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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.
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The End