., Fisher, J., Levitt, J.J., Shenton, M.E., Kikinis, R., Grimson, W.E.L.,Wells, W.M.: A unifying approach to registration, segmentation, and intensitycorrection. In: MICCAI. (2005) 310–3185. Droske, M., Ring, W.: A Mumford-Shah level-set approach for geometric imageregistration. SIAM Appl. Math. (2005) to appear.6. Droske, M., Ring, W., Rumpf, M.: Mumford-shah based registration. Computingand Visualization in Science manuscript (2005) submitted.7. Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity prob-lems. Boll. Un. Mat. Ital. B  6  (1992) 105–1238. Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumpsby elliptic functionals via  Γ  -convergence. Comm. Pure Appl. Math.  43  (1990)999–10369. Mumford, D., Shah, J.: Boundary detection by minimizing functional. In: Pro-ceedings. IEEE conference on Computer Vision and Pattern Recognition, San Fran-cisco, USA (1985)10. Bourdin, B., Chambolle, A.: Implementation of an adaptive Finite-Element ap-proximation of the Mumford-Shah functional. Numer. Math.  85  (2000) 609–64611. Clarenz, U., Henn, S., Rumpf, M. Witsch, K.: Relations between optimizationand gradient flow methods with applications to image registration. In: Proceed-ings of the 18th GAMM Seminar Leipzig on Multigrid and Related Methods forOptimisation Problems. (2002) 11–30
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