Summary of Klein, A., Ghosh, S. S., Bao, F. S., Giard, J., Häme, Y., Stavsky, E., ... & Keshavan, A. (2017). Mindboggling morphometry of human brains. PLoS computational biology, 13(2), e1005350. ## IntroductionBrain images have been used a biomarkers for mental illness, but there are still not a lot of reliable biomarkers out there. A significant impediment to understanding mental health is variation in human brain anatomy etc. The normal variation must first be established to determine what is out of range. To know normal variation, correspondence between brains must first be established, but that is very difficult. registrations methods are highly variable etc. instead neuroanatomists prefer to use high level features, such as distinctive cortical folding patterns etc. To compare these features across individuals we need to quantify them. Current methods for quantification can be via greyscale values in a volume, deformation based morphometry, voxel based morphometry, directly measure shape, volume, surface area, cortical thickness. More subtle shape measures may provide more sensitive and specific biomarkers and combining shape measures in a multivariate analysis could improve results. ## Design and ImplementationMindboggle takes in preprocessed T1 data and outputs volume, surface, and tabular data containing label, feature, and shape information for further analysis. It can be run command line, in python, and via docker. ## Results
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## Hallgrimsson et al. 2015 - Morphometrics, 3D Imaging, and Craniofacial Development (summary)3/13/2018 Summary of: Hallgrimsson, B., Percival, C. J., Green, R., Young, N. M., Mio, W., & Marcucio, R. (2015). Morphometrics, 3D imaging, and craniofacial development. In Current topics in developmental biology (Vol. 115, pp. 561-597). Academic Press. This article/chapter is interesting to me because of its review on morphospaces and morphometrcis. I'll just include notes on the parts particularly relevant to my work. ## Morphometrics and MorphospacesDefinitions: Morphometrics is the quantification and statistical analysis of form.Form is the combination of size and shape of a geometric object in an arbitrary orientation and location.Shape is what remains of the geometry once standardized for size. morphology can be mapped in a systematic way, often in a 'morphospace'. Morphospaces are maps show how shapes are defined by quantitative traits. The morphospace idea is derived from deformation grids from Thompson (1942/61, 1917), but a quantitative morphospace was first used by Raup (1966). The concepts existed but he created the first visual metaphor of the shape relationship between specimens, particularly shell form. His shell morphospace raised questions on why there were no natural specimens in all possible regions of the morphospace. Explanations were that there are developmental constraints and constraints related to shell function. To be useful, morphospaces must have a few key properties: 1. locations and distances in the spaces must have biological meaning. Forms that are similar must cluster togethers and those that are dissimilar must be far apart. 2. directions within the morphospace should have biological meaning. without bio. meaning it is impossible to predict morphologies based on a continuous relationships or determine whether a group of related mutations produce effects int he same direction. 3. colinearlity is required. parralel trajectories in morphospaces should represent comparable shape changes. 4. axes should be independent. Often morphospaces do fail to meet these requirements. ## Approaches to MorphometricsLandmark Based Methodsmost modern morphometric approaches are based on the analysis of landmarks, however, not all are equally useful. Bookstein 1991 classified landmarks into 3 types: Type 1: discrete identifiable points, usually at the intersection of distinct anatomical structures Type 2: points of maximal curvature along definable features. Type 3: defined along extremes that are often defined by other points, Semi-landmarks are a special version of Type 3 landmarks. Semi=landmarks are defined by distributing points across a surface defined by other landmarks. Usually placed in a grid and slid to optimize their position. Geometric Morphometrics (GMM)GMM uses supoerimposition of landmark coordinate data to place individuals into a common morphospace. Typically uses generalized Procrustes Analysis (GPA). This chapter does a nice review of the methods, no need to go into it here. Euclidean Distance Matrix AnalysisEDMA does not use superimposition, instead specimens are represented as a matrix of linear distances between all possible pairs of landmarks. Morphometric distances can be identified as changes in specific linear distances on an object through pairwise comparisons of mean form or shape matrices, using bootstrapping for significance. Image Analysis-Based methods Voxel based, auto landmarking, dense landmarking, statistical shape models ## Quantifying VariationComparing shape and size among groupsVariance-covariance matrix (VCM) - fundamental to morphometric analyses - consists of the set of landmarks by coordinate variances along their diagonal and all of the pairwise covariances in the off diagonal cells. PCA is an intuitive way to express the variation in the matrices. A PCA will use the VCM to create new variable that correspond to progressively smaller proportions of the total variance in the sample. Canonical variates analysis (CVA) orients the data along axes that maximally distinguish groups that are defined a priori. Best used as a data exploration technique. This paper was presented at the MICCAI conference in 2014 paper summary of: Wachinger, C., Golland, P., & Reuter, M. (2014, September). Brainprint: Identifying subjects by their brain. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 41-48). Springer, Cham. ## AbstractIntroducing BrainPrint, a compact and discriminative representation of anatomical structures in the brain. BrainPrint captures shape information of an ensemble of cortical and subcortical structures by solving the 2D and 3D Laplace-Beltrami operator on triangular (boundary) and tetrahedral (volumetric) meshes. We derive a robust classifier for this representation that identifies the subject in a new scan, based on a database of brain scans. In an example dataset containing over 3000 MRI scans, we show that BrainPrint captures unique information about the subject’s anatomy and permits to correctly classify a scan with an accuracy of over 99.8%. All processing steps for obtaining the compact representation are fully automated making this processing framework particularly attractive for handling large datasets. ## IntroductionThis paper is focused on developing a subject-specific brain signature that is stable across time and insensitive to image artifacts. It must also be holistic, so the subject can be identified even if one part changes. To fulfill these requirements, the authors introduce BrainPrint, a holistic representation of brain anatomy, consisting of shape info on cortical and subcortical structures. They use the Laplace-Beltrami operator to calculate an eigenvalue spectrum that represents the shape of the object. The structures they use come from Freesurfer and consist of both the boundary surfaces (pial and white matter) and the individual structures. To get a robust identification, they let each brain structure vote independently for the individual. The classifier they use can identify previously encountered subject with high accuracy and can also determine whether the subject exists in the database already or not. BrainPrint is particularly beneficial for large datasets. First, they segment the anatomical structures from the anatomical T1 image, the they essentially transfer this info into a compact and discriminative representation. The representation takes up very little memory and allows for easier calculations and comparisons that the original whole image. ## Related Work3D objects can be represented as a 3D volume or a 2D boundary (surface) BrainPrint is based off of shape-DNA by Reuter and use both the volume (tetrahedral mesh) and surface meshes. Although shape DNA has been used with single structures before, this is the first paper to apply it to multiple structures and cortical structures. ## Shape DescriptorThe shape descriptor used is the Laplace Beltrami spectrum. See paper for details on its computation. ## ClassifierThey built a classifier that combines results from weaker classifiers working on specific brain structures. They use one that uses all structures and one that combines votes from the structures and uses the mode vote. ## ResultsThey use the ADNI dataset and calculate a shape descriptor for 36 subcortical structures and 8 cortical structures. They also calculate the asymmetry between the left/right cortical structures.
They achieved an accuracy of 99.8% using 50 eigenvalues on all features with the asymmetry features. paper summary of: Wachinger, C., Salat, D. H., Weiner, M., Reuter, M., & Alzheimer’s DiseaseNeuroimaging Initiative. (2016). Whole-brain analysis reveals increased neuroanatomical asymmetries in dementia for hippocampus and amygdala. Brain, 139(12), 3253-3266. ## AbstractStructural magnetic resonance imaging data are frequently analysed to reveal morphological changes of the human brain in dementia. Most contemporary imaging biomarkers are scalar values, such as the volume of a structure, and may miss the localized morphological variation of early presymptomatic disease progression. Neuroanatomical shape descriptors, however, can represent complex geometric information of individual anatomical regions and may demonstrate increased sensitivity in association studies. Yet, they remain largely unexplored. In this article, we introduce a novel technique to study shape asymmetries of neuroanatomical structures across subcortical and cortical brain regions. We demonstrate that neurodegeneration of subcortical structures in Alzheimer’s disease is not symmetric. The hippocampus shows a significant increase in asymmetry longitudinally and both hippocampus and amygdala show a significantly higher asymmetry cross-sectionally concurrent with disease severity above and beyond an ageing effect. Our results further suggest that the asymmetry in these structures is undirectional and that primarily the anterior hippocampus becomes asymmetric. Based on longitudinal asymmetry measures we subsequently study the progression from mild cognitive impairment to dementia, demonstrating that shape asymmetry in hippocampus, amygdala, caudate and cortex is predictive of disease onset. The same analyses on scalar volume measurements did not produce any significant results, indicating that shape asymmetries, potentially induced by morphometric changes in subnuclei, rather than size asymmetries are associated with disease progression and can yield a powerful imaging biomarker for the early presymptomatic classification and prediction of Alzheimer’s disease. Because literature has focused on contralateral volume differences, subcortical disease lateralization may have been overlooked thus far. ## IntroductionThis paper is looking at subcortical structures of the brain using the spectral shape descriptors from Wachinger/Reuter's BrainPrint. This is essentially the Laplace Beltrami spectrum of the shapes. They are also looking at asymmetry of the structures in particular. The author's propose using the BrainPrint method for four reasons: 1. it avoids lateral processing bias as it works on the two hemispheres independently 2. it does not require prior spatial alignment 3. it is a brain-wide analysis 4. it is a within-subject measure that identifies directional and undirectional asymmetry directional asymmetry: hemispheric differences that show a stronger effect on one of the hemispheres. undirectional asymmetry: no consistent hemispheric effect, magnitude of asymmetry independent of direction Their results suggest that there is a strong increase in undirectional asymmetry with the progression of dementia. They used an unstructured multi-cohort longitudinal data from ADNI. Mixed effects models were used to differentiate across- and within-individual variations in asymmetry. ## Materials and MethodsBrainPrint is based on the automated segmentation output from Freesurfer. The subjects in this study were processed with the longitudinal framework in Freesurfer. The marching cubes algorithm was used to create surface and volumetric meshes of the segmented structures. shapeDNA was used as the shape descriptor (aka the Laplace Beltrami Spectrum). They used 50 eigenvalues in this study, normalized by volume/surface area. The eigenvalues of the Laplace Beltrami spectra are isometry invariant, meaning length-preserving deformations will not change the spectrum. This also includes, rigid body motion and reflections. No alignment is needed. The figure below shows the first 6 non-constant eigenfunctions of the hippocampus. The eigenfunctions show natural vibrations of the shape when oscillating at a frequency specified by the square root of the eigenvalues. To localize the shape changes they use the level set analysis for the first eigenfunction as proposed in Reuter et al. 2009. this is show as green curves. They computed the circumference of 100 level sets and avearage among 10. ## AsymmetryBecause shape DNA is invariant to reflections, they directly computed the Mahalanobis distance between the lateralized brain structures. "The asymmetry measure presents a within-subject measure that can identify directional and undirectional asymmetry. The difference of eigenvalues can be used to differentiate directional and undirectional asymmetry. We compute the asymmetry for 11 lateralized structures: cerebral white matter, pial region, cerebellum white/grey matter, lateral ventricles, hippocampus, amygdala, thalamus, caudate, putamen, and accumbens. For white matter and pial region, the analysis is performed on volumetric meshes." ## ResultsThey used 697 individuals from ADNI.
They used a mixed effects model with a global intercept B0, age at baseline B1, years from baseline B2 and diagnosis (control, MCI stable, MCI progressor, Alzheimers AD). To be finished... ## Hu et al. - Visualizing Shape Deformations with Variation of Geometric Spectrum (paper summary)1/4/2018 Paper summary of: Hu, J., Hamidian, H., Zhong, Z., & Hua, J. (2017). Visualizing Shape Deformations with Variation of Geometric Spectrum. IEEE transactions on visualization and computer graphics, 23(1), 721-730. This is a continuation of the MICCAI 2016 paper summarized here. ## IntroductionSee paper for short summary of other shape analysis methods. Shape spectrum is inspired by the Fourier transform in signal processing. basically the same thing but in 3d. See paper for a short review on the publications on the laplace beltrami spectrum. "In this paper, we focus on spectrum alignment of general shapes using the eigenvalue variation in order to quantify the non-isometric deformations between surface shapes. In our approach, shapes are au-tomatically aligned by calculating the metric scaling on both shapes. Our method defines the surface shape deformation by the variation of Laplace-Beltrami spectrum of the shapes and quantifies the multi-scale deformations through the use of different sets of eigenvalues. Compared to the traditional approaches, it can detect and localize small non-isometric deformations in addition to global difference of the shapes, without using any defined landmarks on the manifolds. This is because the spectrum only depends on the intrinsic geometry of the shape and is invariant to spatial translation, rotation, scaling and isometric deformation. This method is computationally affordable and suitable to map surface shapes for non-isometric deformation analysis." Related works: Good review of ICP method and spectral methods. Main contributions of the paper: - We prove a theorem that the shape spectrum is a piecewise analytic to a scale function defined on the Riemann metric of the manifold. Our theorem holds in both continuous domains and discrete triangle meshes and supports the variation of eigen-values for non-isometrically deformed shapes. Therefore, it en-ables a quantitative method for deformation analysis.
- We present a spectrum alignment algorithm for triangle meshes, supporting non-isometric deformation analysis. In the discrete domain, the variation of eigenvalues in terms of the scale vector can be turned into a matrix form, which introduces a linear system. Together with the smoothness and local bound constraints, the linear system can be solved efficiently. After the eigenvalues are matched, the eigenfunction distributions are aligned as well. This means the shape spectrum can be controlled analytically through a scale vector, hence, non-isometric analysis is available within shape spectra.
- Our developed system verifies the spectrum variation the-orem and demonstrates the accuracy and efficiency of the spectral variation algorithm on visualization of non-isometrically deformed shapes. The applications to biomedical imaging problems show that it is a viable solution for morpho-metric analysis and visualization in biomedical applications and clinical diagnoses.
## Methods, Theorems etc.see paper, too hard to summarize adequately ## Experiments They did 3 main 'experiments'. One is the same as the one reported in the 2016 paper so I will not summarize it again. Sensitivity analysis To test the sensitivity of the model they used the Bunny model with 3000 vertices and made a bump on the back of the bunny. They tested the method with 30, 50, and 80 eigenvalues and 10 iterations. The more eigenvalues the better the results. Figure 2 shows the results. They also tested 80 eigenvalues with 1, 5, and 10 iterations. Figure 3 shows the results. ## Alzheimer Study and DiagnosisThey looked at 10 healthy and 10 AD patients with longitudinal data. Globally: The AD hippocampi both right and left are shrunk after one year. The average shrinkage for AD subjects is more than healthy subjects. There is more shrinkage in the left than the right. locally: the middle region (CA1) is affected more serverely in both left and right. see the figure. ## Comparisons to Spatial Registration MethodsThey compare their method to ICP and voxel based methods and find that their method comes out on top and does not need correspondence. See the paper for figures and details.
Paper summary for: Hamidian, H., Hu, J., Zhong, Z., & Hua, J. (2016, October). Quantifying Shape Deformations by Variation of Geometric Spectrum. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 150-157). Springer International Publishing. ## OverviewI am quite excited about this paper. It provides a way to determine local shape differences using the Laplace Beltrami spectrum. The Laplace Beltrami spectrum is typically a global shape analysis method. Here is the abstract: This paper presents a registration-free method based on geometry spectrum for mapping two shapes. Our method can quantify and visualize the surface deformation by the variation of Laplace-Beltrami spectrum of the object. In order to examine our method, we employ synthetic data that has non-isometric deformation. We have also applied our method to quantifying the shape variation between the left and right hippocampus in epileptic human brains. The results on both synthetic and real patient data demonstrate the effectiveness and accuracy of our method. ## IntroductionTwo types of methods for mapping shapes: spatial registration methods and spectral methods spatial registration methods: require landmarks to map two shapes, often labor intensive, landmarks can be difficult to define on some objects spectral methods: do not require landmarks, or alignment, invariant to isometric deformation inspired by Fourier transform in signal processing can only describe the global difference between shapes In this paper, "they focus on spectrum alignment of general shapes using the eigenvalue variation and present a spectral geometry method for localizing and quantifying non-isometric deformations between surface shapes." "A recent study shows that the shape spectrum can be controlled using a function on the Riemann metric" The method described in this paper uses the laplace-beltrami spectrum to detect and localize small deformations in addition to global difference of shape. it is registration free and does not need landmarks. It is invariant to spacial translation, rotation, scaling, and isometric deformation. ## Methodsee paper for detailed description of the methods ## Experiments and ResultsThey tested their method with hippocampi surface meshes. The surfaces have about 5000 vertices. They used the values k = 100 and K = 10 (see paper for details). (100 eigenvalues). They first test their method with synthetic data by generating non-isometric deformations on a single hippocampus. A hippocampus is segmented from a brain MRI image. The surface was then deformed using Blender. They apply their spectrum alignment to the first 100 non-zero eigenvalues. Figure 1 describes the results of these tests: They compare their methods to the non-rigid Iterative Closest Point method and found their method to have higher accuracy (92% vs 81%) (time <20s vs >60s ) ## Epilepsy imaging testThey also apply their method to an epilepsy study, again looking at the hippocampus. Epilepsy may cause shrinkage of the affected hippocampus (one side vs. the other). They applied the method to 20 subjects with epilepsy to quantify the shape variation between left and right hippocampus. The right hippocampi were then mirrored to have the same direction as the left ones. Figure 2 shows the results of the shape comparisons. Columns 1 and 2 show the left and right hippocampi. A and B show examples of 2 different subjects. The third column shows the computed scale function distributions on the left hippocampus surface when mapping from the left to the right. Red means dilating and blue means contraction. White means no distortion. A is for a patient with a left abnormal hippocampus, so mapping from left to right should show expansion. B is for a patient with a right abnormal hippocampus. so it should show contraction. C is showing how the eigenfunction vary after the eigenvalues are mapped to the right one. ## ConclusionThis paper presents a registration free method to quantify the deformation between shapes. It can handle complicated meshes with more than 5000 or 10000 vertices.
Paper summary for: Wachinger, C., Rieckmann, A., & Reuter, M. (2017, September). Latent Processes Governing Neuroanatomical Change in Aging and Dementia. In International Conference on Medical Image Computing and Computer-Assisted Intervention(pp. 30-37). Springer, Cham. ## OverviewIn this article, they look at using the shape of brain structures in combination with behavioral/cognitive data to identify disease-related changes in brain morphology from normal age related changes. This paper uses BrainPrint and does a PLS analysis. ## IntroductionAging is characterized by a multifaceted set of neurobiological changes that occur at different rates in different people with complex and interdependent effects on cognitive decline. Some of these changes are normal, but some are due to specific diseases. Some may be similar to a disease but arise from a different cause. This study focuses on differentiating which morphological brain changes are normal and which are associated with the development of Alzheimer's. Structural changes in the brain do not occur uniformly across the cortical surface. Some regions are likely affected by disease and age related changes, while others may be affect only by aging. Paper gives the example of the striatum - it is affected by aging, but not Alzheimer's disease. In order to take advantage of the heterogeneity of aging and disease-related effects, this paper looks at joint modeling changes across many structures rather than focusing on single structures alone. It is understood that there are many underlying processes that can cause morphological changes - modeled here as latent factors. It is also known that aging and disease can cause different changes in different subregions of specific structures. Volume or surface area measurements of these structures may not show any difference, but there may be noticeable shape changes. 'To obtain a discriminative characterization of neuroanatomy' the authors uses "BrianPrint' (see link above). BrainPrint calculates the Laplace Beltrami Spectrum for 3D surfaces of cortical areas of the brain segmented using Freesurfer. They did a cross-sectional and longitudinal study. They 'identify neuroanatomical processes that are best associated to aging and disease by maximizing the covariance between morphology and response variables, yielding the projection of the data to latent structures. ' See paper for short paragraph on related work. ## MethodThey used BrainPrint to represent the brain morphology based on the automated segmentation of anatomical brain structures with Freesurfer. BrainPrint uses ShapeDNA (the Laplace Beltrami Spectrum). They normalized the spectra by surface area and eigenvalue index. Normalizing by index balances the impact of higher eigenvalues that typically show higher variance. They applied this method to both cross-sectional data and for longitudinal data. Latent Factor Model The looked at the morphology and response variables - age and performance on the mini-mental state examination (MMSE) (clinical screening for loss of memory and intellectual abilities). Objective is to extract the latent variables that account for much of the factor variation in the data. To take the response variables into account they used PLS or projections to latent structures or partial least squares. PLS combines information about both the predictors and the resonses and the correlations among them. See the paper for details of applying the PLS. ## ResultsThey used ADNI data. subjects with baseline scans and with follow up scans after 6, 12, and 14 months. 393 subjects total. The diagnostic groups were Healthy Control (HC), Mild Cognitive Impairment (MCI) and Alzheimer's Disease (AD). Meshes from 23 brain structures were used and 5 eigenvalues. Of the components reported from the PLS ( Iguess it is similar to PCA) for the longitudinal analysis, two were significantly correlated only with Age and two were significantly correlated only with the MMSE Figures 1 and 2 show the 4 processes for the longitudinal data. The first a third are related to progression of dementia. The color of the brain areas are related to importance. The first process shows opposing effects on the hippocampus and amygdala vs. the lateral and third ventricle. The authors suggest this reflects the typical brain changes associated with dementia, shrinkage of the hippocampus and amygdala and expansion of the ventricles. the third process is showing opposing effects on the amygdala vs. hippocampus. This suggests that there are two separable dementia related processes. For the 2nd and 4th processes the weights of the amygdala and hippocmapus are lower than for the dementia processes, and higher for the ventricles. Lastly, they evaluated the predictive performance for the latent factor model compared to traditional multiple linear regression and with volume instead of shape in the PLS model. The mean absolute prediction error was significantly lower in the PLS with shape model than the other 2. Summary notes for: Niethammer, M., Reuter, M., Wolter, F. E., Bouix, S., Peinecke, N., Koo, M. S., & Shenton, M. E. (2007, October). Global medical shape analysis using the Laplace-Beltrami spectrum. In International Conference on Medical Image Computing and Computer-Assisted Intervention(pp. 850-857). Springer, Berlin, Heidelberg. ## OverviewThis is a paper from the MICCAI 2007 conference proposing using the Laplace Beltrami Spectrum as a global shape descriptor for medical shape analysis. This method is particularly useful because it allows for shape comparisons with minimal preprocessing. No registration, mapping, or remeshing is required. They demonstrate the usefulness on a population of female caudate shapes, comparing normal control to subjects with schizotypal personality disorder. ## Motivation and BackgroundBefore this time (2007), morphometric studies of the brain focused on volumetric measurements. Now the method in this paper and other non volumetric methods are popular. This is a GLOBAL method. There is a difference between local shape differences and global shape differences. Local shape comparisons: compare spatially localized areas, straightforward to interpret, but rely on a lot of preprocessing steps. there must be one-to-one correspondence between the two surfaces/areas being compared. The shapes have to be registered and aligned to each other, possibly resampled.Global shape comparisons: cannot necessarily show spatially localized shape changes, but can be used to indicate shape differences between groups. can reduce the preprocessing steps needed. This paper describes a methodology for global shape comparison using the Laplace Beltrami Specturm (LBS) of a Riemannian manifold. Previous methods for global shape analysis include: use of invariant moments, shape index, spherical harmonics (check paper for citations). This method differs from those in these ways: - it works for any reimannian manifold. spherical harmonics requires surfaces with spherical topology - the only preprocessing is the extraction of a surface approximation from a manually segmented binary volume. NO registration, remeshing, or additional mapping required. Invariant to isometries and remeshing.Isometries include rotation, reflection, translation, meaning that it doesnt matter if the object is rotated or reflected etc. the result is the same. ## Shape DNA - the Laplace Beltrami SpectrumThey refer to the LBS as the shape dna of an object. The LBS is a signature computer from the intrinsic geometry of an object. The description in the paper:It is the beginning sequence of the spectrum of the LB operator defined for real valued functions on Riemannian manifolds and can be used to identify and compare objects like surfaces and solids independent of their representation, position and, if desired, independent of their size. The LBS can be regarded as the set of squared frequencies (the so called natural or resonant frequencies) that are associated to the eigenmodes of an oscillating membrane defined on the manifold. (E.g., the eigenmodes of a sphere are the spherical harmonics.) This method was first described in: Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: SPM ’05: ACM Symposium on Solid and Physical Modeling, ACM Press (2005) 101–106 and a first description of the basic ideas were given in: Wolter, F.E., Friese, K.: Local and global geometric methods for analysis, interrogation, reconstruction, modification and design of shape. In: CGI’00. (2000) 137–151 See the paper for a review of the 'basic theory' behind the LBS. The spectrum itself is defined to be the family of eigenvalues of the Laplacian Eigenvalue Problem consisting of a diverging sequence 0 < ev1 < ev2 < etc. with each ev repeated according to its multiplicity.The spectrum is ISOMETRIC INVARIANT. it only depends on the gradient and divergence which are depending only on the Riemannian structure of the manifold (the intrinsic geometry). Scaling an n-dimensional manifold by factor a results in the eigenvalues being scaled by a factor 1/a^2. The eigenvalues can be normalized, allowing shape to be compared regardless of scale and position. Normalization and comparison: When unnormalized spectra are compared, higher eigenvalues will lead to better discrimination between the two groups, but most of that difference will be due to the difference in surface area. By normalizing by surface area you can see if there are additional shape differences. Identical noise levels are require for accurate results. For shapes acquired using the same method (manual segmentation etc.) identical noise levls is a safe assumption. ## Statistical Analysis of groups of LB spectraHere they used a maximum t-statistic permutation test. see paper for more details. ## ResultsPopulations studied:
MR images of 32 females with Schizotypal Personality Disorder (SPD) and 29 female normal controls. Preprocessing: Caudate nucleus delineated manually, isosurfaces extracted using marching cubes. analysis was performed on the surfaces directed and on 'smoothed' surfaces that were smoothed using spherical harmonics. Volume and Area Analysis: SPD subjects have less volume and surface area. smoothing the surfaces did not affect volume much, but did reduce the surface area. however the relationship between the groups was preserved. LBS analysis; they normalized by surface area. A maximum t-statistic permutation test on a 100 dimensional spectral shape descriptor shows significant shape differences for surace area normalization for the unsmoothed surfaces, but not the smoothed surfaces. The use of too many evs can be detrimental as it introduces noise. Higher order modes correspond to higher frequencies and are thus more likely noise. This spring I am going to follow along with the posted video lectures from Justin Solomon's Shape Analysis course from spring 2017. Justin video taped most of his lectures and posted the videos on youtube along with his slides on his course website. The videos can be found here and the course website is here. The specific name and number for the MIT course is 6.838 Shape Analysis. I plan to make a 'blog post' of my notes for each lecture. This is really just to help me keep tract of the information I learn and to force me to take notes. It may also help others learn more about shape analysis, but no guarantees! Click on the category SA Lecture Notes for all the lectures. ## Lecture 1Video for lecture 1
Differential Geometry is the study of smooth manifolds. A manifold is a topological space that locally resembles Euclidian space near each point. A surface is a 2D manifold, meaning that at any and all points on the surface, locally it looks like a Euclidean plane. Here is a useful way to explain it from WolframMathWorld: To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem. Differential Geometry 'Toolbox': Gaussian Curvature Mean Curvature Distances - Geodesic distance - distance along the shape Flows and Vector fields Differential Operators - spectral geometry - hit a shape with a hammer, hear the set of frequencies, based on the set of frequencies can you tell what the shape looks like? turns out yes, because it is really hard to find two objects that 'sound' the same. Computational Toolbox: Many notions of shape: can be a triangular mesh, graph, point cloud, pairwise distance matrix, really can be nearly anything with a notion of proximity/distance/curvature. Triangular mesh: collection of triangles, approximation of a smooth surface where is the curvature on a triangular mesh? the triangles are technically flat... Discrete Differential Geometry: asks the question: can i develop a discrete theory of diff. geometry from the ground up based on triangles and angles, parallel to differential geometry two important things you want: Structure Preservation: keeping properties from the continuos abstraction exactly true in a discretization Convergence: increasing approximation quality as a discretization is refined but you cannot have both perfectly. Have to pick and choose which are needed most for your purposes. |
I will be using this blog space as a repository of notes on various articles and lectures related to my research. Click on the categories below to organize the posts by topic.
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