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Proposed Shape Dissimilarity Measures for Statistical Shape Modeling

This study introduces SDF and SSD measures for shape analysis, with emphasis on invariance and simplicity. Empirical evaluations and comparisons with other models are conducted to demonstrate effectiveness, especially in 3D experiments. The proposed methods show promising results in shape modeling, segmentation, and statistical analysis.

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Proposed Shape Dissimilarity Measures for Statistical Shape Modeling

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  1. Proposed Dissimilarity Measure • We chose the SDF as our shape descriptor because: • Convergence condition of gradient descent methods is satisfied(Huang et al PAMI’06). • Invariance to rotations and translations. • Ability to handle topological changes. • Its relative simplicity. To deal with the dimension added by the SDF definition • Proposed SSD Measure where: or

  2. Convexity of the proposed SSD measure • Empirical Evaluation (2D case): • Pick a shape: • Fix 3 parameters and vary the remaining 2. • The ranges of the 2 unknown parameters are uniformly quantized using 100 samples: • Convexity in full dimensionality is not guaranteed

  3. Empirical Evaluation of the Proposed SSD

  4. Euler Lagrange Equations • For each parameter where: • Implementation consideration: different time steps may need to be used for different parameters

  5. Comparisons with the other models Initial position Isotropic scale-based model VDF-based model 208.67 sec 300.57 sec 141.26 sec 221.35sec Our results 139.67 sec 206.82sec 180.68sec 102.23sec

  6. More comparisons Initial position Isotropic scale-based model VDF-based model 538.67sec 271.57 sec 263.69 sec 219.87 sec 296.67sec 157.76 sec 147.20 sec Our results 169.77 sec

  7. Recovered parameters • GT: Ground truth • M1: Our model • M2:VDF-model • M3: Homogeneous scale-based model

  8. 3D Experiments

  9. Model shape variations using PCA Align Shapes Application: Statistical modeling of shapes Shape Model = Mean Shape + Basic Variations Implicit Rep. Training data

  10. Before alignments Overlap After alignments Overlap Alignments Goal: Establish correspondences among shapes over the training set

  11. Qualitative Evaluation • Correlation Coefficient

  12. Modeling shape variations using PCA • Compute the mean of the aligned data and mean offsets and • SVD of covariance matrix with • New shape, within the variance observed in training set, can be approximated chose k s.t.

  13. First four principal modes Mode 1 Mode 2 Mode 3 Mode 4

  14. Application: Shape-based segmentation • Generate an Active Shape Model (ASM) and use it to locate objects in hard to segment images (Cootes and Taylor’95) Isotropic scale-based model Our model

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