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The relationship between the Pascal Triangle and the Radon Transform of an image �

Copyright 2012, all rights reserved.� 1

3 Copyright 2012, all rights reserved.

1.  The Pascal Triangle of a gray scale image 1.1 Inspiration

Shanshan Huang and I invented the Pascal Triangle in order to gain a geometric interpretation of the content of a gray scale image. We were mostly interested in the geometric features of simple objects, such as

•  Dots, •  Straight line segments, •  Loops, •  Crosses •  Etc.

Our inspiration was the concept of principal curvatures in differential geometry, which can be used to classify surfaces. However, we did not want to rely on derivatives, as these are ill-suited for application to discrete images. Rather, we worked with moments, a discrete analogue, which we could define directly in terms of the data given in the image.

The Pascal triangle of an image is a pyramidal arrangement of moments.

4 Mathematics in Science and Society Lecture, UIUC, Nov. 13, 2012 Copyright 2012, all rights reserved. 4

1.  The Pascal Triangle of a gray scale image 1.2 Definition

Start with a gray scale image with

pixel locations (xk,yk)

pixel intensity wk

for k = 1, …, N

(x1,y1)=(1,1)

Example: N=9 pixels





pixel intensity wk

for k = 1, …, N

(x2,y2)=(1,2)

Example: N=9 pixels





pixel intensity wk

for k = 1, …, N

(x3,y3)=(1,3) Example: N=9 pixels





pixel intensity wk

for k = 1, …, N

w1=0

Example: N=9 pixels





pixel intensity wk

for k = 1, …, N

w1=255

Example: N=9 pixels





pixel intensity wk

for k = 1, …, N

w1=53

Example: N=9 pixels



Write pixel location using complex numbers

(xk,yk) → zk= xk+ i yk

for k = 1, …, N



Write pixel location using complex numbers

(xk,yk) → zk= xk+ i yk

for k = 1, …, N

Compute complex moments

!

µmn = wkzkm

k=1

N

" zk( )n



We form the Pascal triangle of the image by arranging the moments this way:

Moments of order two



We form the Pascal triangle of the image by arranging the moments this way:

Moments of order three



15 Mathematics in Science and Society Lecture, UIUC, Nov. 13, 2012 Copyright 2012, all rights reserved. 15

1.  The Pascal Triangle of a gray scale image 1.3 Relation to Radon transform

16 Mathematics in Science and Society Lecture, UIUC, Nov. 13, 2012

The Radon transform fθ( r ) is the projection of the image intensities onto a straight line through the origin with direction vector eiθ.

z1 z2 z3

Copyright 2012, all rights reserved. 16




θ





θ

!

f" (r) = wkks.t.r=xkcox" +yk sin"

#




Denote by mn(θ) the nth order moment of the Radon transform fθ( r ).

!

mn (") = rnrs.t .f" (r )#0

$ f" (r) = (xk cos" + yk sin")n

k=1

N

$ wk

Lemma: For any n=0,1, 2, …

!

mn (") =12n

nl#

$ % &

' ( µl,n) le

i(n)2l )"

l=0

n

*



Row r of the Pascal triangle corresponds to the Fourier series coefficients of the rth order moment of the Radon transform:

Coefficients of Fourier series of Third order moment of Radon transform of the image



Coefficients of the Fourier series of the Second order moment of Radon transform of image


Row r of the Pascal triangle corresponds to the Fourier series coefficients of the rth order moment of the Radon transform:



I = { (zk,wk) }k=1,…,N

T2N-2 (I) { Mn(θ) }n=0,…,2N-2

fθ (r)


For more details:

M. Boutin, S. Huang, “The Pascal Triangle of a discrete image: definition, properties, and application to shape analysis,” available at http://arxiv.org/abs/1209.4850

A.W. Haddad, S. Huang, M. Boutin and E. J. Delp, ``Detection of symmetric shapes on a mobile device with applications to automatic sign interpretation,'' Proc. of SPIE-IS&T Electronic Imaging 8304, p. 83040G, 2012.


1.  The Pascal Triangle of a gray scale image 1.5 Conclusion

We are grateful for the support provided by the following funding agencies:

NSF Department of Homeland Security Indiana 21st century fund DARPA

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