Analysis of shape Biomedical Image processing course, Yevhen Hlushchuk and Jukka Parviainen

Contents Representation of shapes and contours –signatures –chain coding –segmentation of the contours –polygonal and parabolic modeling –thinning and skeletonization Shape factors –compactness –moments –chord-length statistics

Shape importance in medicine most human organs possess certain reedily identifiable shapes (deviations might be caused by a pathology) very important issue is differentiation between malignant and benign tumours, general rule: benign masses have smooth boundaries and simplper shapes (not so many angles :)

Signatures of contours The most general representation of the contouris in terms (x,y) coordinates. Converting coordinate-based to distances from each contour point to reference point (centroid). Radial distance may also be used but has drawbacks for irregular shapes. FIG 6.2, 6.3 here (benign masses – smooth signatures)

Chain coding relies on specifying the starting point, direction of traversal (clockwise ot counter- clockwise) and movement need to be done to get to the next point (e.g., 1 pixel up, or 1 pixel right). Number of different movements used in the code defines how fine is the representation (compare 4 and 8) Figure 6.5 here

Chain code

Advantages: –more compact representation (2-3 bits per point) –invariant to shift or translation –certain possibilities to scaling and rotattion (by 45 or 90 degrees) –nice to calculate the length of the contour, area of a closed loop, check for multiple loops and closure

Segmentation of the contour Useful step before analysis and modeling Book author’s own example : –locating points of inflections (f’’=0; f’=!0; f’’’=!0) –irrelevant points of inflection (on straight segments) – cumulative sums might help

Inflection points

Polygonal modeling prespicifying the number of segments (e.g., using points of inflections) main criteria – arch-to-chord deviation: –if it exceeds certain threshold the curved part is segmented at the point of the max deviation

Parabolic modeling straight segments may not contribute much to the discrimination between benign and malignant masses After all, classification accuracy was 76% (compared to what? radiologist? or histology? )with a set of 54 contours

Thinning and skeletonization

Shape factors Idea is to encode the nature or form of a conotur using a small number of features, called shape factors Basic properties: –invariance to spatial shift –invariance to rotation –invariance to scaling

Shape factors Compactness is a popular measure of the efficiency of the contour to contain a given area and defined as perimeter in the second power divided by the area contained within the contour (circle is the best here :). Moments of the contours: to the centre of the image, to the centroid of the contour, normalized and so on. High order momens are sensitive to noise (thus different types of normalization on low-order moments have been attempted)

Chord-length statistics One can calculate the mean, deviation, skewness and curtosis for the cord-lengths (Kolgorov-Smirnov statistics). Nice about it: –invariant to spatial shift –invariant to rotation –invariant to scaling Not so nice – ”certain invariance to shape ” (objects with different shapes might still have similar statistics)

Summary (contents) Representation of shapes and contours –signatures –chain coding –segmentation of the contours –polygonal and parabolic modeling –thinning and skeletonization Shape factors –compactness –moments –chord-length statistics