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

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