An Integrated Pose and Correspondence Approach to Image

Matching

Anand Rangarajan

Image Processing and Analysis GroupDepartments of Electrical Engineering and Diagnostic RadiologyYale University

Motivation I

• Human Brain Mapping:– Different subjects.

• Statistical analysis.

• Normal vs. abnormal.

– Different times.• Detect significant change, help diagnosis.

– Different modalities.• Combine complementary information.

Motivation II

• Difficulty : – Variability in pose, size, shape and acquisition.

• Brain registration : – Common coordinate frame.– Data comparable.– Quantitative analysis.

Results

Interactive 3D Sulcal Tracing

Overview

• Extract features: – Sulcal traces represented as point sets.– Labeling, ordering information [optional].

• Jointly solve feature correspondence and spatial mapping.

Overview II

• Part II: Information Analysis: – Measurements. – Learn from the data, construct statistical

models.• e.g., probabilistic atlas for structures / functions.

– Make inference for new data based on the learned models.

• e.g., automated sulcal labeling, segmentation, computer aided diagnosis.

Outline

• Related work.

• The approach.– Point-based representation of sulci.– Robust point matching algorithm.

• Results and examples.

• Future work.

Other Work in Brain Registration

• Voxel-based methods:– Volumetric Warping: Christensen et al., Gee et

al., Collins et al.

• Feature-based methods: – Landmarks: Bookstein.– Curves: Sandor and Leahy, Collins et al.– Surfaces: Thompson et al., Davatzikos et al. – Sulcal Graphs: Lohmann and von Cramon.

Approach Rationale

• Voxel intensity matching does not ensure that corresponding sulci indeed match.

• Landmarks hard to define.

• Extraction, representation and matching of cortical curves / surfaces / graphs is difficult.

Our Approach

Point-based Representation• Hundreds of points, statistically more

robust than just a few landmarks.

• Additional information can be used:– Major sulcal labels.

• Further analyses made easy:– Procrustes mean. – Eigen-analysis of the error covariance matrix.

Our Approach

Robust Point Matching (RPM)

• Estimation : – Correspondence and spatial mapping.

• Softassign:– Soft correspondence.– Allows partial matching, noise.– Less sensitive to local minima.

• Handles outliers.

Robust Point Matching

Alternating Optimization

• When correspondence M is known, standard least squares solution for spatial mapping A.

• When spatial mapping A is fixed, assignment solution for correspondence M.– Softassign - soft correspondence.– Deterministic Annealing - temperature T.

Robust Point Matching Energy Function

Robust Point Matching

Step I. Solve Spatial Mapping

• Given correspondence M, find the optimal spatial mapping A (affine):

• Standard least-squares solution.

• Gradually relaxed regularization on

Robust Point Matching

Part II. Softassign

• Given spatial mapping A, solve the Linear Assignment Problem:

subject to

Robust Point Matching

Step II. Softassign

Two-way constraints

M ij

M ij

M iji

Row Normalization

M ij

M ij

M ijj

Col. Normalization

Positivity

=exp( )QijM ij

•Step I: Mij = exp ( - Qij/T).

•Step II: Double Normalization. Sinkhorn’s Algorithm.

Outlier rejection using slack variables.

Robust Point Matching Part II. Softassign

• Deterministic Annealing :– T as an extra parameter.– F = Eassign - TS =

• Gibbs Distribution :– Positivity ganranteed.– High T, insensitive to Q, uniform M .– Low T, sensitive to Q, binary M .

Robust Point Matching Algorithm Summary

• Start: uniform M, high temperature T.

• Do until final temperature is reached.– Given M, solve for spatial mapping A.– Given A, use Softassign to update M.

• Decrease temperature.

Experiment on Brain Sections

Results of Method

Results

Interactive 3D Sulcal Tracing

Results

RPM Example

Two labeled sulcal point sets, initial position.

RPM without label information

Results

Visual Matching Comparison

Results

Visual Matching Comparison

Quantitative Comparison

Quantitative Comparison

Future Work

• Error measure on the entire volume.

• Fully non-rigid 3D spatial mapping.– Thin-plate spline and correspondence.

• Automated sulcal extraction, Zeng et al.

• Investigate partially labeled case.

• Automated labeling.

• Atlas construction.

The End

Thin-plate-spline Implementation

Thin-plate-spline Implementation

Results

Visual Matching Comparison

TPS