Nonrigid Shape Correspondence using Landmark Sliding, Insertion, and Deletion

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Nonrigid Shape Correspondence using Landmark
Sliding, Insertion, and Deletion

• Theodor Richardson

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Overview

• 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.

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Statistical Shape Analysis (SSA)

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

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Shape Correspondence

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

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The (Landmark-Based) Point-Correspondence Problem

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

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The Three Factors Affecting Correspondence
Accuracy

• There are three main factors that determine the
  accuracy of shape correspondence
• Landmark-correspondence error it is necessary
  to measure the accuracy of the landmark mapping,
• Shape-representation error only when a set of
  landmarks well-represents the underlying contour
  does shape-correspondence equate to
  landmark-correspondence,
• Shape-representation compactness a sparse
  sampling of landmarks is desirable for current
  SSA methods, meaning the fewest number of
  landmarks required is desirable.

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Fixed Landmark Methods

• Many prior methods construct a mapping based on a
  set of pre-sampled landmarks along each shape
  instance.
• These methods tend to use either local or global
  methods of matching one landmark to another.
• Global methods may not utilize local shape
  features to capture the underlying contour
• Local methods may catch local feature information
  but they tend to overlook global positioning

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Nonfixed Landmark Methods

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

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The Landmark Sliding Methods

• The work most closely related to the method of
  correspondence applied in the proposed method is
  landmark sliding.
• Bookstein first proposed the idea of sliding
  landmarks along their tangent directions to
  relocate them to ideal positions to minimize
  thin-plate spline bending energy.

F. L. Bookstein. Principal warps Thin-plate
splines and the decomposition of
deformations. IEEE Trans. PAMI, 11(6)567585,
June 1989. F. L. Bookstein. Landmark methods for
forms without landmarks Morphometrics of
group differences in outline shape. Medical Image
Analysis, 1(3)225243, 1997.
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Landmark Correspondence Error

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

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Shape Representation Error

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

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Shape Representation Compactness

• Shape representation compactness simply requires
  that the landmark set be as small as possible
  while still upholding the criteria of the other
  two factors.
• This will increase shape representation error, so
  a balance must be found to prevent both
  supersampling and undersampling

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An Algorithmic Solution

• Choose one shape instance as the template Vt
• Initialize the landmark sets Vq, q 1, 2, n
• //Main loop
• Repeat while max sliding distance gt 0
• Repeat while alpha gt epsilonH
• Landmark insertion
• Update the template Vt
• Loop over each shape instance
• Landmark sliding
• Update the template Vt
• Repeat while alpha lt epsilonL
• Landmark deletion
• Update the template Vt
• End

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Detecting High Curvature Points

• High curvature points are easily detected by
  human vision they generally represent
  mathematically critical points to defining a
  curve
• These points decrease representation error
• Retaining high curvature points emulates human
  vision shape correspondence
• These points also act as an edge case to the
  sliding algorithm used herein

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High Curvature Case

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

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Landmark Sliding Algorithm

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

S.Wang, T. Kubota, and T. Richardson. Shape
correspondence through landmark sliding. In Proc.
Conf. Computer Vision and Pattern Recog., pages
143150, 2004.
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Topology Preservation

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

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Landmark Insertion/Deletion

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

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Comparison Study

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

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Visual Comparison
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D1 Corpus Callosum
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D1 Corpus Callosum
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D2 - Cerebellum
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D3 - Cardiac
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D4 - Kidney
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D5 - Femur
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Conclusion

• This 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.