Landmark/Image-based Deformable Registration of Gene Expression Data

Uday Kurkure1, Yen H. Le1, Nikos Paragios2, James P. Carson3, Tao Ju4, and Ioannis A. Kakadiaris11University of Houston, Houston, TX, USA 2Ecole Centrale Paris, France

3Pacific Northwest National Laboratory, Richland, WA, USA4Washington University, St. Louis, MO, USA

http://cbl.uh.edu

Abstract

Analysis of gene expression patterns in brain imagesobtained from high-throughput in situ hybridization re-quires accurate and consistent annotations of anatomicalregions/subregions. Such annotations are obtained by map-ping an anatomical atlas onto the gene expression im-ages through intensity- and/or landmark-based registrationmethods or deformable model-based segmentation meth-ods. Due to the complex appearance of the gene expres-sion images, these approaches require a pre-processing stepto determine landmark correspondences in order to incor-porate landmark-based geometric constraints. In this pa-per, we propose a novel method for landmark-constrained,intensity-based registration without determining landmarkcorrespondences a priori. The proposed method performsdense image registration and identifies the landmark cor-respondences, simultaneously, using a single higher-orderMarkov Random Field model. In addition, a machine learn-ing technique is used to improve the discriminating proper-ties of local descriptors for landmark matching by project-ing them in a Hamming space of lower dimension. We qual-itatively show that our method achieves promising resultsand also compares well, quantitatively, with the expert’s an-notations, outperforming previous methods.

1. IntroductionWith the developments in high-throughput in situ hy-

bridization (HITISH) [3], gene expression patterns can beobtained at cellular resolution to explore the functional re-lationship between various genes and disease mechanisms.The gene expression images are generated with differentgene probes that highlight different cells expressing genesat different levels. Determining the correspondence map-ping in these images is necessary for any meaningful in-terpretation of multiple gene expression profiles within thecells. The images can be then organized into a database andqueried for similarities in expression patterns to find poten-

Figure 1. Example images depicting the complex appearance andshape patterns in gene expression images. Each depicted gene ex-pression image is generated with a different gene probe that high-lights different cells expressing gene at different levels.

tial interactive relationships between different genes in thesame anatomical subregion. However, the expression im-ages exhibit significant variations in appearance and shape,and do not have significant anatomical information (Fig. 1).In addition, the acquisition and sectioning process may in-troduce multiple artifacts related to smearing and missingparts. Nevertheless, it is required to determine the corre-spondences of anatomical regions/subregions (Fig. 2) in theexpression images to the annotated anatomical atlas to gainknowledge about which genes are expressed with a par-ticular expression pattern [2] in a specific anatomical re-gion/subregion.

Existing methods for gene expression mapping can bebroadly classified into approaches based on image registra-tion and deformable model fitting. Due to the complexityof the appearance of the gene expression images, intensity-based registration approaches treat this problem as a multi-modal registration problem [9]. However, due to the lack ofinformation in many parts of the images, other approacheshave additionally incorporated geometric constraints usingsigned-distance maps [17] and anatomical landmarks [7, 6].

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Figure 2. Overlay of the boundary contours depicting the anatomi-cal regions annotated manually on the reference image (from [1]).

Though signed-distance maps are effective in constrainingthe solution at the boundaries, they lack information of in-ternal anatomical regions. Bello et al. [1] proposed an at-las deformation method, where a deformable mesh was fit-ted using a statistical shape model, anatomical landmarks,and region boundaries in various stages, successively. Thecorresponding landmarks were detected prior to the fittingprocess via a classification method by computing featuresfrom regions arranged manually for each individual land-mark. Landmarks provide anatomy specific constraints andguide the deformation process in regions with uneven infor-mation. However, most of the landmark-based approachesrequire to determine landmark correspondences a priori ei-ther interactively or by using an automated approach.

In this paper, we present a novel method to determine thecorrespondences for automated region annotation in geneexpression images using a landmark-constrained registra-tion approach based on the Markov Random Field (MRF)framework. The key difference between our approach andprevious approaches is that it solves the landmark corre-spondence and iconic registration problems simultaneouslywhile being rigid transform invariant. Our method does notrequire to determine the landmark correspondences a pri-ori. It only needs to be provided with a few landmark can-didates among which there exists at least one desired candi-date. Our approach is based on an optimization step whereboth landmark correspondences and iconic registration areoptimized through inter-connected variables. We constructa single graphical model that incorporates intensity-basedimage registration in one layer and landmark matching inanother layer. Both layers are connected through a neigh-borhood system to impose geometric constraints on eachother. The rigid transform invariant formulation is obtainedby imposing higher order constraints in the landmark layerthrough a prior geometric model that is learned from the rel-ative statistics of higher order geometry. The landmark can-didates are determined based on their local descriptors’ sim-ilarity with the landmarks in a reference image. However,

owing to the variability within the gene expression images,it is difficult to define appropriate descriptors. Toward this,Hamming embeddings for the descriptors are learned usingsimilarity sensitive hashing [10] for efficient matching.

Our method partially shares the philosophy in terms ofthe interaction between the landmark and the deformationspace with that of Sotiras et al. [13]. However, we in-corporate robust higher order constraints through a learnedprior model instead of simple pair-wise regularization con-straints. Second, our approach is formulated to be transla-tion, rotation, and scale invariant, and therefore, removesthe need of global registration, unlike [13]. Finally, land-mark candidate selection is context specific and determinedthrough similarity sensitive hashing, which enhances thedistinctiveness of the landmark descriptors specific to a par-ticular problem.

The remainder of the paper is structured as follows. InSection 2 we describe the methods for the landmark candi-dates detection and the construction of the graphical modelto solve the correspondence problem. Experiments on thegene expression images and the validation results are pre-sented in Section 3, and Section 4 concludes the paper.

2. Methods

The problem of annotation of anatomical regions andsubregions of gene expression images is formulated as acorrespondence matching problem with respect to an an-notated reference image. In this section, we describe ourformulation of Markov Random Field model that simulta-neously optimizes for landmark correspondences and iconicregistration in a single two-layer graphical model. First,we describe our approach to generate landmark candidatesfrom the gene expression images. Then, we present the de-tails of the construction of each layer of the model and theirinter-connections.

2.1. Landmarks and Descriptors

We define landmarks as points in the gene expression im-ages that are locally salient in terms of appearance or shapecharacteristics irrespective of the gene being expressed.Two types of landmarks were chosen through visual inspec-tion [1]: appearance-based landmarks and boundary-basedlandmarks. Figure 3 depicts the chosen landmarks on aNissl-stained reference image.Appearance-based landmarks: We use local image de-scriptors computed from gradient orientation histograms torepresent the appearance-based landmark features [14]. Inthis representation, commonly known as DAISY, the po-lar Gaussian pooling approach is used to construct the his-tograms. The measure of (dis)similarity between any twopoints is defined as the distance between the distributions ofthe oriented gradients in their descriptors. It has been shown

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Figure 3. Manually selected landmarks in tissue sections that arelocally salient in terms of appearance or shape characteristics. Thenumbers represent the IDs assigned to each landmark location(from [1]).

to outperform other discriminative local image descriptors[16]. In addition, it can be computed very efficiently.

Figure 4(a) depicts sample patches from the gene expres-sion images for landmark ‘0’, whereas Fig. 4(b) depictsthe distance map computed with reference to the landmarkpoint in the reference image. The best candidate for thislandmark can be selected as the point that corresponds to theminimum distance in the distance map. However, it can beobserved from Fig. 4(b) that it may result in false matches,and may not identify the real landmark point in the imagebecause of complex appearance information, which variesfrom image to image.

We employ a machine learning approach to constructefficient similarity measures to reduce both types of er-rors to enhance the (dis)similarity of the descriptor between(dis)similar points. An ensemble classifier based on boost-ing technique is trained on pairs of point descriptors in-stead of single point descriptors. The paired learning isperformed to select similarity-relevant features, and to bi-narize them such that the Hamming distance between thedescriptors of a pair of points is small, if they correspond tothe same landmark.

Specifically, we randomly selected eight gene expres-sion images in addition to the reference image for learn-ing the binary embedding model. We constructed sets ofpositive pairs and negative pairs for each landmark from alocal patch centered at the landmark points. We followeda boosting-based approach [10] to learn the transformationof a high dimensional feature space into a reduced space ofbinary features. The n-dimensional binary Hamming em-bedding is represented as ξ(x) = (ξ1(x), ..., ξn(x)), whereeach dimension is computed by a binary function parame-terized by a projection function ϕ : X → R, or

ξi(x) =

0, if ϕi(x) ≤ 01, otherwise

.

Each such function defines a weak binary classifier on pairs

(a) (b) (c)

Figure 4. Depiction of a single candidate selection for landmark‘0’ in a search window: (a) gene expression image (left column),(b) distance maps using daisy descriptor (middle column), and (c)distance maps using binary embedding (right column). The dis-tance maps are computed by comparing features of each pixel inthe search window with the features of the landmark in the refer-ence image. The symbol ‘o’ depicts the expected location and ‘+’depicts the obtained location as the minima of the distance map.

of points,

hi(x, y) =

+1, if ξi(x) = ξi(y)−1, otherwise

,

or simply,

hi(x, y) = sign(ϕi(x)) · sign(ϕi(y)).

Thus, the Hamming metric between the embeddings ξ(x)and ξ(y) of a pair of points x and y can be computed as

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weighted summation of output of the binary classifiers,

d(ξ(x), ξ(y)) =1

2

n∑i=1

αi −1

2

n∑i=1

αihi(x, y), (1)

where αi > 0 is the weight for the ith dimension. In eachiteration of the boosting, we select a Hamming embeddingξi for which the binary weak classifier hi maximizes theweighted correlation,

ri =∑

wikskhi(xk, yk),

of labels sk for all pairs. Note that the Hamming embeddingξi is a function of the projection map ϕi which is repre-sented by a single feature from the descriptors and a thresh-old ai for that feature. Figure 4(c) depicts the Hammingdistance map computed using Eq. (1) with reference to thelandmark ‘0’ in the reference image (the expected locationand the minimum of the distance map are depicted as ‘o’and ‘+’, respectively). It can be observed that for the de-picted images, the candidates selected using the Hammingembeddings are closer to the expected locations than thecandidates selected using the original features. Additionalcandidates can be selected for each landmark by finding ad-ditional local minima points in the Hamming distance maps.Boundary-based landmarks: The curvature value of theboundary contour at the landmark points is used to repre-sent the boundary-based landmarks. First, the gene expres-sion image is segmented by applying histogram threshold-ing, flood-filling and morphological operations. Then, theboundary of the segmented brain image is obtained usinga border tracing algorithm [12]. The resulting boundary isfurther smoothed with a moving average filter. The curva-ture κi at each point on the smoothed boundary is computedas:

κi =xiyi − yixi

(x2i + y2

i )3/2,

where, (xi, yi) and (xi, yi) are the first order and secondorder derivatives, respectively. The candidate points fora boundary landmark in a given image are determined asthe points with maximum convex or concave curvature (de-pending on the landmark curvature type in the reference im-age) in the local search window.

2.2. Iconic Registration

Consider an image I : Ω→ R to be registered to anotherimage J . Using energy minimization principles, the spatialcorrespondences between the two images can be obtainedby recovering an optimal transformation T (x) from:

E1(T ) =

∫Ω

ψ1(J(x), I(T (x))dx,

where ψ1(.) is a similarity function that defines the relation-ship between the intensity patterns in the two images. For

nonlinear registration, the transformation T (x) is defined interms of the deformation field D(x), or T (x) = x +D(x).To impose smoothness on the deformation field, a regular-ization function is included in the energy function as:

E2(T ) =

∫Ω

ψ2(∇T (x))dx,

where ψ2(.) is a smoothness function. Considering a de-formation grid of control points, C, super-imposed on theimage, the deformation field, D(x), at any point in the im-age can be interpolated from the deformation vectors of thecontrol points:

D(x) =∑c∈C

η(|x− c|)dc,

where η(.) is a weighting function describing the contribu-tion of the control point c at any point x in the image anddc is the displacement vector of the control point c. Theappearance-based energy function thus can be redefined as:

E1(T ) =∑c∈C

∫Ω

η(x− c)ψ1(J(x), I(T (x))dx,

where η(.) determines the influence of a point x on the con-trol point c.

Next, we describe the construction of the imageregistration-based graph layer for the MRF model follow-ing the formulation of Glocker et al. [4]. Consider a graph,Ga (here a refers to the registration layer), whose nodesare the Ma control points C from the registration grid. LetLa = la1 , ... , laHa be a discrete set of Ha labels for thatlayer. These labels correspond to a quantized deformationspace D = d1, ... ,dHa. For a particular node i, a la-bel assignment lak corresponds to the displacement of thenode by dlak

. Thus, the goal of the registration is to find amapping F a : Ga → La that is optimal given some cri-terion to recover the transformation T (x) defined in termsof the deformation field U(x), or T (x) = x + U(x). TheMRF provides an elegant and efficient mathematical frame-work for solving such discrete labeling problems. Any pos-sible assignment fa = fa1 , ... , faMa of labels to the ran-dom variables is called a configuration of F a, and is essen-tially a realization of the field. Note that every configurationfa defines a labeling and Fa denotes the set of all possi-ble configurations. We also define a neighborhood systemN = Ni | ∀i ∈ C for the set of control point nodes C,where Ni is the set of all neighbors of the node i ∈ C. TheMRF energy function for image registration is defined as:

Ea(fa) =∑i∈C

V ai (fai ) +

∑i∈C

∑j∈Ni

V aij(f

ai , f

aj ),

where V ai (·) and V a

ij(·, ·) are the first- and second-orderclique potential functions representing the data and the

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smoothness terms, respectively. The unary or the first-orderpotentials are defined as:

V ai (fai ) =

∫Ω

η(x−c)ψ1(J(x), I(x+U t−1(x)+dfai

))dx,

where ψ1(·) is an image similarity measure. Though thegene expression images belong to the same modality, owingto the variations in the observed information caused by thedifferent types of probing, we can consider that they are ac-quired from different modalities. Such a consideration leadsus to use the popular multi-modality similarity function de-fined in terms of normalized mutual information [8, 15]. Inthis work, we computed the normalized mutual informationusing a simple histogram-based method with 128 bins. Theregularization function for smoothness in the label domainis defined as a function of distance between the deformationvectors of neighboring control points, or

V aij(f

ai , f

aj ) = λaexp

(−‖dfa

i− dfa

j‖2

2(σa)2

),

where λa is a weighting parameter balancing the effects ofthe appearance-based similarity and the smoothness of thedeformation field.

2.3. Landmark Correspondences

Landmarks are generally used in the registration processto constrain the solution at specific known locations in theimage. The landmark-based constraints are imposed on thedeformation field by adding an additional energy term cor-responding to the sum of distances between the correspond-ing landmarks. Note that in such methods the location of thelandmarks in the reference image and the given image areassumed to be known a priori along with their correspon-dences. If the landmark correspondences are not known apriori, a pre-processing step is performed to select the bestlandmark candidate in the given image corresponding to thelandmark in the reference image.

The problem of selecting a single best candidate for eachlandmark can be formulated as a discrete labeling problem.Thus, for the landmark-based graph layer, consider a graphGb (b refers to the landmark layer) whose nodes are the M b

landmark points Q = q1, ... ,qMb in the reference im-age, and the discrete labels Lb correspond to the Hb can-didate points pi

1, ... ,piHb in the given image for each

landmark qi. A label assignment f bi = lbk to a landmarkqi corresponds to the selection of the candidate pi

k as thebest candidate for that landmark. The MRF energy functionfor landmark matching is defined as:

Eb(f b) =∑i∈Q

V bi (f bi ) +

∑c∈C

V bc (f bc ),

where V bi (·) and V b

c (·) are the unary and higher order po-tential functions, respectively, for the landmark graph layer

and f bc is the configuration of the triplets forming a clique cfrom a set of cliques C.

The unary potential for the landmarks is defined by ψ3(.)which measures the candidacy strengths of the landmarkcandidates based on the Hamming metric or the curvaturedepending on the type of the landmark:

V bi (f bi ) = ψ3

(pifbi

).

The higher order potential encodes the penalty functionof assigning a triplet of labels to a triplet of connectednodes. The interdependencies between the locations of thevarious landmark triplets can be modeled using the proba-bility distribution of the relative lengths of triangles formedbetween the landmarks. Specifically, given a triplet of land-mark points i, j, and k belonging to a clique c, the shape ofthe triangle formed by them can be defined by the relativelengths of any two sides of the triangle. The relative lengthfor a particular side is defined by the ratio of the length ofthat side over the total length of all three sides. Thus, werepresent a triplet forming the clique c by a two element de-scriptor rc = (rij , rjk), where rst is the relative length ofthe side defined by the points s and t.

Using such representation, we can capture spatial vari-ability that is translation, rotation, and scale invariant bylearning the distributions for each rc from a set of trainingimages. Thus, we learn the distribution, ρc(rc) using a mul-tivariate Gaussian distribution,N (rc|µc,Σc), where µc andΣc are the mean and the covariance matrix learned from thetraining set, respectively. This prior captures the local rela-tionship between the landmark locations and constrains thespace in which a triplet of landmarks can co-exist. This isespecially advantageous when the strength of landmark can-didacy is not strong enough. Thus, the higher order cliquepotential is defined to incorporate this prior knowledge toimpose a global prior cost on the locations of the landmarkcandidates as:

V bc (f bc ) = λb

(1− ρc(rc(f bc ))

),

where λb is a positive weight, rc(fbc ) maps the locations

of the triplet c to the two element descriptor formed fromrelative lengths of the sides.

2.4. Combining Image Registration and LandmarkCorrespondences

In this section, we describe how to connect the two graphlayers such that the control points and the landmark pointsinfluence each other to obtain optimal labeling configura-tion. The MRF framework allows us to add an additionallayer to the graph. For a given landmark point qi in thereference image, selection of a particular candidate pointpik determines the displacement of that point in the given

image. In other words, the point defined as a landmark

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in the reference image is then known to be displaced byuk,i = pi

k − qi in the given image. Thus, the controlpoints that are in the vicinity of the landmark point shouldbe assigned labels that correspond to the displacement thatis most similar to uk,i in terms of magnitude and direction.Similarly, the assignments of the displacement labels to thecontrol points in the vicinity of the landmark points shouldalso influence the selection of the landmark candidate basedon its displacement vector. Such interactions between thelandmark points and the control points can be defined by apair-wise potential function which is a function of the dis-tance between the displacements of the control points andthe landmark points. Thus, we define the pairwise potentialfunction between a landmark point qi and a control pointcj as:

V abij (f bi , f

aj ) = λabφ(qi, cj)exp

(−‖uk,i − dfa

j‖2

2(σab)2

),

where,

φ(qi, cj) = 1− exp

(−‖qi − cj‖2

2(σr)2

),

is a weighing function based on the distance between thelandmark point and the control point. The weighting func-tion φ(·) can also be used to to determine the edges be-tween the two layers of the graph. We define a local neigh-borhood system for the landmarks with respect to the con-trol points to impose geometrical constraints in order topreserve the local deformations. Specifically, a landmarkpoint qi is paired with all the control points for whichNab(qi) = cj |φ(qi, cj) < ε, where ε is a threshold.Based on the threshold ε, we can increase or decrease thenumber of edges between the two layers of the graph. More-over, such neighborhood system avoids undesired regular-ization, especially at the lower grid resolution.

2.5. MRF Optimization

Finally, we need to define a way to optimize the proposedhigher-order MRF energy function. Toward this, we adoptthe general framework proposed by Ishikawa [5]. In thismethod, the optimization is performed by: (i) transform-ing the multi-label MRF problem to a binary-MRF prob-lem, and (ii) transforming the higher-order MRF to a first-order MRF. The optimization of the binary MRF is per-formed using a well-known quadratic pseudo-boolean opti-mizer (QPBO) [11] implemented by Vladimir Kolmogorov.

3. Results and DiscussionWe evaluated our method on the 2D gene expression im-

ages that were acquired as part of gene expression study [2]and they were provided by Bello et al. [1]. These images are

sagittal sections of postnatal day 7 mouse brains at standardsection 9, each revealing the expression of a single gene af-ter in situ hybridization. The images were acquired using alight microscope at 3.3 µm per pixel resolution resulting inapproximately 2400 × 4000 pixels image size. The imageswere scaled down to 25% of the original size for computa-tion purposes.

To capture the local deformations, the control point gridresolution was successively increased at each iteration. Ateach grid resolution, multiple optimization cycles were usedwith successively decreasing maximum displacement rangefor each control point. The maximum displacement rangewas sampled to provide a total of 25 labels or possible dis-placements for each control point. The landmark candidatepoints in a given test image were selected by dividing thelocal search window into 3 × 3 subregions and picking thetwo best candidates from each subregion that have the mini-mum Hamming distance with respect to the landmark pointin the reference image.

To demonstrate the advantage of the dual graph layers forregistration, we compare, qualitatively and quantitatively,the proposed method with iconic registration performedwithout the landmark layer. Figure 5 depicts selected qual-itative results for both methods using checkerboard visual-ization of the registered gene expression images.

Table 1 depicts the mean distance error in pixels for eachlandmark averaged over two cohorts of images - all the 100images (Cohort A) and a subset of 53 images that excludestraining images and images with severe boundary defor-mity or tear (Cohort B). Note that the proposed method per-forms comparable to or better than the iconic registrationmethod. The errors for the boundary landmarks, especiallylandmarks 1 and 2, are higher than those for the appearance-based internal landmarks. This can be attributed to the highvariability in the local shape and position of the anatomicalstructures in the mouse brain.

We also compare the performance of the proposedmethod with the results provided by Bello et al. [1] in termsof landmark distance errors in pixels. Figure 6 depicts thescatter plots of distance errors for selected landmarks com-paring the two methods for Cohort B gene expression im-ages. In the scatter plots, the data points below the diagonalrepresent the images where the proposed method outper-forms the previous method. For few boundary landmarks,our method has slightly higher error in individual imagesbut the overall mean error is still significantly lower than theoverall mean error of the previous method. Since the bound-ary landmarks are described by a weak curvature-based fea-ture, there is possibility of unreliable candidates which mayproduce worse results. Thus the proposed method has alandmark bias which can be put to good use by improv-ing the candidate detection (e.g., by fusing the shape andappearance information for describing the boundary land-

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Figure 5. Checkerboard visualization of the registered images obtained from the image registration without (top) and with (bottom) thelandmark graph layer on selected gene expression images.

REG LM+REGID A B A B

0 9.49 9.60 5.33 5.841 20.73 7.68 16.04 7.002 17.29 15.04 13.70 10.983 10.92 6.83 9.29 6.184 9.44 9.35 6.25 6.485 9.19 7.38 9.52 7.636 12.17 11.11 9.78 8.627 8.80 9.71 5.99 6.348 9.42 6.63 8.29 5.349 5.48 6.19 5.51 6.19

Table 1. Average distance error (in pixels) for each landmark loca-tion compared to the manual annotations. The corresponding lo-cations of the landmarks were obtained using only the registrationlayer (REG) and both registration and landmark layers in tandem(LM+REG). Columns A and B refer to results from cohorts A andB, respectively.

marks). Incorporating measures of confidence on reliabilityof the candidates and suitably modifying the graph struc-ture to accommodate missing landmarks can alleviate theproblem of missing correspondences.

4. Conclusion

In this paper, we have presented a novel method to ob-tain correspondences between gene expression images us-ing a landmark-constrained image registration method. Weformulate the registration and landmark matching problemin a single MRF model as a discrete labeling problem. Ourmethod does not assume the landmark correspondences tobe known prior to registration. The geometric relationships

of the landmarks are coded as higher-order spatial priors im-posing translation, rotation, and scale-invariant constraintsthat are learned from few training images. Furthermore, alandmark specific similarity model is learnt using a boostingapproach enhancing the descriptors’ discriminative proper-ties for landmark candidates selection. Finally, our methodachieves lower errors for correspondence mapping as com-pared to other methods on a challenging dataset of gene ex-pression images.

References[1] M. Bello, T. Ju, J. P. Carson, J. Warren, W. Chiu, and I. A.

Kakadiaris. Learning-based segmentation framework for tis-sue images containing gene expression data. IEEE Transac-tions on Medical Imaging, 26:728–744, 2007. 1090, 1091,1094, 1096

[2] J. P. Carson, T. Ju, M. Bello, C. Thaller, J. Warren, I. A.Kakadiaris, W. Chiu, and G. Eichele. Automated pipelinefor atlas-based annotation of gene expression patterns: Ap-plication to postnatal day 7 mouse brain. Methods, 50:85–95,Aug. 2010. 1089, 1094

[3] J. P. Carson, C. Thaller, and G. Eichele. A transcriptome atlasof the mouse brain at cellular resolution. Current Opinion inNeurobiology, 12(5):562–565, Oct. 2002. 1089

[4] B. Glocker, N. Komodakis, G. Tziritas, N. Navab, andN. Paragios. Dense image registration through MRFs andefficient linear programming. Medical Image Analysis,12(6):731–741, Dec. 2008. 1092

[5] H. Ishikawa. Transformation of general binary MRF mini-mization to the first order case. IEEE Transactions on PatternAnalysis and Machine Intelligence, Mar. 2010. 1094

[6] M. Jagalur, C. Pal, E. Learned-Miller, R. Zoeller, andD. Kulp. Analyzing in situ gene expression in the mousebrain with image registration, feature extraction and blockclustering. BMC Bioinformatics, 21:1–21, 2007. 1089

1095

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6. Comparison of the distance errors in pixels between ourmethod (vertical axis) and [1] (horizontal axis) for: (a) landmark0, (b) landmark 1, (c) landmark 2, (d) landmark 5, (e) landmark 6,(f) landmark 7, (g) landmark 8, and (h) landmark 9.

[7] T. Lin, C. L. Guyader, E.-F. Lee, I. D. Dinov, P. M. Thomp-son, A. W. Toga, and L. A. Vese. Gene to mouse atlas regis-tration using a landmark-based nonlinear elasticity smoother.In Proc. SPIE Medical Imaging: Image Processing, pages 1–16, Lake Buena Vista, FL, Feb. 7–12 2009. 1089

[8] F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, andP. Suetens. Multimodality image registration by maximiza-tion of mutual information. IEEE Transactions on MedicalImaging, 16(2):187–198, April 1997. 1093

[9] L. Ng, S. D. Pathak, C. Kuan, C. Lau, H. Dong, A. Sodt,C. Dang, B. Avants, P. Yushkevich, J. C. Gee, D. Haynor,E. Lein, A. Jones, and M. Hawrylycz. Neuroinformaticsfor genome-wide 3D gene expression mapping in the mouseBrain. IEEE/ACM Transactions on Computational Biologyand Bioinformatics, 4(3):382–393, 2007. 1089

[10] L. Ren, G. Shakhnarovich, J. Hodgins, H. Pfister, and P. Vi-ola. Learning silhouette features for control of human mo-tion. ACM Transactions on Graphics, 24(4):1303–1331,2005. 1090, 1091

[11] C. Rother, V. Kolmogorov, V. Lempitsky, and M. Szummer.Optimizing binary MRFs via extended roof duality. In Proc.IEEE Conference on Computer Vision and Pattern Recogni-tion, pages 1–8, Minneapolis, MN, June 17-22 2007. 1094

[12] M. Sonka, V. Hlavac, and R. Boyle. Image processing, anal-ysis and machine vision. Brooks/Cole Publishing Company,Pacific Grove, USA, 2nd edition, 1999. 1092

[13] A. Sotiras, Y. Ou, B. Glocker, C. Davatzikos, and N. Para-gios. Simultaneous geometric-iconic registration. In Proc.Medical Image Computing and Computer-Assisted Interven-tion, pages 676–683, Beijing, China, 2010. 1090

[14] E. Tola, V. Lepetit, and P. Fua. DAISY: An efficientdense descriptor applied to wide baseline stereo. IEEETransactions on Pattern Analysis and Machine Intelligence,32(5):815–830, May 2010. 1090

[15] P. A. Viola and W. M. Wells III. Alignment by maximizationof mutual information. International Journal of ComputerVision, 24(2):137–154, Sep. 1997. 1093

[16] S. Winder and M. Brown. Learning local image descriptors.In Proc. IEEE Conference on Computer Vision and PatternRecognition, pages 1–8, Minneapolis, MN, Jun. 18-23 2007.1091

[17] Z. Yi and S. Soatto. Correspondence transfer for the regis-tration of multimodal images. In Proc. IEEE InternationalConference on Computer Vision, pages 1–8, Rio de Janeiro,Brazil, Oct. 14 - 21 2007. 1089

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