Nonrigid Shape Correspondence using Landmark Sliding, Insertion, and Deletion Theodor Richardson.

Nonrigid Shape Nonrigid Shape Correspondence using Correspondence using

Landmark Sliding, Landmark Sliding, Insertion, and DeletionInsertion, and Deletion

Theodor RichardsonTheodor Richardson

22 Theodor RichardsonTheodor Richardson

OverviewOverview

Statistical Shape Analysis (SSA) is growing in usage (mainly to develop models for better image segmentation)Accurate SSA methods depend upon an accurate shape correspondence.To address this problem, a novel, nonrigid, landmark-based method to correspond a set of 2D shape instances is presented. Unlike prior methods, the proposed method combines three important factors in measuring the shape-correspondence error:

landmark-correspondence error, shape-representation error, and shape-representation compactness.


Statistical Shape Analysis (SSA)Statistical Shape Analysis (SSA)

Most anatomical Most anatomical structures possess a structures possess a unique shape.unique shape.This shape is often used This shape is often used in medical imaging for in medical imaging for purposes of automated purposes of automated diagnosis.diagnosis.SSA can build models of SSA can build models of such shapes for use in such shapes for use in guiding shape guiding shape extraction/image extraction/image segmentation.segmentation.


Shape CorrespondenceShape Correspondence

SSA relies upon an SSA relies upon an accurate mapping across accurate mapping across a set of shape instances.a set of shape instances.Constructing this Constructing this mapping is the shape mapping is the shape correspondence problem.correspondence problem.A shape is defined as a A shape is defined as a continuous curve, also continuous curve, also referred to as a contour.referred to as a contour.SSA utilizes a finite SSA utilizes a finite sampling of each curve sampling of each curve called a landmark set.called a landmark set.


The (Landmark-Based) Point-The (Landmark-Based) Point-Correspondence ProblemCorrespondence Problem

The discrete form of The discrete form of shape correspondence is shape correspondence is often called the point-often called the point-correspondence problem.correspondence problem.The desired outcome of The desired outcome of this correspondence is a this correspondence is a mapping from any point mapping from any point along one shape instance along one shape instance to an equivalent point to an equivalent point along all other shape along all other shape instances.instances.Human vision can solve Human vision can solve this problem for high this problem for high curvature points.curvature points.


The Three Factors Affecting The Three Factors Affecting Correspondence AccuracyCorrespondence Accuracy

There are three main factors that determine the There are three main factors that determine the accuracy of shape correspondence:accuracy of shape correspondence: Landmark-correspondence error – it is necessary to Landmark-correspondence error – it is necessary to

measure the accuracy of the landmark mapping,measure the accuracy of the landmark mapping, Shape-representation error – only when a set of Shape-representation error – only when a set of

landmarks well-represents the underlying contour landmarks well-represents the underlying contour does shape-correspondence equate to landmark-does shape-correspondence equate to landmark-correspondence,correspondence,

Shape-representation compactness – a sparse Shape-representation compactness – a sparse sampling of landmarks is desirable for current SSA sampling of landmarks is desirable for current SSA methods, meaning the fewest number of landmarks methods, meaning the fewest number of landmarks required is desirable.required is desirable.


Fixed Landmark MethodsFixed Landmark Methods

Many prior methods construct a mapping based Many prior methods construct a mapping based on a set of pre-sampled landmarks along each on a set of pre-sampled landmarks along each shape instance. shape instance.

These methods tend to use either local or global These methods tend to use either local or global methods of matching one landmark to another.methods of matching one landmark to another. Global methods may not utilize local shape features Global methods may not utilize local shape features

to capture the underlying contourto capture the underlying contour Local methods may catch local feature information but Local methods may catch local feature information but

they tend to overlook global positioningthey tend to overlook global positioning


Nonfixed Landmark MethodsNonfixed Landmark Methods

The fixed landmark methods have a major The fixed landmark methods have a major drawback; there is no way to overcome a drawback; there is no way to overcome a poor initialization of the landmark points.poor initialization of the landmark points.Nonfixed landmark methods allow Nonfixed landmark methods allow landmarks to travel from their original landmarks to travel from their original position to an optimal location.position to an optimal location.The machine learning techniques for The machine learning techniques for correspondence are a subset of this correspondence are a subset of this group, including MDLgroup, including MDL


The Landmark Sliding MethodsThe Landmark Sliding Methods

The work most closely related to the The work most closely related to the method of correspondence applied in the method of correspondence applied in the proposed method is landmark sliding.proposed method is landmark sliding.

Bookstein first proposed the idea of sliding Bookstein first proposed the idea of sliding landmarks along their tangent directions to landmarks along their tangent directions to relocate them to ideal positions to relocate them to ideal positions to minimize thin-plate spline bending minimize thin-plate spline bending energy*.energy*.

*F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations.IEEE Trans. PAMI, 11(6):567–585, June 1989.*F. L. Bookstein. Landmark methods for forms without landmarks: Morphometrics of groupdifferences in outline shape. Medical Image Analysis, 1(3):225–243, 1997.


Landmark Correspondence ErrorLandmark Correspondence Error

The model chosen for The model chosen for representing the representing the landmark landmark correspondence error correspondence error is the thin-plate spline is the thin-plate spline bending energy bending energy proposed by proposed by Bookstein.Bookstein.Bending energy is Bending energy is invariant to affine invariant to affine transformations.transformations.


Shape Representation ErrorShape Representation Error

Shape representation error is the measure of Shape representation error is the measure of data loss in representing a continuous curve with data loss in representing a continuous curve with a finite number of landmarks.a finite number of landmarks.


Shape Representation Shape Representation CompactnessCompactness

Shape representation compactness simply Shape representation compactness simply requires that the landmark set be as small requires that the landmark set be as small as possible while still upholding the criteria as possible while still upholding the criteria of the other two factors.of the other two factors.

This will increase shape representation This will increase shape representation error, so a balance must be found to error, so a balance must be found to prevent both supersampling and prevent both supersampling and undersamplingundersampling


An Algorithmic SolutionAn Algorithmic SolutionChoose one shape instance as the template VtInitialize the landmark sets Vq, q = 1, 2, … n//Main loopRepeat while max sliding distance > 0

Repeat while alpha > epsilonHLandmark insertion

Update the template VtLoop over each shape instance

Landmark slidingUpdate the template VtRepeat while alpha < epsilonL

Landmark deletionUpdate the template Vt

End


Detecting High Curvature PointsDetecting High Curvature Points

High curvature points are High curvature points are easily detected by human easily detected by human vision; they generally vision; they generally represent mathematically represent mathematically critical points to defining a critical points to defining a curvecurveThese points decrease These points decrease representation errorrepresentation errorRetaining high curvature points Retaining high curvature points emulates human vision shape emulates human vision shape correspondencecorrespondenceThese points also act as an These points also act as an edge case to the sliding edge case to the sliding algorithm used hereinalgorithm used herein


High Curvature CaseHigh Curvature Case

The local maxima for the curvature plot are The local maxima for the curvature plot are subjected to a threshold of the maximum subjected to a threshold of the maximum difference in unsigned curvature. difference in unsigned curvature. Points above this threshold are retained as Points above this threshold are retained as critical correspondence landmarks (CCLs)critical correspondence landmarks (CCLs)CCLs are prevented from sliding and maintain CCLs are prevented from sliding and maintain equivalent points in all shape instances to equivalent points in all shape instances to preserve correspondencepreserve correspondenceIf a CCL is not present in all shape instances, If a CCL is not present in all shape instances, the placeholder for the CCL is allowed to slide to the placeholder for the CCL is allowed to slide to conform to the shape instances that have the conform to the shape instances that have the fixed CCL.fixed CCL.


Landmark Sliding AlgorithmLandmark Sliding Algorithm

The landmark sliding algorithm addresses the landmark-The landmark sliding algorithm addresses the landmark-correspondence accuracy.correspondence accuracy.Landmarks slide along their estimated tangent directions.Landmarks slide along their estimated tangent directions.The offset landmarks are then projected back onto the original The offset landmarks are then projected back onto the original curve to preserve shape representation*.curve to preserve shape representation*.Allowable landmark sliding distance is determined by the Allowable landmark sliding distance is determined by the curvature at the starting position for each landmark.curvature at the starting position for each landmark.Sliding is optimized by quadratic programming (minimizing a Sliding is optimized by quadratic programming (minimizing a quadratic function)quadratic function)

*S.Wang, T. Kubota, and T. Richardson. Shape correspondence through landmark sliding. InProc. Conf. Computer Vision and Pattern Recog., pages 143–150, 2004.


Topology PreservationTopology Preservation

For the landmark correspondence to represent For the landmark correspondence to represent the underlying shape correspondence, the the underlying shape correspondence, the topology of the underlying shape must be topology of the underlying shape must be preserved. This means that landmarks should preserved. This means that landmarks should not be allowed to slide past each other or move not be allowed to slide past each other or move in a way that breaks the flow of the underlying in a way that breaks the flow of the underlying shape contour.shape contour.

This is accomplished by a constraint bounding This is accomplished by a constraint bounding the allowed sliding length.the allowed sliding length.


Landmark Insertion/DeletionLandmark Insertion/Deletion

Landmark insertion: When the mean alpha Landmark insertion: When the mean alpha value is above epsilon, the representation error value is above epsilon, the representation error is too high; to counter this, a new landmark is is too high; to counter this, a new landmark is inserted in the gap between landmarks inserted in the gap between landmarks contributing most to the representation error.contributing most to the representation error.Landmark deletion: When the mean alpha value Landmark deletion: When the mean alpha value is below epsilon, the representation error is too is below epsilon, the representation error is too low; therefore a landmark is deleted from the low; therefore a landmark is deleted from the span of the curve contributing the least amount span of the curve contributing the least amount of representation error.of representation error.These processes are opposites and require two These processes are opposites and require two separate epsilon values to prevent oscillation.separate epsilon values to prevent oscillation.


Comparison StudyComparison Study

Our method was compared to the Our method was compared to the implementation of the Minimum Description implementation of the Minimum Description Length (MDL) method over five data sets. Each Length (MDL) method over five data sets. Each data set was run with three initializations for data set was run with three initializations for each algorithm to compare the statistical results.each algorithm to compare the statistical results.


Visual ComparisonVisual Comparison


D1 – Corpus CallosumD1 – Corpus Callosum


D1 – Corpus CallosumD1 – Corpus Callosum


D2 - CerebellumD2 - Cerebellum


D3 - CardiacD3 - Cardiac


D4 - KidneyD4 - Kidney


D5 - FemurD5 - Femur


ConclusionConclusionThis method considers three important factors in modeling the shape-correspondence error:

landmark-correspondence error, representation error, and representation compactness.

These three factors are explicitly handled by the landmark sliding, insertion, and deletion operations, respectively. The performance of the proposed method was evaluated on five shape-data sets that are extracted from medical images and the results were quantitatively compared with an implementation of the MDL method. Within a similar allowed representation error, the proposed method has a performance that is comparable to or better than MDL in terms of

(a) average bending energy, (b) principal variances in SSA, (c) representation compactness, and (d) algorithm speed.

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Nonrigid Shape Correspondence using Landmark Sliding, Insertion, and Deletion Theodor Richardson.

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