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ABSTRACT In this paper, we propose a method for non-rigid image registration based on a spatially adaptive stochastic model. A smoothness constraint is imposed on the deformation field between the two images which is assumed to be a random variable
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  A SPATIALLY ADAPTIVE HIERARCHICAL STOCHASTIC MODEL FORNON-RIGID IMAGE REGISTRATION Evangelos Fotiou , Christophoros Nikou and Nikolaos Galatsanos University of Ioannina,Department of Computer Science,PO Box 1185, 45110 Ioannina, Greece, { efotiou, cnikou, galatsanos } ABSTRACT In this paper, we propose a method for non-rigid image reg-istration based on a spatially adaptive stochastic model. Asmoothness constraint is imposed on the deformation fi eldbetween the two images which is assumed to be a randomvariable following a Gaussian distribution, conditioned onthe observations and maximum a posteriori (MAP) estima-tion is employed to evaluate the model parameters. Further-more, the model is enriched by considering the deformation fi eld to be spatially adaptive by assuming different densityparameters for each image location. These parameters areassumed random variables generated by a Gamma distribu-tion, which is conjugate to the Gaussian, leading to a modelthat can be estimated. Numerical experiments are presentedthat demonstrate the advantages of this model. 1. INTRODUCTION Image registration is the process of determining an appropri-ate transformation function which, applied to the coordinatesof one image (source image), aligns it with another (targetimage). It is used either in an intra or an inter-subject level.The involved transformations are divided into two main cat-egories: rigid and non-rigid. Rigid transformations preservethe distance between all points in an image and can be rep-resented by shifting and rotating a cartesian system. In thenon-rigid case, straight lines are mapped to curves, increas-ing the degrees of freedom of the problem and, consequently,making it more dif  fi cult to solve. Medical imaging is one of the main, if not the most important, application fi elds of non-rigid image registration [8].Non-rigid image registration methods [15] may begrouped in two main subcategories, according to their the-oretical foundation. In the fi rst subcategory the deformationis modeled using a set of basis functions. These can be B-splines [18], wavelets [21] or radial basis functions [11]. Onthe other hand, there are methods that connect the image datawith a physically deforming system. They use concepts fromcontinuum mechanics to model the source image as a ver-sion of the target, embedded into a deformable medium. Themedium is then deformed subject to internal forces whichresult in the recon fi guration to its srcinal state (target im-age). Well known methods are based on physical modelssuch as linear elasticity [9], viscus fl uid fl ow [5] and optical fl ow [19].Non-rigid image registration yields a nonlinear ill-posedinverse problem that require regularization [20]. Minimiza-tion of the energy functional in the intensity based methods,locates the optimal transformation function out of the spaceof all admissible solutions. Even when local minima areavoided, the transformation does not necessarily preserve thetopology of the image data. Topology preserving mappingis required especially in inter-subject registration of medicalimages, since anatomical structures have the same topologyfor any individual. As a result, further constraints have tobe introduced, in order to restrict the space of admissible so-lutions to those which provide anatomically coherent trans-formations. In the physical models such as linear elastic-ity, these constraints arise naturally, while another popularchoice is the Laplacian constraint [2]. Once the constraintis chosen, it is minimized simultaneously with the energyfunctional [7]. However, for large non-rigid transformations,this leads to large computational cost, since the constraint, inthe linear regularization models, generally increases propor-tionally to the deformation magnitude. Multiresolution tech-niques, where coarse to fi ne strategies are used, have beenreported to reduce the computational demands [16]. Even inthat case, the task of estimating a transformation which pro-vides both satisfactory and topology preserving mapping isnontrivial. Inducing further restrictions to the above models,such as setting the transformation to be diffeomorphic, seemsto result in topology preserving transformations [14]. How-ever, in this case, the need to track the discrete Jacobian of the transformation can be an important drawback. Similar inspirit approaches were also proposed in [12, 1, 17].In the present study, we propose a method motivated bythe constrained optical fl ow formulation for image registra-tion [13]. At fi rst, we impose a smoothness constraint on thedeformation fi eld. Also, the deformation fi eld is assumed tobe a random variable following a Gaussian pdf and we recurto the maximum a posteriori (MAP) formulation in order toevaluate the model parameters. Moreover, we assume a more fl exible model by considering different pdf parameters foreach image location, thus, making the model spatially adap-tive . In the remainder of this paper, we present our spatiallyadative model in section 2, numerical results and discussionon the advantages and of the proposed model with respect toboth the constrained optical fl ow and the simple non adaptivemodel are presented in section 3. 2. REGISTRATION METHOD2.1 Problem formulation Let A ( s ) = A (  x  ,  y ) be the target image and B ( s ) = B (  x  ,  y ) the source image to be aligned with A . We assume that weare dealing with a single-modal image registration problem.Our objective is to estimate a displacement fi eld u , which  minimizes the energy function E  between the two images: E  ( u ) =   Ω   B ( s + u ( s )) −  A ( s )  2 d s (1)where Ω is the bounded domain de fi ned by the images and s = (  x  ,  y ) is the position vector for a given pixel. Under theassumption that the target image is a geometrically deformedversionofthesourceimagewithi.i.d. (independentandiden-tically distributed) Gaussian noise added to each pixel, mini-mization of (1) yields to the optimal solution in the maximumlikelihood  sense.Vectorizing the two images, we get two N  -dimensionaldiscrete signals a ( i ) and b ( i ) where i ∈  I  ⊂ Z  N  , and I  is an N  -dimensional discrete interval representing the set of all pixelcoordinates in the image. The deformation fi eld is also vec-torized in the form u = ( u x u y ) T , where u x , u y ∈ R N . Forsimplicity from now on u ( s ) = u .First, we summarize the optical fl ow model for imageregistration. In this model we start by retaining the fi rst-orderterms of the Taylor expansion of the intensity function in thesource image. b ( s + u ) = b ( s )+ u T  ∇ b ( s )+ ε  ( s ) , (2)where ε  ( s ) the residual of the Taylor expansion and is as-sumed small. We assume b ( s + u ) − a ( s )  0 , (3)or b ( s )+ u T  ∇ b ( s ) − a ( s ) = − ε  ( s ) , (4)or d  ( s )+ u T  ∇ b ( s ) = − ε  ( s ) , (5)where d  ( s ) = b ( s ) − a ( s ) , is the intensity difference betweensource and target image. Using this approximation we obtaina displacement fi eld to be applied in the source image. This fi eld implies a displacement in the direction of  ∇ b ( s ) and itsorientation is + ∇ b ( s ) if  b ( s ) < a ( s ) and − ∇ b ( s ) otherwise.No displacement occurs wherever the two intensities match.The main disadvantage of this model is that since thereare no constraints in the deformation fi eld, equation (5) doesnot have a unique solution. In other words, from N  observa-tions stacked in vector d , 2  N  parameters stacked in vectors u x , u y have to be estimated. This is an ill-posed problemthat requires regularization [20]. In this study, we apply theLaplacian operator on the displacement fi eld and then usethe Euclidian norm of the product for regularization. In sub-section 2.2 we present a generalization of the above modelwhich uses concepts from stochastic estimation theory.Let g ( s ) = ∇  B ( s ) and Q = ∇ 2 be the Laplacian oper-ator. Applying the optical fl ow model using, in the sensementioned above, the Laplacian operator as a constraint forTikhonov regularization, we come up with the followingminimization problem. u = argmin u [  d + Gu  2 + α   Qu  2 ] (6)where α  is a weight factor, d = ( d  1 d  2 ... d   N  ) T  (7)is a vector containing the temporal image differences and G = diag { g  x  1 , g  x  2 , ..., g  xN  , g  y 1 , g  y 2 , ..., g  yN  } (8)is a matrix containing the image gradient with respect to thehorizontal and vertical directions. Taking the above formu-lation into account and minimizing equation (6) with respectto u yields: ( GG T  + α  Q T  Q ) u = − Gd . (9)Due to the high dimensionality of the involved matrices,the above linear system may be approximately solved us-ing the Conjugate Gradients Squared method (CGS). Oncewe obtain the deformation fi eld u , we compute the deformedsource image b ∗ through cubic interpolation. The algorithmiterates over time, producing an increasing deformation fi eldafter each iteration. At the end of each iteration the deformedimage b ∗ is resampled and used as the source image for thenext iteration. 2.2 Spatially adaptive model It is well known that Tikhonov regularization has also a sto-chastic interpretation using MAP estimation and introducingan appropriate prior [10]. In this framework, we assign aGaussian prior probability distribution p ( u ) to the elementsof the deformation fi eld. In our case, the observed data set isthe vector with elements the differences in intensity for eachpixel d = ( d  1 d  2 ... d   N  ) T  . The likelihood function p ( d | u ) isrelated to the posterior probability p ( u | d ) through Bayes’theorem:  p ( u | d ) ∝ p ( d | u )  p ( u ) (10)We can now determine u by fi nding the most probable valueof  u given the observations by MAP estimation.The Gaussian pdf for a N  -dimensional vector x of con-tinuous variables is expressed by N    ( x ; μ  , Σ ) = 1 ( 2 π  ) N2 | Σ | 12 exp − 12 ( x − μ  ) T Σ − 1 ( x − μ  ) (11)where N  is the dimensionality of the vector and μ  , Σ are themean vector and covariance matrix respectively. The inverseof the covariance matrix is known as the precision matrix.The prior pdf of  u is:  p ( u ) = N    ( u ; 0 , ( α  Q T Q ) − 1 ) (12)where α  = ( α   x  α   y ) T  , whereas the conditional pdf:  p ( d | u ) = N    ( d ; − Gu , γ  − 1 I ) . (13)Parameters α  and γ  , which control the precision and, con-sequently, the distribution of model parameters, are called hyperparameters . From this point we can proceed in twoways.The simplest is to assume that these hyperparameters are spatially constant  . This in effect implies that the statistics of both the residual ε  ( s ) in (2) and Qu are Gaussian indepen-dent identically distributed. This is clearly an oversimpli fi -cation of reality. The residuals at areas of large deformationswill be larger since the linearization used in (2) would be lessaccurate than in areas of small deformations. Furthermore, in  areas where we have abrupt changes of  u the values of  Qu will be larger than in areas of smooth changes of  u .However, in this case it is very simple to obtain the MAPestimates. Indeed by minimizing the negative log-posteriordistribution: − ln  p ( u | d ) (14)with respect to u , we obtain the solution ( γ  GG T + α  Q T Q ) u = − γ  Gd (15)The above linear system is approximately solved iterativelyusing the CGS method. This time though, apart from thesource image, we also update the values for the parameters γ  , α  , through the following equations, obtained by minimiz-ing again (14) with respect to γ  , α   x  and α   y respectively: γ  = N   d + Gu  2 , (16) α   x  = N  − 1  Qu x  2 , (17)and α   y = N  − 1  Qu y  2 . (18)In order to overcome the above mentioned dif  fi cultiesof the spatially invariant model we impose spatial adaptiv-ity. In other words, the parameters of our model are spa-tially varying taking different values for each pixel loca-tion and it is possible to express them by the following vec-tors: α  x = ( α  x1 α  x2 ... α  xn ) T , α  y = ( α  y1 α  y2 ... α  yn ) T and γ  = ( γ  1 γ  2 ... γ  n ) T . Equations (12), (13) then become:  p ( u ) = N    ( u ; 0 , ( Q T AQ ) − 1 ) (19)and  p ( d | u ) = N    ( d ; − Gu , Γ − 1 ) (20)where A = diag { α   x  1 , α   x  2 , ..., α   xN  , α   y 1 , α   y 2 , ..., α   yN  } (21)and Γ = diag { γ  1 , γ  2 , ..., γ   N  , γ  1 , γ  2 , ..., γ   N  } . (22)In such case, the proposed model will have 3  N  parametersto be estimated from N  observations. Clearly such modelalthough it has many advantages and capture the spatiallyvarying nature of the residual and Qu also over fi ts the dataand can not generalize [4]. For this purpose, we follow theBayesian paradigm and add one more layer to our model.In other words, we assume ( γ  i , α   xi , α   yi ) i = 1 ,...  N  to be ran-dom variables and to follow a Gamma pdf parameterized by l , m ,  p , q  x  , q  y :  p ( γ  i ) ∝ γ  l − 22 i exp − m ( l − 2 ) γ  i (23)  p ( α   xi ) ∝ α   p − 22  xi exp − q  x  (  p − 2 ) α   xi (24)and  p ( α   yi ) ∝ α   p − 22  yi exp − q  y (  p − 2 ) α   yi . (25)We choose the Gamma distribution for the hyperparametersbecause it is the conjugate prior for the precision of a univari-ate Gaussian [4]. Furthermore, we chose this parametrizationfor the Gamma pdf because of its intuitive interpretation [6].Minimizing (14) this time leads to the following linear sys-tem of equations: ( GΓG T + Q T AQ ) u = − GΓd , (26)with the following update scheme for the spatially variant hyperparameters : γ  i = l − 1 ( d + Gu ) 2 i + 2 m ( l − 2 ) (27) α   xi = p − 1 ( Qu x ) 2 i + 2 q  x  (  p − 2 ) (28)and α   yi = p − 1 ( Qu y ) 2 i + 2 q  y (  p − 2 ) . (29)The role of the Gamma pdf becomes apparent by observingthe above equations. For example when the deformation fi eldbecomes smooth the fi rst term of the denominator of (28) and(29) becomes zero. Thus, without the Gamma pdf ( q  x  , q  y = 0and p = 2) the estimates for α   xi and α   yi become unstable.Parameters l and p may take values within the interval ( 2 , + ∞ ) . Theirchoiceaffectsourmodelinthefollowingway.Consider for example equation (28): when p takes very closeto 2, the second term in the denominator ensures that α   xi de-pends on only on ( Qu x ) 2 i and thus the Gamma hyperprioris non-informative since the estimates of  α   xi and α   yi dependonly on the data. On the other hand, if we assign a largevalues to p , the second term in the denominator of (28) and(29) dominates. Then, the estimates of  α   xi and α   yi do notdepend on the data and have the same value for all spatiallocations. As a result our model degenerates to a spatiallyinvariant model. Parameters m , q  x  , q  y are extracted from thedata, since it turns out that they are proportional to the vari-ances of  ( d + Gu ) , ( Qu x ) and ( Qu y ) respectively. 3. EXPERIMENTAL RESULTS Intensity similarity measures between the deformed sourceimage and the target image are widely used criteria whileevaluating a deformable registration algorithm. The maindisadvantage of these criteria is that they do not provide anyinformation about the topology preservation during the trans-formation. In this study, we used an alternative evaluationscheme, which combines the quantitative measurement of the intensity differences, with the qualitative estimation of the topological characteristics of the transformation.A deformation fi eld u is topologically coherent when itis ”smooth”. In that case the deformation vector for neigh-boring pixels must be similar. As a result, the spatial gradientof the fi eld should not take large values. Based on the above,we de fi ne the smoothness of the deformation fi eld as: S  = 1  ∇ u  (30)We also de fi ne an error  in the intensity matching between thedeformed source image and the target image as: E  = 1  N   N  ∑ i = 1 | b ∗ i − a i | (31)  where we remind that b ∗ is the deformed source image.Given (30) and (31) the fraction E S  will clearly take smallervalues, as the deformation either introduces smaller error orbecomes smoother.In this experiment, we use a 2D image, which is a brainMRI slice as the target image. The target image is deformedusing a known non-linear formula. We produced two sourceimages, one with relatively small deformation and one witha relatively large deformation. We registered the source im-ages to the target image using the stationary and the non sta-tionary algorithms described in section 2.At fi rst, we perform one iteration of the constrained op-tical fl ow equation (9), selecting one value for the constantweight α  . The resulting deformation fi eld is then used to ini-tialize the hyperparameters (27)-(29) for the non-stationaryequation (26), which we let it iterate until convergence. Theresults of this scheme are shown in fi gures 1 and 2, for thesmall and the large deformation respectively. We observethat there is a signi fi cant improvement in the intensity differ-ences for both deformations. For the large deformation, wenotice that the algorithm fails to fully align the two images inregions where the initial error is too large. This is caused bytheregularizationterm, whichdoesnotallownon-continuousmappingbetweenneighboringpixels. Areductioninthecon-straint weight would produce a result with less error, but nottopologically coherent. For the small deformation, it is hardto obtain a good visualization of the differences between theimages. Nevertheless, there are many non-zero values anda deformation does indeed take place, as it is con fi rmed bythe differences in the fi nal result when we additionally let theconstrained optical fl ow algorithm converge.We also calculate the values of the fraction E S  for bothmethods. The results, forvarious initialvalues of  α  , areillus-trated in fi gure 3, for the small (a) and the large (b) deforma-tion. It becomes clear from the above fi gure that applying thenon-stationary method to the problem provides better results,as far as our evaluation criterion is concerned. It is notablethat independently from the initial, arbitrary, choice of  α  ,spatial variance seems to either reduce the error of the regis-trationorincreasethesmoothnessofthedeformation fi eld, orboth. In that way, the bias introduced by the non-automaticinitialization of the constraint weight factor (9) may be re-duced. Moreover, we observed that the execution time isshorter for the non-stationary method, up to a percentage of 70%. This can prove very crucial during the registration of high-dimensional images, especially in the 3D case. 4. CONCLUSION In this paper, we presented a method for non rigid imageregistration based on a spatially variant model. The modelimposes smoothness on the deformation fi eld which is as-sumed to be a random variable conditioned on the obser-vations (inter-image differences). The obtained registrationerrors are inferior to the errors provided by the simple sta-tionary model. Future directions of this study are the exten-sion of the method to 3D MRI registration where the modelnon stationarity will provide a more clear advantage. Also,comparison with other state of the art methods [3, 5], thoughdif  fi cult due to absence of golden standards and ground truth,is a key element and an issue of ongoing research. REFERENCES [1] S. Allassonniere, Y. Amit, and A. Trouv´e. Towardsa coherent statistical framework for dense deformabletemplate estimation. Journal of the Royal StatisticalSociety (B) , 69(1):3–29, 2007.[2] Y. Amit. A nonlinear variational problem for im-age matching. SIAM Journal of Scienti  fi c Computing ,15:207–224, 1994.[3] M. F. Beg, M. I. Miller, L. Younes, and A. Trouv´e.Computing metrics via geodesics on fl ows of diffeo-morphisms. International Journal of Computer Vision ,61(2):139–157, 2005.[4] C.M.Bishop. PatternRecognitionandMachineLearn-ing . Springer Science and Business Media, LLC, NewYork, USA, 2006.[5] Y. Cao, M. I. Miller, R. L. Winslow, and L. Younes.Large deformation diffeomorphic metric mapping of vector fi elds. IEEE Transactions on Medical Imaging ,24(9):1216–1230, 2005.[6] J. Chandas, N. P. Galatsanos, and A. Likas. Bayesianrestoration using a new hierarchical directional contin-uous edge image prior. IEEE Transactions on ImageProcessing , 15(10):2987–2997, 2006.[7] G. E. Christensen, M. I. Miller, M. W. Vannier, andU. Grenander. Individualizing neuro-anatomical atlasesusing a massively parallel computer. IEEE Computer  ,pages 32–38, January 1996.[8] W. Crum, T. Hartkens, and D. Hill. Non-rigid imageregistration: theory and practice. British Journal of Ra-diology , 77:S140–S153, 2004.[9] C. Davatzikos and R. N. Bryan. Using a deformablesurface model to obtain a shape representation of the cortex. IEEE Transactions on Medical Imaging ,15:785–795, 1996.[10] G. Demoment. Image reconstruction and restora-tion: overview of common estimation structures andproblems. IEEE Transactions on Signal Processing ,37:2024 – 2036, 1989.[11] M. Fornefett, K. Rohr, and H. S. Steihl. Radial ba-sis functions with compact support for elastic registra-tion of medical images. Image and Vision Computing ,19:87–96, 2001.[12] C.A. Glasbey and K.V. Mardia. A penalized likelihoodapproach to image warping. Journal of the Royal Sta-tistical Society (B) , 63(3):465–514, 2001.[13] W. Lu, M. L. Chen, G. H. Olivera, K. J. Ruchala, andT. R. Mackie. Fast free-form deformable registrationvia calculus of variations. Physics in Medicine and Bi-ology , 49:3067–3087, 2004.[14] S. Marsland and C. J. Twining. Constructing diffeo-morphic representations for the groupwise analysis of nonrigid registrations of medical images. IEEE Trans-actions on Medical Imaging , 8:1006–1020, 2004.[15] J. Modersitzki. Numerical methods for image registra-tion . Oxford University Press, 2003.[16] O. Musse, F. Heitz, and J. P. Armspach. Topology pre-serving deformable image matching using constrainedhierarchical parametric models. IEEE Transactions on  (a) (b) (c) (d) (e)Figure 1: Registration example with relatively small image deformations. (a) Reference image, (b) image to be registered tothe reference image, (c) difference between the unregistered images, (d) registered image, (e) difference between the registeredimage and the reference image.(a) (b) (c) (d) (e)Figure 2: Registration example with relatively large image deformations. (a) Reference image, (b) image to be registered tothe reference image, (c) difference between the unregistered images, (d) registered image, (e) difference between the registeredimage and the reference image. 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 α    E  r  r  o  r   /   S  m  o  o   t   h  n  e  s  s Small deformation StationaryNon − Stationary10 0 10 1 10 2 10 3 10 4 10 2 10 3 10 4 α    E  r  r  o  r   /   S  m  o  o   t   h  n  e  s  s Large deformation StationaryNon − Stationary (a) (b)Figure 3: Registration error over smoothness curves for various values of the parameter α  for the stationary and the nonstationary registration models. (a) Small deformation ( fi g. 1), (b) large deformation ( fi g. 2).  Image Processing , 10:1081–1093, 2001.[17] F. Richard, A. Samson, and C. Cu´enod. A SAEM al-gorithm for the estimation of template and deformationparameters in medical image sequences. Technical re-port, Universit´e Paris Descartes, MAP-5, 2007.[18] D. Rueckert, L. I. Sonoda, C. Hayes, D. L. G. Hill,M. O. Leach, and D. J. Hawkes. Nonrigid registrationusing free-form deformations: Application to breastMR images. IEEE Transactions on Medical Imaging ,18:712–721, 1999.[19] J. P. Thirion. Image matching as a diffusion process:An analogy with Maxwells demons. Medical Image Analysis , 2:243–260, 1998.[20] A. Tikhonov and V. Arsenin. Solution of Ill-posed Problems . Winston & Sons, Washington, USA, 1977.[21] Y. T. Wu, T. Kanade, C. C. Li, and J. Cohn. Image reg-istration using wavelet-based motion model. Interna-tional Journal of Computer Vision , 38:129–152, 2000.
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