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Mutual Information-based 3D Surface Matching with Applications to Face Recognition and Brain Mapping Yalin Wang Ming-Chang Chiang Paul M. Thompson Mathematics Department Laboratory of Neuro Imaging Laboratory of Neuro Imaging UCLA UCLA School of Medicine UCLA School of Medicine ylwang@math.ucla.edu mcchiang@ucla.edu thompson@loni.ucla.edu Abstract 1. Introduction Face recognition and many medical imaging applica- Many face recognition algorithms have been proposed in tions require the computation of dense correspondence vec- the last few decades [24]. Several approaches (e.g., “eigen- tor ﬁelds that match one surface with another. In brain face” methods) encode patterns of geometric and intensity imaging, surface-based registration is useful for tracking variation between faces, and compute metrics to determine brain change, and for creating statistical shape models of the degree of differences between individual faces. Related anatomy. Based on surface correspondences, metrics can work has focused on image matching for tracking facial also be designed to measure differences in facial geome- features in video images. However, all 2D (image-based) try and expressions. To avoid the need for a large set of face recognition systems are somewhat sensitive to facial manually-deﬁned landmarks to constrain these surface cor- expressions and illumination conditions. 3D geometric sur- respondences, we developed an algorithm to automate the face matching can solve these problems and may offer better matching of surface features. It extends the mutual infor- recognition performance. mation method to automatically match general 3D surfaces Surface models are also widely used in medical imaging (including surfaces with a branching topology). We use dif- to assist in data visualization, nonlinear image registration, feomorphic ﬂows to optimally align the Riemann surface and surface-based signal processing or statistics. Surface structures of two surfaces. First, we use holomorphic 1- models are often generated in computational anatomy stud- forms to induce consistent conformal grids on both sur- ies to support computations, e.g., when statistically com- faces. High genus surfaces are mapped to a set of rectan- bining or comparing 3D anatomical models across subjects, gles in the Euclidean plane, and closed genus-zero surfaces or mapping functional imaging parameters onto anatomical are mapped to the sphere. Next, we compute stable geo- surfaces. Often the comparison of data on two anatomi- metric features (mean curvature and conformal factor) and cal surfaces is required, and a correspondence ﬁeld must be pull them back as scalar ﬁelds onto the 2D parameter do- computed to register one surface nonlinearly onto the other. mains. Mutual information is used as a cost functional to Multiple surfaces can be registered nonlinearly to construct drive a ﬂuid ﬂow in the parameter domain that optimally a mean shape for a group of subjects, and deformation map- aligns these surface features. A diffeomorphic surface-to- pings can encode shape variations around the mean. This surface mapping is then recovered that matches surfaces in type of deformable surface registration has been used to de- 3D. Lastly, we present a spectral method that ensures that tect developmental and disease effects on brain structures the grids induced on the target surface remain conformal such as the corpus callosum and basal ganglia [26], the hip- when pulled through the correspondence ﬁeld. Using the pocampus [7], and the cortex [27]. Nonlinear matching of chain rule, we express the gradient of the mutual informa- brain surfaces can also be used to track the progression tion between surfaces in the conformal basis of the source of neurodegenerative disorders such as Alzheimer’s dis- surface. This ﬁnite-dimensional linear space generates all ease [7], to measure brain growth in development [26], and conformal reparameterizations of the surface. Illustrative to reveal directional biases in gyral pattern variability [19]. experiments apply the method to face recognition and to the Surface registration has numerous applications, but a registration of brain structures, such as the hippocampus direct mapping between two 3D surfaces is challenging in 3D MRI scans, a key step in understanding brain shape to compute. Often, higher order correspondences must alterations in Alzheimer’s disease and schizophrenia. be enforced between speciﬁc anatomical points, curved landmarks, or subregions lying within the two surfaces. One common way to achieve this is to ﬁrst map each of 1.1. Previous Work the 3D surfaces to canonical parameter spaces such as a sphere [11, 3] or a planar domain [20]. The surface cor- Some researchers [17, 21] incorporate a 3D model in respondence problem can then be addressed by computing face recognition research. Bronstein et al. [5] propose a a ﬂow in the parameter space of the two surfaces [26, 9], 3D face recognition approach based on geometric invari- which induces a correspondence ﬁeld in 3D. Furthermore, ants to compute isometric deformations. Several variational correspondences may be determined by using a minimum or PDE-based methods have been proposed for match- description length (MDL) principle, based on the compact- ing surfaces. Surfaces may be represented by parametric ness of the covariance of the resulting shape model [10]. meshes [11], level sets, or both representations [20]. An- Anatomically homologous points can then be forced to genent et al. [2, 1] represent the Laplace-Beltrami operator match across a dataset. Recently, Twining et al. [28] pro- as a linear system and implement a ﬁnite element approxi- posed a theoretical framework to unify groupwise image mation for parameterizing brain/colon surfaces via confor- registration and average model construction. In their ap- mal mapping. Gu et al. [13] found a unique conformal map- proach, an information-based model of the correspondences ping between any two genus zero manifolds by minimizing among a group of images becomes a part of the registration the harmonic energy of the map. Gu and Vemuri [12] also process. matched 3D shapes by ﬁrst conformally mapping them to By the Riemann uniformization theorem, all surfaces a canonical domain and aligning their 2D representations can be conformally embedded in a sphere, a plane or a over the class of diffeomorphisms. They demonstrated their hyperbolic space. The resulting embeddings form special algorithm on genus zero closed surfaces. groups. Using holomorphic 1-forms and critical graphs, The mutual information (MI) method [30, 29] measures global conformal parameterization [15] can also be used to the statistical dependence of the voxel intensities between conformally map any high genus surface (i.e., a surface with two images. This measure of agreement can be used to branching topology) to a set of rectangular domains in the tune the parameters of a registration transform such that Euclidean plane. In this paper, we show how to use con- MI is maximal when the two images are optimally aligned. formal parameterizations to assist in the matching of arbi- The MI method has been successful for rigid [31] and non- trary 3D face and anatomical surfaces. Mutual information rigid [22, 25] image registration. Here, we generalize it to is used to drive a diffeomorphic ﬂuid ﬂow that is adjusted match 3D surfaces. For MI to work, a monotonic map- to ﬁnd appropriate surface correspondences in the param- ping in grayscales between images is not required, so im- eter domain. In this study, we chose the mean curvature ages from different modalities can be registered [18]. Her- and the conformal factor of the surfaces as the differetial mosillo et al. [16] adopted linear elasticity theory to reg- geometric features to be aligned, as these are intrinsic and ularize the variational maximization of MI. D’Agostino et stable. These choices are purely illustrative. In fact, any al. [8] extended this approach to a viscous ﬂuid scheme al- scalar ﬁelds deﬁned on the surfaces could be matched, e.g., lowing large local deformations, while maintaining smooth, cortical thickness maps, or even functional imaging signals one-to-one topology [6]. or metabolic data. Since conformal mapping and ﬂuid reg- istration techniques generate diffeomorphic mappings, the 1.2. Basic Idea 3D shape correspondence established by composing these mappings is also diffeomorphic (i.e., provides smooth one- Suppose ¡ ¢ is an oriented surface. The map from to ¡ £ to-one correspondences). a local coordinate § ¨¡ plane is a conformal map when ¥ ¦¤ © ¥ We also present a spectral approach for ensuring that the the ﬁrst fundamental form satisﬁes: % ! &© $#¤ © "§ ¦¤ © ! ! "§ grid induced on the target surface by the correspondence © . Here ¦¤ is called the conformal factor, a func- ! '§ ﬁeld, remains conformal. Grid orthogonality is advanta- tion that scales the metric at each point § ¨¡ . We say #¤ ¡ geous for accurate numerical discretization of PDEs or for ©¥ ¦¤ is a conformal coordinate of ¥ . Locally, each ( signal processing on the resulting surface meshes. For high surface patch is covered by a conformal coordinate chart. genus surfaces, the global conformal parameterization is not For high genus surfaces, the local conformal parameteriza- unique and all the conformal parameterizations form a lin- tion can be extended to cover the whole surface. By the ear space. The degrees of freedom in this space of con- Riemann-Roch theorem and the circle-valued Morse theo- formal grids are tuned to maximize the mutual information rem, a high genus surface ( ) can be completely cov- 2 0 31) energy of features between the two surfaces. Because the ered by a set of non-overlapping segments. Each segment conformal structure is intrinsic and the conformal parame- can be conformally mapped to a rectangle. With the Gauss terization continuously depends on the Riemannian metric and Codazzi equations, one can prove that a closed surface ! "§ ! "§ on the surface, our method is also stable and computation- ¦¤ 4 in with conformal parameter 75 6 is uniquely ¦¤ '§ ! ally efﬁcient. determined by its conformal factor #¤ and its mean cur- ! "§ vature ! ¤ "§ ¦98 "§ , up to a rigid motion. We call a tuple of ! ¤ ! ¦@8 "§ ¦¤ and a conformal representation of the sur- face . We can solve the surface registration problem ¦¤ 4 by computing intrinsic geometric features from the confor- mal mesh, and aligning them in the parameterization do- main. To align these scalar ﬁelds, we use a ﬂuid registration technique in the parameter domain that is driven by mutual information. With conformal mapping, we essentially con- vert the surface registration problem into an image registra- Figure 1. Illustrates surface conformal struc- tion problem, for which MI methods are especially advan- ture. (a) shows the cut we introduce on a tageous. Finally, by invoking the surface partitioning tech- face surface. The face becomes an open nique induced by holomorphic 1-forms, our surface-based boundary genus one surface. (b) shows a mutual information method works on general surfaces with global conformal parameterization of the sur- arbitrary topologies. face. (c) shows the horizontal trajectory. (d) shows the rectangle to which the face sur- 2. Theoretical Background face is conformally mapped. 2.1. Global Conformal Parameterization Suppose , are two surfaces. Locally they can be ¡ ¡© ¡ map it to a square as in (d) and get its conformal parameter- © ¥ § ¦© 4 © ¥ § BA ¡ 4 A §¡ ization. represented as , , where are ¥¨¤ ¡ ¥ #¤ © ¥ ¦¤¥ 6H5 FG© 5ED© 4 4 their local coordinates, and are vector- C § An atlas is a collection of consistent coordinate charts on ¡ valued functions. The ﬁrst fundamental form of r a§ A q§ih"§ 'cW aYWX USQ p `Y is S Q USQ a manifold, where transition functions between overlapping 2 , where cW fdccW gb e b ` . ) V $ ) TSRP ¡© ¥ ¥ Q I coordinate charts are smooth. Similarly, the ﬁrst fundamental form of ©@A S Q TSQ is deﬁned in We treat © 5 as isomorphic to the complex plane, where the same way: $ V &USQ I ©© ¡ ¥ ¥ s) . Deﬁne a map- ! "§ v ! qh % ! w x§ the point is equivalent to ¦¤ !h z1w , and is #¤ ping between two surfaces. Using lo- © BA F A C vut equivalent to yv § . Let be a surface in § with an 6 |{ © 5 F@© fPt cal coordinates, can be represented as , F |~CG¨v 5 C § ¡ © § t© § ¡ ¡ atlas , where ~¤ ¨v |z} is a chart, and ~ ¨v q¤ . Then any tangent vector " © ¥ #¤ t ¡ ¥ #¤ q¤ § ¡ t ¥ ¥ t maps an open set to the complex plane . G ~ tw on will be mapped to a tangent vector A © $ $#¤ ¥ ¥ An atlas is called conformal if (1). each chart § ~ ¨v z¤ on , y 9x y 9x © BA G` b y ` ` 3`cy i is a conformal chart. Namely, on each chart, the ﬁrst funda- b R ` i i (1) mental form can be formulated as v v v ¨ © ¨#¤ u© s¡ y ; (2). the transition maps i ` F £~ 1 q¤ C 1cv ~ v v & The length of is . We use the length t w § w¡t s ) I w t ~¤ E~ zv are holomorphic. of to deﬁne the length of ¥© V $#¤ . Namely, we deﬁne ¥ t w A chart is compatible with a given conformal atlas if a new metric for t which is induced by the mapping ¡ A adding it to the atlas again yields a conformal atlas. A con- and the metric on . We call this metric the pull-back formal structure (Riemann surface structure) is obtained by metric, and denote it by w t . Replacing in the above ©© fet 9A d © adding all compatible charts to a conformal atlas. A Rie- equation by (1), we get the analytic formula for the pull- mann surface is a surface with a conformal structure. back metric,r $k j k j One coordinate chart in the conformal structure in- y yq qq ¦" W W b ` b m m p ljv i m k i a¨hyg troduces a conformal parameterization between a surface b b cW cW (2) gb db Tgd oTgd n patch and the image plane. The conformal parameteriza- tion is angle-preserving and intrinsic to the geometry, and We call a conformal mapping, if there exists a positive is independent of the resolution and triangulation. t §¡ ¡© su © ¥ § ¡ ¦¤ t © sfet scalar function § ¡ © ¥ ¦¤ , such that ¥ . ¥ © d Locally, a surface patch is covered by a conformal coor- where © ¥ #¤ is called the conformal factor. ¥ ¡ dinate chart. For high genus surfaces, the local conformal Intuitively, all the angles on are preserved on . A © A parameterization can be extended to cover the whole sur- Figure 1 shows a conformal mapping example. Figure 1 face except at several points. These exceptional points are shows a face surface. We introduce a cut on the nose (a) and called zero points. By the Riemann-Roch theorem, there are r w r change it to a genus one surface. We illustrate the conformal zero points on a global conformal structure of a genus ) parameterization via the texture mapping of a checkerboard ) closed surface. By the circle-valued Morse theorem, the in (b). By tracing a horizontal trajectory (c), where the ini- iso-parametric curves through the zero points segment the tial tracing point was manually selected, we conformally whole segment the whole surface to patches, where each patch is either a topological disk, or a cylinder. The seg- tion for a high genus surface is fundamentally determined mentation is determined by the conformal structure of the by the choice of the holomorphic 1-form. surface and the choice of the global conformal parameteri- Figure 3 shows two different conformal parameteriza- zation. tions of a lateral ventricle surface of a HIV/AIDS patient Figure 2 shows an example of the conformal parameter- subject. (a) shows a uniform parameterization result and ization of a dog surface model. We introduce 3 cuts on the (b) shows a nonuniform parameterization result. Note that surface and change it to a genus 2 surface. The computed although both of these are conformal, one has greater area conformal structure is shown in (a). (b) shows a partition of distortion than the other. the dog surface, where each segment is labeled by a unique color. (d) shows the parameterization domain. Each rectan- gle is the image, in the parameterization domain, of a sur- face component in (c). Figure 3. Shows a uniform (left panel) and a non-uniform (right panel) global conformal parameterization for the same surface, the Figure 2. Illustrates conformal parameteriza- lateral ventricles of the human brain. tion for a high genus surface. (a) shows the conformal structure for a model of a dog. Af- ter introducing cuts between both ears and on the bottom, we turn the dog model into an 2.3. Conformal Representation of a General Sur- open boundary genus 2 surface. (b) shows a face partition of this surface, with a unique color For a general surface , we can compute conformal co- labeling each part. (c) shows the parameter- ordinates ¦¤ ! "§ to parameterize . Based on these coordi- ization domain. Each surface component in nates, one can derive scalar ﬁelds including the conformal (b) is conformally mapped to a rectangle in ! '§ ! "§ factor,#¤ , and mean curvature, , of the surface ! '§ ¦98¤ (c). The color scheme shows the association position vector : #¤ between elements in (b) and (c). © q m iq Vv m ¨ W W ¤ ¤ § ¥ §¥ cW cW (3) § ¦ ¥ « q ¨ ¬ m ª © « q m aq W ® W m q ¡ m § ¥ § ¥ Y cW (4) § ¥ W § ¥ 2.2. Optimal Global Conformal Parameterization We can regard the tuple as the conformal represen- D8 ¤ § ! "§ tation of . We have the following theorem [14]. ¦¤ ! '§ Given a Riemann surface § with a conformal atlas Theorem: A closed surface in with confor- 65 #¤ , a holomorphic 1-form is deﬁned by a fam- ! '§ § i § v q} ~¤ mal parameter is uniquely determined by its confor- # !'¤§ ! '§ ily , such that (1). i v q} ~¤ , where u #¤ t v v mal factor #¤ and its mean curvature up to '§ #98 is holomorphic on , and (2). if v ! ¤ ~ aiy t v #¤e rigid motions. A simply connected surface with a "#§ ¤ 4 ! is the coordinate transformation on "iy ¤ eE~ ~ , then boundary in and conformal parameter 5 6 is deter- ¦¤ ¤ t , i.e., the local representation of the ! '§ v lt v¤ mined by its conformal factor ! "§ and its mean curva- #¤ differential form satisﬁes the chain rule. ture¦98 ¤ and the boundary position. For a Riemann surface with genus , all holomor- ) Clearly, various ﬁelds of scalars or tuples could be used phic 1-forms on form a complex -dimensional vector r ¡ ) to represent surfaces in the parameter domain. Because the space ( real dimensions), denoted by ) . The con- ¤ ¡ conformal structure is intrinsic and independent of the data formal structure of a higher genus surface can always be resolution and triangulation, we use the conformal repre- ! "§ ! "§ represented in terms of a holomorphic 1-form basis, which r sentation, ¦¤ and , represent the 3D surfaces. ¤ ¦@8 is a set of functions ) F ¡ C Q ¢ '§ © 5 h . 2 r i§ a§ r ) This representation is stable and computationally efﬁcient. ee £e Any holomorphic 1-form is a linear combination of these Figure 4 illustrates computed conformal factor and mean functions. The quality of a global conformal parameteriza- curvature indexed by color on a hippocampal surface. q Q qm ´ ³ Í ¡ "h h§¡ ""h q m Q h £`´ ³ ` Q m ³ ) #Ì c© »Ï Ï ¡ , ¤ ¤ £© ¤ Ï 7Î Q ¡ Ò Ñ e¨r Vr Ô © xw Ò h Ò Ñ l¨r Ò Ô h $r ¡© xw i Ð q¤ © ¤ ¹ Ó © £V¥ 'e z¤ e © Ð ¢$¥ ae ¤ ¹ Ó as the Parzen kernel, and “ ” denotes convolution. Õ 3. The Surface Mutual Information Method for an Arbitrary Genus Surface Figure 4. Illustrates the computed conformal Next, suppose we want to match two high genus surfaces factor and mean curvature on a hippocampal (i.e., surfaces with the same branching topology). To ap- surface. On the left are two views of the hip- ply our surface mutual information method piecewise, we pocampal surface, colored according to the ﬁrst compute the conformal representations of the two sur- conformal factor. The right two are colored faces based on a global conformal parameterization. Mu- by mean curvature. tual information driven ﬂows are then applied to align the computed conformal representations, while enforcing con- straints to guarantee continuity of the vector-valued ﬂow at the patch boundaries. When the chain rule is used, we can 2.4. Mutual Information for Surface Registration further optimize the mutual information matching results by We now describe the mutual information functional used "§ ! '§ ! optimizing the underlying global conformal parameteriza- to drive the scalar ﬁelds and ¦¤ into correspon- ¤ #98 tion. ¡ e dence, effectively using the equivalent of a 2D image reg- Let and be two surfaces we want to match and the © ¡ ¡ istration in the surface parameter space (i.e., in conformal conformal parameterization of is , conformal param- ¡ ¡ Ö ¡ eterization for is , and are rectangles coordinates). Let and be the target and the deforming ¯ ©¯ © © ¤ Ö £Ö © ¤$Ö© ¡ template images respectively, and . Let F 5 C © § © 9£¯ ¡ ¯ 5 in © 5. Instead of ﬁnding the mapping from to © © 5 ¡ be the common parameter domain of both surfaces directly, we can use mutual information method to ﬁnd a ¡ ¡ (if both are rectangular domains, the target parameter do- diffeomorphism × F eRÖ , such that the diagram be- © × C main can ﬁrst be matched to the source parameter domain low commutes: Ù ¡ xØ © Ö . Then the map can be Ö Ö using a 2D diagonal matrix). Also, let be a deformation obtained from the following commutative diagram, vector ﬁeld on . The MI of the scalar ﬁelds (treated as 2D ¡ Ü images) between the two surfaces is deﬁned by q `¤ ¤ Û Q y Q y q m Q ` Q q m £m ³ ¸¶q Q Q m f £ q m ° ´ ·µ ²± ` Q £³ ` Q ³ p ´ ` ´³ ¥ (5) `Þ Þ (8) Ý Ý h w h ¡ "h ¡ º ¡ "h Ú Ú Û ßÞ where ¥¤ © ¯ º h #$f¤ w R c© ¤ ¥ ¹uh »#¤ #¤ t '¨"¢¹ , ¡ ¥¡ ¯ R h § ¡ ¤ h c© and © ¦f¯ ¥¤ © and ¥ ¯ '»¦¤ ¤ . º £© ¤ ¥ ¹ ¡ ¡ ¡ We adopted the framework of D’Agostino et al. [8] to © &àx Ö Ö Ö . Because , and Ö Ö £Ö © are all diffeomor- phisms, is also a diffeomorphism. maximize MI with viscous ﬂuid regularization. Brieﬂy, the deforming template image was treated as embedded in a compressible viscous ﬂuid governed by Navier-Stokes 3.1. Mutual Information Contained in Maps be- equation for conservation of momentum [6], simpliﬁed to a tween High Genus Surfaces linear PDE: § ! ¿ ¿ ½ % ! ¿ ½ ! ¼ A global conformal parameterization for a high genus ¦¤ @% e ¤ 9 ¤ % © e¾ ¥ Á À À (6) surface can be obtained by integrating a holomorphic 1- ½ ! form . Suppose is a holomor- "§ Q £} h § r a§ r a§ Here is the deformation velocity, and and are the 2 £e ee ») viscosity constants. Following the derivations in [8], we phic 1-form basis, where an arbitrary holomorphic 1-form Q Q ¡ p Q© I take the ﬁrst variation of with respect to , and use the ¦»¯ ¤ has the formula . Assuming the target Parzen window method [23] to estimate the joint probability ""h h§¡ § surface’s parameterization is ﬁxed, the mutual information density function (pdf) . The driving force c© ¤ ¥ ¹ ¦¤ Á ¥ energy between it and the source surface’s parameteriza- that registers features in the 2D surface parameter space is tion is denoted by , which is a function of the lin- ¤ ¶vá given by Q q m ¡ q m ear combination of coefﬁcients . The necessary condi- m ##q Êq m q m É m È Ç Q Q ´ q tion for the optimal holomorphic 1-form is straightforward, ¥ b ° ¥ b ° b z° ®eQ cWÆ W d ` Å HÃ Ä ¥ b Â (7) h '§ â¨ § r i§ r i§ 2 d cW W . If the Hessian matrix £e ee ) g ¨ â W d W¨ W ¤ where "¨"h ¼ § h§¡ is the Ë area of the param- is positive deﬁnite, then will reach the minimum. If the á eter 2 domain © ¤ ¥ ¡ % Hessian matrix is negative deﬁnite, will be maximized. á w Our surface mutual information method depends on the 9. If , return t. Otherwise, assign ¨ñ ó° á ° å á á û ÷å á selection of holomorphic 1-form . To get an optimal sur- to and repeat steps through . ÷ á ü ý face mutual information matching result, we need ﬁnd the optimal holomorphic 1-form for mutual information metric. Currently, we use the following numerical scheme in Q Q ¡ p ©Q Ù Suppose a holomorphic function , our goal aää¸§ I "§ Q r §ããã h step : ü is to ﬁnd a set of coefﬁcients that maxi- ) 2 1. Compute and ! Ô° , å á h "§ Q s Ô $ $ Ô ° å á mize the mutual information energy, . We can solve ° Há å 2 ri¸ãääã£i§ § ã § r ; ) this optimization problem numerically as follows: r §ããã § r h aää¸a§ "§ Q Ô ! UrUrr 2. Compute ; )"§ 2 s y y ¨y 3. Compute ) § ã r raäã¸äã£§a§ 2 h Q Ô° with s $ ó× å á ò ´ ëlq uêìÉ` ê ê é æ y â æ y â G m y ´ uêì ê rUUrr ` UrUr¨ r y ï q ´ ê"ê í ì Equation 9. è è çâ è æ É uêì ê ðq z ¨ y RqÉ ê"ê î í ì ` uêì § ¥ í (9) ! "§ y where is the conformal coordinate. ¦¤ Once we compute , we can use "§ æ ¨ yâ h è 2 r §ããã § r aä¸ä£a§ ) d steepest descent to optimize the resulting mutual informa- tion. A complete description of the surface mutual informa- tion method follows. Algorithm 1 Surface Mutual Information Method (for sur- faces of arbitrary genus) ¡ Input (mesh and ,step length , energy differ- A © A ò ñ ence threshold ), ¡ ¨ñ á Output( ) where minimizes the surface A C |óò F © A ò mutual information energy. 1. Compute global conformal parameterization of two aää¸a§ 'i§ q§ Q Q ¡ p ©Q ô S r §ããã § r hõ r p surfaces, ) ¡ 2 2 I , where Figure 5. Matching surface features in 2D pa- is the surface genus number of two surfaces and , © A rameter domains using Mutual Information. ) A i¸ä¸i§ "i§ z§ QS r §ããã § r h§ r p Geometric features on 3D hippocampal sur- and ) are the coefﬁcients of a 2 2 linear combination of holomorphic function basis elements. faces (the conformal factor and mean curva- The steps include computing the homology basis, cohomol- ture) were computed and compound scalar ogy basis, harmonic 1-form basis and holomorphic 1-form ﬁelds (e.g., 8xconformal factor + mean cur- basis [15]. vature) were mapped to a 2D square by 2. Compute holomorphic ﬂow segmentation of the target conformal ﬂattening. In the 2D parameter surface, © A , from the global conformal parameterization, domain, data from a healthy normal sub- © , which conformally maps the 3D surface to a set of rect- ject (the template, leftmost column) was reg- angles in the Euclidean plane. istered to data from several patients with 3. Compute 2D conformal representation for the target Alzheimer’s disease (target images, second ! '§ ! '§ ! "§ column). Each mapping can be used to ob- surface, and © #¤ , where is the confor- #¤ © 8 ¦¤ mal coordinate; tain a reparameterization of the 3D surface of 4. Compute holomorphic ﬂow segmentation of the source ¡ ¡ the normal subject, by convecting the origi- ! "§ nal 3D coordinates along with the ﬂow. The surface, ! , and 2D conformal representation '§ A ¡ ¦¤ and ; #¤ 8 deformed template images are shown in the 5. Apply the mutual information method to op- third and fourth (gridded) columns. The grids timize the correspondence between two surfaces, show how the ﬂuid transform expands some '§ ¡ ! ! ¡ F " "§ ¦¤ 8 § q§ "§ p ! § "§ © ! C öò i§ r highly curved features to match similar fea- "§ #¤ S ¤ ! i§ r p q§ and 2 " ¦¤ © 8 ¦¤ ¤ ¦¤ 8 2 ; and compute mutual information en- tures. Importantly, there are some consis- ergy á ÷å ; ° tent 3D geometric features that can be re- 6. Compute derivative . ó× ò identiﬁed in the 2D parameter domain; e.g., 7. Update the global conformal parameterization of ¡ ¡ bright areas (arrows) correspond to high cur- source surface, , by changing the coefﬁcients A !¤ vature features in the head of the hippocam- ! pus. òñ ¤ò . u fó× 8. Compute mutual information energy , with steps a"§ ø ú§ ù á . 4. Experimental Results by the joint distribution of features lying in both surfaces. This is a natural idea, in that it uses conformal parameteri- To make the results easier to illustrate, we chose to en- zation to transform a surface matching problem into an im- code the proﬁle of surface features using a compound scalar age registration problem. Whether or not this approach pro- ! "§ ÿ ! "§ ! '§ vides a more relevant correspondences than those afforded function #98 % ¦¤ $P ¦þ ¤ ¤ . We linearly nor- malized its dynamic range to the pixel intensity range 0 to by other criteria (minimum description length, neural nets, 255. Several examples are shown, mapping one face to the or hand landmarking) requires careful validation for each other face and one hippocampal surface to another. Face application. Optimal correspondence depends more on util- mapping is useful for face recognition. The deformable sur- ity for a particular application than on anatomical homol- face registration of hippocampus is important for tracking ogy. Because different correspondence principles produce developmental and degenerative changes in the hippocam- different shape models, we plan to compare them in future pus, as well as computing average shape models with ap- for detecting group differences in brain structure, and indi- propriate boundary correspondences. Figure 5 shows the vidual differences in face recognition applications. . If sta- matching ﬁelds for several pairs of hippocampal surfaces, tistical power is increased in group comparisons, this would establishing correspondences between distinctive features. support the use of correspondence ﬁelds established by mu- ! tual information on surfaces. The velocity ﬁeld in Eqn 6 was computed iteratively by convolution of the force ﬁeld with a ﬁlter kernel derived by Bro-Nielsen and Gramkow [4]. The viscosity coefﬁcients ½ and were set to 0.9 and 6.0 respectively. The deforma- ! tion ﬁeld in the parameter domain ( ) was obtained from by Euler integration over time, and the deformed template image was regridded when the Jacobian determinant of the w deformation mapping at any point in ¥ was smaller than 0.5 [6]. At each step, the joint pdf was updated and the MI re-computed. Iterations were stopped when MI was no longer monotonically increasing or when the number of it- Ò erations reached 350. The Parzen parameter was set to 10 for smoothing the joint pdf. In Figure 6, we show face sur- faces and hippocampal surfaces to be matched, where the face surface was built with a high resolution, real-time 3D face acquisition system [32] and hippocampal surfaces were built from 3D MRI scans of the brain. Speciﬁcally, (a), (b) Figure 6. Surface matching results from our and (e), (f) show two surfaces to be matched; (c), (d) and method. Panels (a), (b), (c) and (d) show the (g) and (h) show 3D vector displacement map, connecting surfaces being matched. (a) and (b) are two corresponding points on the two surfaces. (c) and (g) are be- face surfaces. (e) is a normal subject’s hip- fore and (d) and (h) after reparameterization of the source pocampal surface and (d) is an Alzheimer’s surface using a ﬂuid ﬂow in the parameter domain. These disease patient’s hippocampal surface. We more complex 3D vector ﬁelds store information on geo- ﬂow the surface from (a) to (b) and from (e) metrical feature correspondences between the surfaces. to (f), respectively. Panels (c), (d), (g) and (h) show the 3D vector displacement map, con- necting corresponding points on the two sur- 5. Conclusions and Future Work faces, (c) and (g) before and (d) and (h) af- ter reparameterization of the source surface We extended the mutual information method to match using a ﬂuid ﬂow in the parameter domain. general surfaces. This is useful for face recognition and has After reparameterization, a leftward shift in numerous applications in medical imaging. Our examples the vertical isocurves adds a larger tangen- of matching various hippocampal surfaces are relevant for tial component to the vector ﬁeld. Even so, mapping how degenerative diseases affect the brain, as well the deformed grid structure remains close to as building statistical shape models to detect the anatomical conformal. These more complex 3D vector effects of disease, aging, or development. The face and hip- ﬁelds store information on geometrical fea- pocampus are used as speciﬁc examples, but the method is ture correspondences between the surfaces. general and is applicable in principle to other surfaces. Surface-based mutual information automates the match- ing of surfaces by computing a correspondence ﬁeld guided References [18] B. Kim, J. Boes, K. Frey, and C. Meyer. Mutual informa- tion for automated unwarping of rat brain autoradiographs. [1] S. Angenent, S. Haker, R. Kikinis, and A. Tannenbaum. NeuroImage, 5(1):31–40, 1997. Nondistorting ﬂattening maps and the 3D visualization of [19] G. Kindlmann, D. Weinstein, A. Lee, A. Toga, and colon CT images. IEEE TMI, 19:665–671, 2000. P. Thompson. Visualization of anatomic covariance tensor [2] S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis. ﬁelds. In Proc. IEEE EMBS, San Francisco, CA, 2004. Conformal geometry and brain ﬂattening. 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