The Anatomical Equivalence Class Formulation and its Application to Shape-based Computational Neuroanatomy

Transcription

1 The Anatomical Equivalence Class Formulation and its Application to Shape-based Computational Neuroanatomy Sokratis K. Makrogiannis, PhD From post-doctoral research at SBIA lab, Department of Radiology, Univ. of Pennsylvania during

2 Outline Computational Neuroanatomy Introduction Existing morphological descriptors Open problems Joint deformation and residual representation Description of proposed descriptor Equivalence class formulation-generation of class members Distances between eq. classes Validation-Statistical analysis Subspace decomposition Main idea-anatomical Equivalence Classes Decomposition to principal subspace and the complementary orthogonal Validation Conclusion and Discussion

3 Introduction The objective of this research is to study and analyze the morphometry of the human brain in order to characterize group differences and detect abnormalities caused by aging and/or brain diseases. The contemporary imaging technology offers in vivo and non-invasive means of studying and analyzing the brain structures. This facilitates the analysis of data from many individuals and longitudinal studies which may reduce the effects of intersubject variabilities. Those approaches belong to the general category of Computational Anatomy.

4 Computational Anatomy Its basic principle is that morphological characteristics of an individual anatomy are represented by a shape transformation that maps a template of the anatomy of interest to the individual anatomy. Following this normalization volumetric or deformation field quantities are typically measured in order to distinguish between healthy and diseased subjects. Main principles are to: Define a reference template as a measurement unit Warp template to subject, by applying a shape transformation Compare different subjects by their respective transformations.

5 Registration Approaches The key component of these methods is the stage of finding the spatial transformation from a volumetric image to another. Literature of registration algorithms is richmethods of this category can be divided into Feature-matching Image-matching. Advantage of the first category is that they use anatomically distinct features, however the image-based employ the complete dataset information to find the requested deformation.

6 Shape-based Limitations Computational anatomy approaches rely on the accuracy of the spatial normalization process to correctly estimate the shape features. Perfect anatomical correspondence is difficult to be reached because: Registration algorithms extract features from visual information only, which does not guarantee correspondence of anatomies In many cases a meaningful correspondence cannot be defined due to basic structural differences in the brain. The accuracy of those approaches is therefore limited by the accuracy of the deformation process. Frequently registration residuals provide substantial information that is typically not considered.

7 Deformation-based Limitations A perfect correspondence between different individuals is difficult to be reached. Example

8 The general framework An image descriptor for morphometric analysis of brain structures is presented that combines shape and residual information to derive a complete representation of the subject variability. The efficacy of this descriptor will be evaluated in terms of separating groups of normal and diseased subjects on labelled data. A subspace decomposition algorithm will be presented to learn distances between brain anatomies.

9 Scheme Outline Acquired Data Validation Classifier Training & Testing Pre-processing Segmentation Spatial Normalization Feature extraction & selection Basic Methodology Segmentation of subject images Spatial Normalization to the common template Calculation of image descriptors Statistical Analysis to estimate class separation on labelled data Classification of un-labelled data

10 Spatial Normalization and Extracted Features Subject Normalized Subject Jacobian Determinant Residual Image Template

11 Outline Computational Neuroanatomy Introduction Existing morphological descriptors Open problems Joint deformation and residual representation Description of proposed descriptor Equivalence class formulation-generation of class members Distances between eq.. classes Validation-Statistical analysis Subspace decomposition Main idea-anatomical Equivalence Classes Decomposition to principal subspace and complementary orthogonal Validation Conclusion and Discussion

12 Joint Warping and Residual Representation In a spatial normalization framework a morphological representation A is provided by the warping h and residual complement R. Pairs of (h,r) completely reconstruct the target anatomy, even when a perfect anatomical correspondence cannot be found. Combining two different entities is a challenging task, since they have completely different properties. Moreover, in many practical applications, especially in neuroimaging, we are interested primarily in the spatial distribution of brain tissue. From this perspective, a specific combination of h and R is constructed which is directly related to the spatial distribution of anatomical tissues in the target anatomy.

13 Tissue Density Maps (TDMs) The considered image descriptor is based on the concept of mass preservation during the shape transformation from the subject to template or the reverse. When a shape transformation is applied between two images, the tissue density increases in the areas of contraction, while decreasing when expansion is applied, so that the total mass of the structure remains constant. The tissue density map D associated with each tissue type is calculated by: Principle

14 Spatial Transform and TDMs Tissue density in the template stereotaxic space is proportional to the volume of the respective original (subject) structure. Subject A Subject B Template

15 Equivalence Class Representation The tissue density map calculation incorporates the deformation and residual information and consequently forms a lossless descriptor. Furthermore, since for each warping exists a unique residual image, this transform produces equivalence classes in the tissue density feature space providing an original formulation for morphometric analysis. Distances between anatomies can be defined by the minimum distance between the eq. sub-classes elements. D S2 S1 h

16 Morphological Descriptor Validation In order to assess the performance of the considered descriptor it was compared with the Jacobian of Deformation field feature in the following context: Statistical testing Classification Accuracy The equivalence class formulation was examined by producing deformation fields and TDMs for multiple registration accuracies and measuring the statistical performance.

17 Statistical Testing At the first stage the statistical significance of image descriptor was evaluated by applying a t-test to the employed feature vectors coming from different subject groups. The null hypothesis is that the two groups come from the same population. The resulting p-values represent the probability of error rejecting the null hypothesis of no difference between the two groups when hypothesis is true. Two different approaches were used here to apply the hypothesis test. First the feature vectors were averaged over a ROI and the t-test was applied to the average vectors. The second approach was to apply the statistical test voxel-wise and calculate the geometric average of p- values.

18 Statistical Significance Jacobian of Deformation Field TDMs geom. average of p values geom. average of p values p-values 1.00E E E E E E E E σ p-values 1.00E E E E E E E E σ E-08 λ 1.00E-08 λ

19 Classification Accuracy The separation capabilities of the considered descriptors are compared in the context of classification accuracy. Therefore, pattern classifiers are built upon the two descriptors using different feature vectors. In order to reduce the high dimensionality of the data the PCA approach is employed here that seeks the linear projections that represent the data optimally in the least squares sense. By selecting the direction of the more extended spread of the data set the dimensionality of the data is reduced. The k Nearest Neighbor (k-nn) scheme is used for classification of our labeled data. It originates from non parametric probability density estimation using window functions by growing the estimation volume according to the number of total samples. The cross-validation is carried out by the leave-one-out approach. This process is repeated until all samples have been excluded once and the classification accuracy is finally estimated.

20 Classification Accuracy- Jacobian of Deformation Field One λ The 2-D synthetic dataset consists of 40 images divided into two groups, with 20 subjects each. One of the groups corresponds to normal cases and the other to simulated atrophy. Class. Methods: 1) voting to the k NNs of each sample. 2) voting to Class. Method 1 results of all samples that originate from the same subject. 3) find minimum group-wise distance. CA CA λ Multiple λ s k k Class. Rule 1 Class. Rule 2 Single λ-average

21 Classification Accuracy-TDM The multi-scale version of the tissue density map-based descriptor produces better separation than single scale processing. This is further enhanced when the different samples from the same subject are grouped in the nearest neighbor finding process (second classification rule in figure). Descriptor CA CA One λ Multiple λ s λ k k Class. Rule 1 Class. Rule 2 Single λ-average

22 3-D Data: Warping and Residual The residual image carries significant information that cannot be expressed by shape-based only analysis. A complete representation therefore required for accurate morphometric analysis. TDMs are employed here to combine those two features. Information Top left: Template Top right: Subject Bottom left: Normalized Template Bottom right: Residual Image

23 TDMs: Multiple Levels of Registration Rigidity Elements of equivalence sub-classes are generated by smoothing out the deformation field and then calculating the Tissue Density Maps

24 Elements of Anatomical Equivalence Class Warping and Residual information is complementary: For smoother deformations the residual carries most of the morphological information As the normalized subject resembles the template, deformation is more substantial.

25 Classification Accuracy-TDM Our morphometric descriptor was also tested on a volumetric dataset of 38 normal controls and 23 schizophrenia patients. We concluded that the multi-scale representation considerably improves the class separation especially for smaller values of k. The slightly better performance of the second classification rule implies the construction of equivalence classes by the tissue density-based feature. Descriptor CA CA One λ ls Multiple λ s k k Τhree ls-c. Rule 1 Τhree ls-c. Rule 2 Single ls

26 Outline Computational Neuroanatomy Introduction Existing morphological descriptors Open problems Joint deformation and residual representation Description of proposed descriptor Equivalence class formulation-generation of class members Distances between eq. classes Validation-Statistical analysis Subspace decomposition Main idea-anatomical Equivalence Classes Decomposition to principal subspace and complementary orthogonal Validation Conclusion and Discussion

27 Subspace Decomposition for Morphological Representation TDMs provide a joint warping and residual representation that completely reconstructs the target anatomy and divides the feature space into eq. classes of pairs (h,r). As a next step we estimate a subspace of the tissue density map space that approximates the range of variation of TDMs mainly due to the registration rigidity parameter λ, denoted by Anatomical Equivalence Class (AEC). Our aim is to remove this variation by projecting D to the orthogonal to principal subspace. This yields morphological representations that are largely independent to λ and facilitate the comparisons between different subjects.

28 Subspace Decomposition Process The mean TDMs are first estimated over each subject. Each sample is centered relative to the mean sample of the subject it belongs to. We use PCA to estimate the hyperplane that approximates the variation of TDMs because of λ, denoted here as principal subspace. The projections of original samples to the orthogonal to principal subspace are calculated.

29 Subject Centering and PCA The mean TDMs are first estimated over each subject. Each sample is centered relative to the mean sample of the subject it belongs to.

30 AEC Decomposition We use PCA to estimate the hyperplane that approximates the variation of TDMs because of λ, denoted here as principal subspace. The projections of original samples to the orthogonal to principal subspace are calculated.

31 Orthogonal Subspace Properties The orthogonal components remove much of the variability due to λ, and therefore yield tissue density measurements that represent true morphological variability and not variability due to the position of a measurement along its AEC. We use those projections as our morphological measurements for any subsequent morphological analysis. The validation of this scheme follows the same principles with our previous analyses.

32 Data Experiments Synthetic 1-D Data of time pulses Volumetric Dataset with Simulated Local Atrophy Validation Criteria Statistical tests Classification rates

33 1-D Problem The equivalence relation used to generate the equivalence classes was the property of similar pulse area. The intrinsic dimensionality of this manifold is 1 and the confounding factor is the pulse area. This is in straight analogy with the tissue density maps concept where the total volume of a 3-D image is preserved within the AEC.

34 Decomposition Results \

35 3-D Data: Simulation of Local Brain Atrophy We picked 30 subjects from a normal cohort, i.e. female gender and age under 40 years and divided this set into two groups of equal population. We selected a voxel located inside the precentral gyrus of our atlas and determined spherical area around this point. Spatially normalized the template to each subject and calculated the location of the spherical area on the subjects.

36 Local PCA and Decomposition TDM-one subject Local Feature Patch Original TDM OAEC

37 T-test on Original Features To ensure that the test data are not statistically significant we apply statistical testing to the original Tissue Density Maps and Jacobians. Those tests indicate that there is no obvious difference between the examined populations p-values Tissue Density Maps-Original λ Jacobian-Original λ σ σ p-values

38 Eigenvalues of Morphological Principal Components Eigenvalues of morphological features are depicted using varying Gaussian kernel widths. A moderate smoothing operation reduces the discontinuities of the Eigenvalue distribution.

39 T-test on Projected Features Statistical results of the AEC principal components and their orthogonal complement.

40 Classification Rates The classification rates using projections the AEC concept are compared with the Jacobian-based feature using single or multiple λs and different distance measures.

41 Conclusions A novel morphological descriptor was presented that overcomes weaknesses of traditional shapebased analysis. It utilizes a joint warping and residual representation by estimating the local tissue density and divides the feature space into equivalence classes of subject anatomies. This representation produces improved group separation compared to the shape-based feature provided by the Jacobian of the deformation field. A subspace decomposition approach of the tissue density maps is also proposed to learn distances between brain anatomies. This represents a linear manifold embedding algorithm in the space of morphological representations and yields promising results on simulated and real datasets.

42 Thanks to: Section of Biomedical Image Analysis Department of Radiology-School of Medicine Univ. of Pennsylvania Team members: C. Davatzikos, PhD D. Shen, PhD B. Karacali, PhD R. Verma, PhD