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. 2011 Dec;15(6):814-29.
doi: 10.1016/j.media.2011.06.003. Epub 2011 Jul 28.

Morphological appearance manifolds for group-wise morphometric analysis

Nai-Xiang Lian  1 , Christos Davatzikos
Affiliations

Morphological appearance manifolds for group-wise morphometric analysis

Nai-Xiang Lian et al. Med Image Anal. 2011 Dec.

Abstract

Computational anatomy quantifies anatomical shape based on diffeomorphic transformations of a template. However, different templates warping algorithms, regularization parameters, or templates, lead to different representations of the same exact anatomy, raising a uniqueness issue: variations of these parameters are confounding factors as they give rise to non-unique representations. Recently, it has been shown that learning the equivalence class derived from the multitude of representations of a given anatomy can lead to improved and more stable morphological descriptors. Herein, we follow that approach, by approximating this equivalence class of morphological descriptors by a (nonlinear) morphological appearance manifold fitting to the data via a locally linear model. Our approach parallels work in the computer vision field, in which variations lighting, pose and other parameters lead to image appearance manifolds representing the exact same figure in different ways. The proposed framework is then used for group-wise registration and statistical analysis of biomedical images, by employing a minimum variance criterion to perform manifold-constrained optimization, i.e. to traverse each individual's morphological appearance manifold until group variance is minimal. The hypothesis is that this process is likely to reduce aforementioned confounding effects and potentially lead to morphological representations reflecting purely biological variations, instead of variations introduced by modeling assumptions and parameter settings.

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Figures

Figure 1
Figure 1
Morphological Appearance Manifold
Figure 2
Figure 2
Manifold of varying (a) intermediate templates or (b,c) regularization parameters. (b,c) represent different images. Each sample on manifolds represents a CMD.
Figure 3
Figure 3
Scatter plot between distance of the TDMs and distance of the residuals of 10000 groups of randomly selected samples
Figure 4
Figure 4
Simulated 2D shapes resembling sulci of the human brain. Some shapes (a) don't, and some shapes (b) do have thinning in the middle portion (middle third) of the ribbon
Figure 5
Figure 5
The minimum p-value vs. the value of μ for simulated 2D subjects
Figure 6
Figure 6
The simulated atrophy at the superior temporal gyrus (right image).
Figure 7
Figure 7
The root-MSE of linear curve fitting vs. the value of μ for simulated 3D MR images
Figure 8
Figure 8
Manifold-constrained optimization: the AEC of each individual resembles a manifold embedded in RN, where N is the dimensionality of the measurement space. An optimal member of each person's AEC is found by locally approximating the structure of this manifold with PCA, and iteratively traversing the manifold until a certain criterion of minimum variance is met. This procedure removes variations that are introduced by confounding variables during the calculation of a template transformation.
Figure 9
Figure 9
Original shapes (first row), optimal λ & intermediate templates (second row) and the global template (last in second row)
Figure 10
Figure 10
Maps of standard deviation across subjects with (a) fixed regularization λ = 1, (b) fixed regularization λ = 35, (c) optimal fixed regularization λ = 12, (d) manifold mean, and (e) optimized signature.
Figure 11
Figure 11
Compare our solution with (a) fixing regularization for each shape and (b) 10000 groups of randomly selected samples.
Figure 12
Figure 12
Templates (a) T1, (b) T2 and (c) T3.
Figure 13
Figure 13
The P-maps of simulated centered shapes using (a) Jacobian determinant λ = 1, Combined Jacobian and residual with regularization (b) λ = 1, (c) λ = 35, and optimized combined Jacobian and residual using (d) global and (e-h) local PCA approximation, (d,e,g,h) L2-norm and (f) L1-norm distance criterion, (d-f) T1, (g) T2 and (h) T3 global template.
Figure 14
Figure 14
The P-maps of simulated shifted shapes using (a) Jacobian determinant λ = 1, Combined Jacobian and residual with regularization (b) λ = 1, (c) λ = 35, and optimized combined Jacobian and residual using (d) global and (e-h) local PCA approximation, (d,e,g,h) L2-norm and (f) L1-norm distance criterion, (d-f) T1, (g) T2 and (h) T3 global template.
Figure 15
Figure 15
Ground truth for the simulated 2D shapes of low morphological variability. Atrophy of 5% was introduced in the two regions
Figure 16
Figure 16
The P-maps for the shapes of Fig. 15 using (a) Jacobian determinant, Combined Jacobian and residual with (b) λ = 1, (c) λ = 35, (d) global PCA and (e-f) local PCA approximation, (e) L1-norm and (d,f) L2-norm distance criterion.
Figure 17
Figure 17
(a) An example of shapes with two sulcus (b) Global template used
Figure 18
Figure 18
The P-maps of simulated two-convolution shapes using (a) Jacobian determinant λ = 1, (b) Combined Jacobian and residual with λ = 1, (c) whole image optimization, piecewise optimization based on (d) two pieces, (e) four pieces, and (f) six pieces.
Figure 19
Figure 19
Comparison of longitudinal profiles - the curve of mean TDM vs. years between most conforming CMD and OMS in the region of (a) Posterior Cingulate, (b) Hippocampus, (c) Superior temporal gyrus.
Figure 20
Figure 20
Comparison of longitudinal profiles between most conforming CMD and OMS in the region without atrophy.
Figure 21
Figure 21
(a) Global template and corresponding P-maps of 3D MR brain images using (b) Jacobian, (c) TDM with λ = 1. Combined features with (d) λ = 1, (e) λ = 6, and optimized combined features using (f) global and (g-h) local PCA approximation, and (g) L1-norm and (f,h) L2-norm distance criterion.
Figure 22
Figure 22
P-maps of (a) TDM with λ = 1, (b) combined feature with λ = 1, and (c) optimized combined feature.
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