Sharif University of Technology

A modified algorithm to obtain Translation, Rotation & Scale

invariant Zernike Moment shape Descriptors

G.R. AmayehDr. S. Kasaei

A.R. Tavakkoli

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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

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Shape Descriptors

Fig.1: Same regions. Fig.2: Same contours.

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Zernike & Pseudo-Zernike Moments

Zernike Moments of Order n, with m-repetition:

Zernike Moment’s Basis Function jm

mnmnmn eRVyxV )(),(),( ,,,

CircleUnit mnmn dydxyxVyxfn

Z ,,1 *

,,

evenismn

nmWhere

(1)

(2)

(3)

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Zernike & Pseudo-Zernike Moments

Zernike Moment Radial Polynomials:

Pseudo-Zernike Radial Polynomials:

MZforS

mnS

mnS

snR

mn

s

SnS

mn

2

||

0

2,

)!2

||()!

2

||(!

)!()1(

MZPsforSmnSmnS

snR

mn

s

SnS

mn .)!||()!||(!

)!12()1(||

0,

(4)

(5)

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A Cross Section ofRadial 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.

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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.

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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.

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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.

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Normalization Method

Algorithm: Translate image’s center of mass to origin.

Scale image:

0,0

1,0

0,0

0,1 ,m

my

m

mxwhere

)6(, yyxxf

0,0

)7(,m

awherea

y

a

xf

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Normalization Method

Fig.9 : From left to right, Original, Translated, & Scaled images

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Normalization Method

Fig.10 : From left to right, original image & normalized images with different s.

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Normalization Method Drawbacks

Interpolation Errors: Down sampling image leads to loss of

data. Up sampling image adds wrong

information to image.

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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:0,0

,, m

ZZ mn

mn (8)

Fig.11 : Images & fitted circles.

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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.

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Proposed Method Algorithm:

Computing a Grid Map. Performing translation and scale on the map

indexes.

Fig.12: Mapping.

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Proposed Method Translate origin of coordination system to the center

of mass

(9)

yyy

xxx

Fig(13). Translation of Coordination Origin.

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Proposed Method Scale coordination system

yay

xax(10)

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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.

),(),( 2 yxfayxg ),( yx ),( yxg

),( yxg

),(a

y

a

xf

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Proposed Method

Fig.15 : From left to right, original image & normalized images with different s.

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Proposed Method Special case

Fig.17 : Zernike moments by proposed method & IZM (Improved ZM with out normalization ).

Fig.16 : Original image.

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Experimental Results

Fig.16 : Original image & 70% scaled image.

Fig.17 : Error of Zernike moments between original image & scaled image.

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Experimental Results

Fig.18 : Original image & 55 degree rotated image.

Fig.19 : Error of Zernike moments between original image & rotated image.

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Experimental Results

Fig.21 : Error of Zernike moments between original & scaled images.

Fig.20 : Original image & 120% scaled image.

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Experimental Results

Fig.23 : Error of Zernike moments between original image & rotated image.

Fig.21 : Original image & 40 degree rotated image.

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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