Analysis of shape

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

2. 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

3. 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 :)

4. 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)

5. 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

6. Chain code

7. 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

8. 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

9. Inflection points

10. 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

12. 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

13. Thinning and skeletonization

14. 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

15. 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)

16. 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)

17. 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