Sharif University of Technology A modified algorithm to obtain Translation, Rotation & Scale invariant Zernike Moment shape Descriptors G.R. Amayeh Dr. S. Kasaei A.R. Tavakkoli

2 Introduction  Shape is one of the most important features to human for visual distinguishing system.  Shape Descriptors Contour-Base  Using contour information  Neglect image details Region-Base  Using region information

3 Shape Descriptors Fig.1: Same regions.Fig.2: Same contours.

4 Zernike & Pseudo-Zernike Moments  Zernike Moments of Order n, with m- repetition: Zernike Moment’s Basis Function (1) (2) (3)

5 Zernike & Pseudo-Zernike Moments  Zernike Moment Radial Polynomials:  Pseudo-Zernike Radial Polynomials: (4) (5)

6 A Cross Section of Radial Polynomials of ZM & PsZM Fig.3 : ZM (blue) & Ps. ZM (red) of 4-order with repetition 0. Fig.5 : ZM (blue) & Ps. ZM (red) of 5-order with repetition 1. Fig.4 : ZM (blue) & Ps. ZM (red) of 6-order with repetition 4. Fig.6 : ZM (blue) & Ps. ZM (red) of 7-order with repetition 3.

7 3-D Illustration of Radial Polynomials of ZM & Ps.ZM Fig.7 : Radial polynomial of ZM of 7-order with repetition 1. Fig.8 : Radial polynomial of Ps. ZM of 7-order with repetition 1.

8 Zernike Moments Properties  Invariance Properties: Zernike Moments are Rotation Invariant  Rotation changes only moment’s phase.  Variance Properties: Zernike Moments are Sensitive to Translation & Scaling.

9 Achieving Invariant Properties  What is needed in segmentation problem? Moments need to be invariant to rotation, scale and translation.  Solution to achieve invariant properties Normalization method. Improved Zernike Moments without Normalization (IZM). Proposed Method.

10 Normalization Method  Algorithm: Translate image’s center of mass to origin. Scale image:

11 Normalization Method Fig.9 : From left to right, Original, Translated, & Scaled images 

12 Normalization Method Fig.10 : From left to right, original image & normalized images with different s.

13 Normalization Method Drawbacks  Interpolation Errors: Down sampling image leads to loss of data. Up sampling image adds wrong information to image.

14 Improved Zernike Moments without Normalization  Algorithm: Translate image’s center of mass to origin. Finding the smallest surrounding circle and computing ZMs for this circle. Normalize moments: (8) Fig.11 : Images & fitted circles.

15 Drawbacks  Increased Quantization Error. Since the SSC of images have a small number of pixels, image’s resolution is low and this causes more QE.

16 Proposed Method  Algorithm: Computing a Grid Map. Performing translation and scale on the map indexes. Fig.12: Mapping.

17 Proposed Method Translate origin of coordination system to the center of mass (9) Fig(13). Translation of Coordination Origin.

18 Proposed Method Scale coordination system (10)

19 Proposed Method  Computing Zernike Moment in new coordinate for where.  We can show that the moments of in the new coordinate system are equal to the moments of in the old coordinate system.

20 Proposed Method Fig.15 : From left to right, original image & normalized images with different s. 

21 Proposed Method  Special case Fig.17 : Zernike moments by proposed method  & IZM (Improved ZM with out normalization ). Fig.16 : Original image.

22 Experimental Results Fig.16 : Original image & 70% scaled image. Fig.17 : Error of Zernike moments between original image & scaled image.

23 Experimental Results Fig.18 : Original image & 55 degree rotated image. Fig.19 : Error of Zernike moments between original image & rotated image.

24 Experimental Results Fig.21 : Error of Zernike moments between original & scaled images. Fig.20 : Original image & 120% scaled image.

25 Experimental Results Fig.23 : Error of Zernike moments between original image & rotated image. Fig.21 : Original image & 40 degree rotated image.

26 Conclusions  Principle of our method is same as the Normalization method.  Does not resize the original image. No Interpolation Error.  Reduces the quantization error. (using beta parameter)  Trade off Between QE and power of distinguishing.  Has all the benefits of both pervious methods.

The End