{"title": "Wavelet based multi-scale shape features on arbitrary surfaces for cortical thickness discrimination", "book": "Advances in Neural Information Processing Systems", "page_first": 1241, "page_last": 1249, "abstract": "Hypothesis testing on signals de\ufb01ned on surfaces (such as the cortical surface) is a fundamental component of a variety of studies in Neuroscience. The goal here is to identify regions that exhibit changes as a function of the clinical condition under study. As the clinical questions of interest move towards identifying very early signs of diseases, the corresponding statistical differences at the group level invariably become weaker and increasingly hard to identify. Indeed, after a multiple comparisons correction is adopted (to account for correlated statistical tests over all surface points), very few regions may survive. In contrast to hypothesis tests on point-wise measurements, in this paper, we make the case for performing statistical analysis on multi-scale shape descriptors that characterize the local topological context of the signal around each surface vertex. Our descriptors are based on recent results from harmonic analysis, that show how wavelet theory extends to non-Euclidean settings (i.e., irregular weighted graphs). We provide strong evidence that these descriptors successfully pick up group-wise differences, where traditional methods either fail or yield unsatisfactory results. Other than this primary application, we show how the framework allows performing cortical surface smoothing in the native space without mappint to a unit sphere.", "full_text": "Wavelet based multi-scale shape features on arbitrary\n\nsurfaces for cortical thickness discrimination\n\nWon Hwa Kim\u2020\u00b6\u2217 Deepti Pachauri\u2020 Charles Hatt\u2021\n\nMoo K. Chung\u00a7 Sterling C. Johnson\u2217\u00b6 Vikas Singh\u00a7\u2020\u2217\u00b6\n\n\u2020Dept. of Computer Sciences, University of Wisconsin, Madison, WI\n\n\u00a7Dept. of Biostatistics & Med. Informatics, University of Wisconsin, Madison, WI\n\n\u2021Dept. of Biomedical Engineering, University of Wisconsin, Madison, WI\n\n\u00b6Wisconsin Alzheimer\u2019s Disease Research Center, University of Wisconsin, Madison, WI\n\n\u2217GRECC, William S. Middleton VA Hospital, Madison, WI\n\n{wonhwa, pachauri}@cs.wisc.edu\n\n{hatt, mkchung}@wisc.edu\n\nscj@medicine.wisc.edu\n\nvsingh@biostat.wisc.edu\n\nAbstract\n\nHypothesis testing on signals de\ufb01ned on surfaces (such as the cortical surface) is\na fundamental component of a variety of studies in Neuroscience. The goal here\nis to identify regions that exhibit changes as a function of the clinical condition\nunder study. As the clinical questions of interest move towards identifying very\nearly signs of diseases, the corresponding statistical differences at the group level\ninvariably become weaker and increasingly hard to identify. Indeed, after a mul-\ntiple comparisons correction is adopted (to account for correlated statistical tests\nover all surface points), very few regions may survive. In contrast to hypothesis\ntests on point-wise measurements, in this paper, we make the case for perform-\ning statistical analysis on multi-scale shape descriptors that characterize the local\ntopological context of the signal around each surface vertex. Our descriptors are\nbased on recent results from harmonic analysis, that show how wavelet theory\nextends to non-Euclidean settings (i.e., irregular weighted graphs). We provide\nstrong evidence that these descriptors successfully pick up group-wise differences,\nwhere traditional methods either fail or yield unsatisfactory results. Other than\nthis primary application, we show how the framework allows performing cortical\nsurface smoothing in the native space without mappint to a unit sphere.\n\n1\n\nIntroduction\n\nCortical thickness measures the distance between the outer and inner cortical surfaces (see Fig.\n1). It is an important biomarker implicated in brain development and disorders [3]. Since 2011,\nmore than 1000 articles (from a search on Google Scholar and/or Pubmed) tie cortical thickness to\nconditions ranging from Alzheimer\u2019s disease (AD), to Schizophrenia and Traumatic Brain injury\n(TBI) [9, 14, 13]. Many of these results show how cortical thickness also correlates with brain\ngrowth (and atrophy) during adolescence (and aging) respectively [22, 20, 7]. Given that brain\nfunction and pathology manifest strongly as changes in the cortical thickness, the statistical analysis\nof such data (to \ufb01nd group level differences in clinically disparate populations) plays a central role\nin structural neuroimaging studies.\nIn typical cortical thickness studies, magnetic resonance images (MRI) are acquired for two popu-\nlations: clinical and normal. A sequence of image processing steps are performed to segment the\ncortical surfaces and establish vertex-to-vertex correspondence across surface meshes [15]. Then, a\ngroup-level analysis is performed at each vertex. That is, we can ask if there are statistically signi\ufb01-\ncant differences in the signal between the two groups. Since there are multiple correlated statistical\n\n1\n\n\fFigure 1: Cortical thickness illus-\ntration: the outer cortical surface (in\nyellow) and the inner cortical sur-\nface (in blue). The distance between\nthe two surfaces is the cortical thick-\nness.\n\ntests over all voxels, a Bonferroni type multiple comparisons correction is required [4]. If many\nvertices survive the correction (i.e., differences are strong enough), the analysis will reveal a set of\ndiscriminative cortical surface regions, which may be positively or negatively correlated with the\nclinical condition of interest. This procedure is well understood and routinely used in practice.\nIn the last \ufb01ve years, a signi\ufb01cant majority of research has shifted\ntowards investigations focused on the pre-clinical stages of dis-\neases [16, 23, 17]. For instance, we may be interested in iden-\ntifying early signs of dementia by analyzing cortical surfaces\n(e.g., by comparing subjects that carry a certain gene versus\nthose who do not). In this regime, the differences are weaker,\nand the cortical differences may be too subtle to be detected.\nIn a statistically under-powered cortical thickness analysis, few\nvertices may survive the multiple comparisons correction. An-\nother aspect that makes this task challenging is that the cortical\nthickness data (obtained from state of the art tools) is still in-\nherently noisy. The standard approach for \ufb01ltering cortical sur-\nface noise is to adopt an appropriate parameterization to model\nthe signal followed by a diffusion-type smoothing [6]. The pri-\nmary dif\ufb01culty is that most (if not all) widely used parameteri-\nzations operate in a spherical coordinate system using spherical\nharmonic (SPHARM) basis functions [6]. As a result, one must\n\ufb01rst project the signal on the surface to a unit sphere. This \u201cballooning\u201d process introduces serious\nmetric distortions. Second, SPHARM parameterization usually suffers from ringing artifacts (i.e.,\nGibbs phenomena) when used to \ufb01t rapidly changing localized cortical measurements [10]. Third,\nSPHARM uses global basis functions which typically requires a large number of terms in the ex-\npansion to model cortical surface signals to high \ufb01delity. Subsequently, even if the globally-based\ncoef\ufb01cients exhibit statistical differences, interpreting which brain regions contribute to these vari-\nations is dif\ufb01cult. As a result, the coef\ufb01cients of the model cannot be used directly in localizing\nvariations in the cortical signal.\nThis paper is motivated by the simple observation that statistical inference on surface based signals\nshould be based not on a single scalar measurement but on multivariate descriptors that characterize\nthe topologically localized context around each point sample. This view insures against signal noise\nat individual vertices, and should offer the tools to meaningfully compare the behavior of the signal\nat multiple resolutions of the topological feature, across multiple subjects. The ability to perform\nthe analysis in a multi-resolution manner, it seems, is addressable if one makes use of Wavelets\nbased methods (e.g., scaleograms [19]). Unfortunately, the non-regular structure of the topology\nmakes this problematic. In our neuroimaging application, samples are not drawn on a regular grid,\ninstead governed entirely by the underlying cortical surface mesh of the participant. To get around\nthis dif\ufb01culty, we make use of some recent results from the harmonic analysis literature [8] \u2013 which\nsuggests how wavelet analysis can be extended to arbitrary weighted graphs with irregular topol-\nogy. We show how these ideas can be used to derive a wavelet multi-scale descriptor for statistical\nanalysis of signals de\ufb01ned on surfaces. This framework yields rather surprising improvements in\ndiscrimination power and promises immediate bene\ufb01ts for structural neuroimaging studies.\nContributions. We derive wavelet based multi-scale representations of surface based signals. Our\nrepresentation has varying levels of local support, and as a result can characterize the local context\naround a vertex to varying levels of granularity. We show how this facilitates statistical analysis of\nsignals de\ufb01ned on arbitrary topologies (instead of the lattice setup used in image processing).\n\n(i) We show how the new model signi\ufb01cantly extends the operating range of analysis of cortical\nsurface signals (such as cortical thickness). At a pre-speci\ufb01ed signi\ufb01cance level, we can detect\na much stronger signal showing group differences that are barely detectable using existing\napproaches. This is the main experimental result of this paper.\n\n(ii) We illustrate how the procedure of smoothing of cortical surfaces (and shapes) can completely\nbypass the mapping on to a sphere, since smoothing can now be performed in the native space.\n\n2\n\n\f2 A Brief Review of Wavelets in Signal Processing\n\nRecall that the celebrated Fourier series representation of a periodic function is expressed via a su-\nperposition of sines and cosines, which is widely used in signal processing for representing a signal\nin the frequency domain and obtaining meaningful information from it. Wavelets are conceptually\nsimilar to the Fourier series transform, in that they can be used to extract information from many\ndifferent kinds of data, however unlike the Fourier transform which is localized in frequency only,\nwavelets can be localized in both time and frequency [12] and extend frequency analysis to the no-\ntion of scale. The construction of wavelets is de\ufb01ned by a wavelet function \u03c8 (called an analyzing\nwavelet or a mother wavelet) and a scaling function \u03c6. Here, \u03c8 serves as a band-pass \ufb01lter and\n\u03c6 operates as a low-pass \ufb01lter covering the low frequency components of the signal which cannot\nbe tackled by the band-pass \ufb01lters. When the band-pass \ufb01lter is transformed back by the inverse\ntransform and translated, it becomes a localized oscillating function with \ufb01nite duration, providing\nvery compact (local) support in the original domain [21]. This indicates that points in the original\ndomain which are far apart have negligable impact on one another. Note the contrast with Fourier\nseries representation of a short pulse which suffers from issues due to nonlocal support of sin(\u00b7)\nwith in\ufb01nite duration.\nFormally, the wavelet function \u03c8 on x is a function of two parameters, the scale and translation\nparameters, s and a\n\n\u03c8s,a(x) =\n\n1\na\n\n\u03c8(\n\nx \u2212 a\n\ns\n\n)\n\n(1)\n\nVarying scales control the dilation of the wavelet, and together with a translation parameter, con-\nstitute the key building blocks for approximating a signal using a wavelet expansion. The function\n\u03c8s,a(x) forms a basis for the signal and can be used with other bases at different scales to decom-\npose a signal, similar to Fourier transform. The wavelet transform of a signal f (x) is de\ufb01ned as the\ninner product of the wavelet and signal and can be represented as\n\nWf (s, a) = (cid:104)f, \u03c8(cid:105) =\n\n(2)\nwhere Wf (s, a) is the wavelet coef\ufb01cient at scale s and at location a. The function \u03c8\u2217 represents\nthe complex conjugate of \u03c8. Such a transform is invertible, that is\n\n)dx\n\ns\n\nx \u2212 a\n\nf (x)\u03c8\u2217(\n\n(cid:90)\n\n1\na\n\nWf (s, a)\u03c8s,a(x)da ds\n\n(3)\n\n(cid:90)(cid:90)\n\nf (x) =\n\n1\nC\u03c8\n\nwhere C\u03c8 = (cid:82) |\u03a8(j\u03c9)|2\n\n|\u03c9|\n\nd\u03c9 is called the admissibility condition constant, and \u03a8 is the Fourier\n\ntransform of the wavelet [21], and the \u03c9 is the domain of frequency.\nAs mentioned earlier, the scale parameter s controls the dilation of the basis and can be used to pro-\nduce both short and long basis functions. While short basis functions correspond to high frequency\ncomponents and are useful to isolate signal discontinuities, longer basis functions corresponding to\nlower frequencies, are also required to to obtain detailed frequency analysis. Indeed, wavelets trans-\nforms have an in\ufb01nite set of possible basis functions, unlike the single set of basis functions (sine\nand cosine) in the Fourier transform. Before concluding this section, we note that while wavelets\nbased analysis for image processing is a mature \ufb01eld, most of these results are not directly applicable\nto non-uniform topologies such as those encountered in shape meshes and surfaces in Fig. 1.\n\n3 De\ufb01ning Wavelets on Arbitrary Graphs\n\nNote that the topology of a brain surface is naturally modeled as a weighted graph. However, the\napplication of wavelets to this setting is not straightforward, as wavelets have traditionally been\nlimited to the Euclidean space setting. Extending the notion of wavelets to a non-Euclidean setting,\nparticularly to weighted graphs, requires deriving a multi-scale representation of a function de\ufb01ned\non the vertices. The \ufb01rst bottleneck here is to come up with analogs of dilation and translation on the\ngraph. To address this problem, in [8], the authors introduce Diffusion Wavelets on manifolds. The\nbasic idea is related to some well known results from machine learning, especially the eigenmaps\nframework by Belkin and Niyogi [1].\nIt also has a strong relationship with random walks on a\nweighted graph. Brie\ufb02y, a graph G = (V, E, w) with vertex set V , edge set E and symmetric edge\n\n3\n\n\fweights w has an associated random walk R. The walk R, when represented as a matrix, is conjugate\nto a self adjoint matrix T , which can be interpreted as an operator associated with a diffusion process,\nexplaining how the random walk propagates from one node to another. Higher powers of T (given\nas T t) induce a dilation (or scaling) process on the function to which it is applied, and describes\nthe behavior of the diffusion at varying time scales (t). This is equivalent to iteratively performing a\nrandom walk for a certain number of steps and collecting together random walks into representatives\n[8]. Note that the orthonormalization of the columns of T induces the effect of \u201ccompression\u201d, and\ncorresponds to downsampling in the function space [5]. In fact, the powers of T are low rank (since\nthe spectrum of T decays), and this ties back to the compressibility behavior of classical wavelets\nused in image processing applications (e.g., JPEG standard). In this way, the formalization in [8]\nobtains all wavelet-speci\ufb01c properties including dilations, translations, and downsampling.\n\n3.1 Constructing Wavelet Multiscale Descriptors (WMD)\n\nVery recently, [11] showed that while the orthonormalization above is useful for iteratively obtain-\ning compression (i.e., coarser subspaces), it complicates the construction of the transform and only\nprovides limited control on scale selection. These issues are critical in practice, especially when\nadopting this framework for analysis of surface meshes with \u223c 200, 000 vertices with a wide spec-\ntum of frequencies (which can bene\ufb01t from \ufb01ner control over scale). The solution proposed in [11]\ndiscards repeated application of the diffusion operator T , and instead relies on the graph Laplacian\nto derive a spectral graph wavelet transform (SGWT). To do this, [11] uses a form of the wavelet\noperator in the Fourier domain, and generalizes it to graphs. Particularly, SGWT takes the Fourier\ntransform of the graph by using the properties of the Laplacian L (since the eigenvectors of L are\nanalogous to the Fourier basis elements). The formalization is shown to preserve the localization\nproperties at \ufb01ne scales as well as other wavelets speci\ufb01c properties. But beyond constructing the\ntransform, the operator-valued functions of the Laplacian are very useful to derive a powerful multi-\nscale shape descriptor localized at different frequencies which performs very well in experiments.\nFor a function f (m) de\ufb01ned on a vertex m of a graph, interpreting f (sm) for a scaling constant s,\nis not meaningful on its own. SGWT gets around this problem by operating in the dual domain \u2013 by\ntaking the graph Fourier transformation. In this scenario, the spectrum of the Laplacian is analogous\nto the frequency domain, where scales can be de\ufb01ned (seen in (6) later). This provides a multi-\nresolution view of the signal localized at m. By analyzing the entire spectra at once, we can obtain\na handle on which scale best characterizes the signal of interest. Indeed, for graphs, this provides\na mechanism for simultaneously analyzing various local topologically-based contexts around each\nvertex. And for a speci\ufb01c scale s, we can now construct band-pass \ufb01lters g in the frequency domain\nwhich suppresses the in\ufb02uence of scales s(cid:48) (cid:54)= s. When transformed back to the original domain, we\ndirectly obtain a representation of the signal for that scale. Repeating this process for multiple scales,\nthe set of coef\ufb01cients obtained for S scales comprises our multiscale descriptor for that vertex.\nGiven a mesh with N vertices, we \ufb01rst obtain the complete orthonormal basis \u03c7l and eigenvalues\n\u03bbl, l \u2208 {0, 1,\u00b7\u00b7\u00b7 , N \u2212 1} for the graph Laplacian. Using these bases, the forward and inverse graph\nFourier transformation are de\ufb01ned using eigenvalues and eigenvectors of L as,\n\nN(cid:88)\n\nN\u22121(cid:88)\n\n\u02c6f (l) = (cid:104)\u03c7l, f(cid:105) =\n\n\u03c7\u2217\nl (n)f (n), and f (n) =\n\n\u02c6f (l)\u03c7l(n)\n\n(4)\n\nn=1\n\nl=0\n\nUsing the transforms above, we construct spectral graph wavelets by applying band-pass \ufb01lters at\nmultiple scales and localizing it with an impulse function. Since the transformed impulse function in\nthe frequency domain is equivalent to a unit function, the wavelet \u03c8 localized at vertex n is de\ufb01ned\nas,\n\n\u03c8s,n(m) =\n\ng(s\u03bbl)\u03c7\u2217\n\nl (n)\u03c7l(m)\n\n(5)\n\nwhere m is a vertex index on the graph. The wavelet coef\ufb01cients of a given function f (n) can be\neasily generated from the inner product of the wavelets and the given function,\n\nWf (s, n) = (cid:104)\u03c8s,n, f(cid:105) =\n\ng(s\u03bbl) \u02c6f (l)\u03c7l(n)\n\n(6)\n\nN\u22121(cid:88)\n\nl=0\n\nN\u22121(cid:88)\n\nl=0\n\n4\n\n\fThe coef\ufb01cients obtained from the transformation yield the Wavelet Multiscale Descriptor (WMD)\nas a set of wavelet coef\ufb01cients at each vertex n for each scale s.\n\nWMDf (n) = {Wf (s, n)|s \u2208 S}\n\n(7)\nIn the following sections, we make use of the multi-scale descriptor for the statistical analysis of\nsignals de\ufb01ned on surfaces(i.e., standard structured meshes). We will discuss shortly how many of\nthe low-level processes in obtaining wavelet coef\ufb01cients can be expressed as linear algebra primi-\ntives that can be translated on to the CUDA architecture.\n\n4 Applications of Multiscale Shape Features\n\nIn this section, we present extensive experimental results demonstrating the applicability of the\ndescriptors described above. Our core application domain is Neuroimaging. In this context, we \ufb01rst\ntest if the multi-scale shape descriptors can drive signi\ufb01cant improvements in the statistical analysis\nof cortical surface measurements. Then, we use these ideas to perform smoothing of cortical surface\nmeshes without \ufb01rst projecting them onto a spherical coordinate system (the conventional approach).\n\n4.1 Cortical Thickness Discrimination: Group Analysis for Alzheimer\u2019s disease (AD) studies\n\nTable 1: Demographic details and baseline cognitive sta-\ntus measure of the ADNI dataset\nADNI data\n\nAs we brie\ufb02y discussed in Section 1, the identi\ufb01cation of group differences between cortical surface\nsignals is based on comparing the distribution of the signal across the two groups at each vertex. This\ncan be done either by using the signal (cortical thickness) obtained from the segmentation directly,\nor by using a spherical harmonic (SPHARM) or spherical wavelet approach to \ufb01rst parameterize and\nthen smooth the signal, followed by a vertex-wise T\u2212test on the smoothed signal. These spherical\napproaches change the domain of the data from manifolds to a sphere, introducing distortion. In\ncontrast, our multi-scale descriptor is well de\ufb01ned for characterizing the shape (and the signal) on\nthe native graph domain itself. We employ hypothesis testing using the original cortical thickness\nand SPHARM as the two baselines for comparison when presenting our experiments below.\nData and Pre-processing. We used Magnetic Resonance (MR) images acquired as part of the\nAlzheimer\u2019s Disease Neuroimaging Initiative (ADNI). Our data included brain images from 356\nparticipants: 160 Alzheimer\u2019s disease subjects (AD) and 196 healthy controls (CN). Details of the\ndataset are given in Table1.\nThis dataset was pre-processed using a stan-\ndard image processing pipeline, and the\nFreesurfer algorithm [18] was used to seg-\nment the cortical surfaces, calculate the cor-\ntical thickness values, and provide vertex to\nvertex correspondences across brain surfaces.\nThe data was then analyzed using our algo-\nrithm and the two baselines algorithms men-\ntioned above. We constructed WMDs for each vertex on the template cortical surface at 6 different\nscales, and used Hotelling\u2019s T 2\u2212test for group analysis. The same procedure was repeated us-\ning the cortical thickness measurements (from Freesurfer) and the smoothed signal obtained from\nSPHARM. The resulting p-value map was corrected for multiple comparisons over all vertices using\nthe false discovery rate (FDR) method [2].\nFig. 2 summarizes the results of our analysis. The \ufb01rst row corresponds to group analysis using the\noriginal cortical thickness values (CT). Here, while we see some discriminative regions, group dif-\nferences are weak and statistically signi\ufb01cant in only a small region. The second row shows results\npertaining to SPHARM, which indicate a signi\ufb01cant improvement over the baseline, partly due to\nthe effect of noise \ufb01ltering. Finally, the bottom row in Fig. 2 shows that performing the statistical\ntests using our multi-scale descriptor gives substantially larger regions with much lower p-values.\nTo further investigate this behavior, we repeated these experiments by making the signi\ufb01cance level\nmore conservative. These results (after FDR correction) are shown in Fig. 4. Again, we can directly\ncompare CT, SPHARM and WMD for a different FDR. A very conservative FDR q = 10\u22127 was\nused on the uncorrected p-values from the hypothesis test, and the q-values after the correction were\nprojected back on the template mesh. Similar to Fig. 2, we see that relative to CT and SPHARM,\nseveral more regions (with substantially improved q-values) are recovered using the multi-scale de-\nscriptor.\n\nCategory\n# of Subjects\nAge\nGender (M/F)\nMMSE at Baseline\nYears of Education\n\n-\n5.13\n-\n3.09\n3.23\n\nAD (mean) AD (s.d.) Ctrl (mean) Ctrl (s.d.)\n160\n75.53\n86 / 74\n21.83\n13.81\n\n196\n76.09\n101 / 95\n28.87\n15.87\n\n-\n7.41\n-\n5.98\n4.61\n\n5\n\n\fTo quantitatively compare the behavior above, we evaluated the uncorrected p-values over all ver-\ntices and sorted them in increasing order. Recall that any p-value below the FDR threshold is con-\nsidered signi\ufb01cant, and gives q-values. Fig. 3 shows the sorted p-values, where blue/black dotted\nlines are the FDR thresholds identifying signi\ufb01cant vertices.\n\nFigure 2: Normalized log scale p-values after FDR correction at q = 10\u22125, projected back on a brain mesh\nand displayed. Row 1: Original cortical thickness, Row 2: SPHARM, Row 3: Wavelet Multiscale descriptor.\n\nAs seen in Figs. 2, 3 and 5, the number of signi\ufb01-\ncant vertices is far larger in WMD compared to CT and\nSPHARM. At FDR 10\u22124 level, there are total 6943 (CT),\n28789 (SPHARM) and 40548 (WMD) vertices, showing\nthat WMD \ufb01nds 51.3% and 17.9% more discriminative\nvertices over CT and SPHARM methods. In Fig. 5, we\ncan see the effect of FDR correction. With FDR set to\n10\u22123, 10\u22125 and 10\u22127, the number of vertices that sur-\nvives the correction threshold decreases to 51929, 28606\nand 13226 respectively.\nFinally, we evaluated the regions identi\ufb01ed by these tests\nin the context of their relevance to Alzheimer\u2019s disease.\nWe found that the identi\ufb01ed regions are those that might\nbe expected to be atrophic in AD. All three methods iden-\nti\ufb01ed the anterior entorhinal cortex in the mesial temporal\nlobe, but at the prespeci\ufb01ed threshold, the WMD method\nwas more sensitive to changes in this location as well as\nin the posterior cingulate, precuneus, lateral parietal lobe,\nand dorsolateral frontal lobe. These are regions that are commonly implicated in AD, and strongly\ntie to known results from neuroscience.\nRemarks. When we compare two clinically different groups of brain subjects at the opposite ends of\nthe disease spectrum (AD versus controls), the tests help identify which brain regions are severely\naffected. Then, if the analysis of mild AD versus controls reveals the same regions, we know that the\nnew method is indeed picking up the legitimate regions. The ADNI dataset comprises of mild (and\nrelatively younger) AD subjects, and the result from our method identi\ufb01es regions which are known\nto be affected by AD. Our experiments suggest that for a study where group differences are expected\nto be weak, WMDs can facilitate identi\ufb01cation of important variations which may be missed by the\ncurrent state of the art, and can improve the statistical power of the experiment.\n\nFigure 3: Sorted p-values from statisti-\ncal analysis of sampled vertices from left\nhemisphere using cortical thickness (CT),\nSPHARM, WMD for FDR q = 10\u22123\n(black) and q = 10\u22124 (blue).\n\n6\n\n\fFigure 4: Normalized log scale p-values after FDR correction on the left hemisphere with q = 10\u22127 on\ncortical thickness (left column) , SPHARM (middle column), WMD (right column) repectively, showing both\ninner and outer sides of the hemisphere.\n\nFigure 5: Normalized log scale p-values showing the effect of FDR correction on the template left hemisphere\nusing WMD with FDR q = 10\u22123 (left column), q = 10\u22125 (middle column) and q = 10\u22127 (right column)\nrepectively, showing both inner and outer sides of the hemisphere.\n4.2 Cortical Surface Smoothing without Sphere Mapping\n\nExisting methods for smoothing cortical surfaces and the signal de\ufb01ned on it, such as spherical\nharmonics, explicitly represent the cortical surface as a combination of basis functions de\ufb01ned over\nregular Euclidean spaces. Such methods have been shown to be quite powerful, but invariably cause\ninformation loss due to the spherical mapping. Our goal was to evaluate whether the ideas introduced\nhere can avoid this compromise by being able to represent (and smooth) the signal de\ufb01ned on any\narbitrarily shaped mesh using the basis in Section 3.1 .\nA small set of experiments were performed to evaluate this idea. We used wavelets of varying\nscales to localize the structure of the brain mesh. An inverse wavelet transformation of the resultant\nfunction provides the smooth estimate of the cortical surface at various scales. The same process can\nbe applied to the signal de\ufb01ned on the surface as well. Let us rewrite (3) in terms of the graph Fourier\nbasis, 1\nCg\ncase, the set of scales directly control the spatial smoothness of the surface. In contrast, existing\nmethods introduce an additional smoothness parameter (e.g., \u03c3 in case of heat kernel). Coarser\nspectral scales overlap less and smooth higher frequencies. At \ufb01ner scale, the complete spectrum is\nused and recovers the original surface to high \ufb01delity. An optimal choice of scale removes noisy high\nfrequency variations and provide the true underlying signal. Representative examples are shown in\nFig. 6 where we illustrate the process of reconstructing the surface of a brain mesh (and the cortical\nthickness signal) from a coarse to \ufb01ner scales.\n\n(cid:17) \u02c6f (l)\u03c7l(m) which sums over the entire scale s. Interestingly, in our\n\n(cid:16)(cid:82) \u221e\n\n(cid:80)\n\nl\n\n0\n\ns\n\ng2(s\u03bbl)\n\nds\n\n7\n\n\fThe \ufb01nal reconstruction of the sample brain surface from inverse transformation using \ufb01ve scales\nof wavelets and one scaling function returns total error of 2.5855 on x coordinate, 2.2407 in y\ncoordinate and 2.4594 in z coordinate repectively over entire 136228 vertices. The combined error\nof all three coordinates per vertex is 5.346 \u00d7 10\u22125, which is small. Qualitatively, we found that the\nresults compare favorably with [6, 24] but does not need a transformation to a spherical coordinate\nsystem.\n\nFigure 6: Structural smoothing on a brain mesh. Top row: Structural smoothing from coarse to \ufb01ner scales,\nBottom row: Smoothed cortical thickness displayed on the surface.\nImplementation. Processing large surface meshes with \u223c 200000 vertices is computationally in-\ntensive. A key bottleneck is the diagonalization of the Laplacian, which can be avoided by a clever\nuse of a Chebyshev polynomial approximation method, as suggested by [11]. It turns out that this\nprocedure basically consists of n iterative sparse matrix-vector multiplications and scalar-vector\nmultiplications, where n is the degree of the polynomial.\nWith some manipulations (details in the code release),\nthe processes above translate nicely on to the GPU ar-\nchitecture. Using the cusparse and cublas libraries,\nwe derived a specialized procedure for computing the\nwavelet transform, which makes heavy use of commod-\nity graphics-card hardware. Fig. 7 provides a comparison\nof our results to the serial MATLAB implementation and\ncode using the commercial Jacket toolbox, for processing\none brain with 166367 vertices over 6 wavelet scales as\na function of polynomial degree. We see that a dataset\ncan be processed in less than 10 seconds (even with high\npolynomial order) using our implementation.\n\nFigure 7: Running times to process a single\nbrain dataset using native MATLAB code,\nJacket, and our own implementation\n\n5 Conclusions\n\nWe showed that shape descriptors based on multi-scale representations of surface based signals are\na powerful tool for performing multivariate analysis of such data at various resolutions. Using a\nlarge and well characterized neuroimaging dataset, we showed how the framework improves statis-\ntical power in hypothesis testing of cortical thickness signals. We expect that in many cases, this\nform of analysis can detect group differences where traditional methods fail. This is the primary\nexperimental result of the paper. We also demonstrated how the idea is applicable to cortical sur-\nface smoothing and yield competitive results without a spherical coordinate transformation. The\nimplementation will be publicly distributed as a supplement to our paper.\n\nAcknowledgments\n\nThis research was supported by funding from NIH R01AG040396, NIH R01AG021155, NSF RI\n1116584, the Wisconsin Partnership Proposal, UW ADRC, and UW ICTR (1UL1RR025011). The\nauthors are grateful to Lopa Mukherjee for much help in improving the presentation of this paper.\n\n8\n\n\fReferences\n\n[1] M. Belkin and P. Niyogi. Laplacian Eigenmaps for dimensionality reduction and data representation.\n\nNeural Computation, 15(6):pp. 1373\u20131396, 2003.\n\n[2] Y. Benjamini and Y. Hochberg. Controlling the false discovery rate: A practical and powerful approach\n\nto multiple testing. Journal of the Royal Statistical Society, 57(1):pp. 289\u2013300, 1995.\n\n[3] R. Brown, N. Colter, and J. Corsellis. Postmortem evidence of structural brain changes in Schizophrenia\ndifferences in brain weight, temporal horn area, and parahippocampal gyrus compared with affective\ndisorder. 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Grant, Y. Qi, et al. Cortical surface shape analysis based on spherical wavelets. Med. Imaging,\n\nIEEE Trans. on, 26(4):582 \u2013597, 2007.\n\n9\n\n\f", "award": [], "sourceid": 608, "authors": [{"given_name": "Won", "family_name": "Kim", "institution": null}, {"given_name": "Deepti", "family_name": "Pachauri", "institution": null}, {"given_name": "Charles", "family_name": "Hatt", "institution": null}, {"given_name": "Moo.", "family_name": "Chung", "institution": null}, {"given_name": "Sterling", "family_name": "Johnson", "institution": null}, {"given_name": "Vikas", "family_name": "Singh", "institution": null}]}