Pseudo-Zernike polynomials

They are an orthogonal set of complex-valued polynomials

defined as

:$$

V_{nm}(x,y) = R_{nm}(x,y)e^{jm\arctan(\frac{y}{x})},

where $x^2+y^2\backslash leq\; 1,\; n\backslash geq\; 0,\; |m|\backslash leq\; n$ and orthogonality on the unit disk is given as

:$$

\int_0^{2\pi}\int_0^1 r [V_{nl}(r\cos\theta,r\sin\theta)]^* \times

V_{mk}(r\cos\theta,r\sin\theta)\,dr\,d\theta =

\frac{\pi}{n+1}\delta_{mn}\delta_{kl},

where the star means complex conjugation, and

r^2 = x^2+y^2, $x=r\backslash cos\backslash theta$, $y=r\backslash sin\backslash theta$

are the standard transformations between polar and Cartesian coordinates.

The radial polynomials $R\_\{nm\}$ are defined as

R_{nm}(r) = \sum_{s=0}^{n-|m|}D_{n,|m|,s}\ r^{n-s}

with integer coefficients

:$$

D_{n,|m|,s} = (-1)^s\frac{(2n+1-s)!}{s!(n-|m|-s)!(n+|m|-s+1)!}.

Examples are:

$R\_\{0,0\}\; =\; 1$

$R\_\{1,0\}\; =\; -2+3\; r$

$R\_\{1,1\}\; =\; r$

$R\_\{2,0\}\; =\; 3+10\; r^2-12\; r$

$R\_\{2,1\}\; =\; 5\; r^2-4\; r$

$R\_\{2,2\}\; =\; r^2$

$R\_\{3,0\}\; =\; -4+35\; r^3-60\; r^2+30\; r$

$R\_\{3,1\}\; =\; 21\; r^3-30\; r^2+10\; r$

$R\_\{3,2\}\; =\; 7\; r^3-6\; r^2$

$R\_\{3,3\}\; =\; r^3$

$R\_\{4,0\}\; =\; 5+126\; r^4-280\; r^3+210\; r^2-60\; r$

$R\_\{4,1\}\; =\; 84\; r^4-168\; r^3+105\; r^2-20\; r$

$R\_\{4,2\}\; =\; 36\; r^4-56\; r^3+21\; r^2$

$R\_\{4,3\}\; =\; 9\; r^4-8\; r^3$

$R\_\{4,4\}\; =\; r^4$

$R\_\{5,0\}\; =\; -6+462\; r^5-1260\; r^4+1260\; r^3-560\; r^2+105\; r$

$R\_\{5,1\}\; =\; 330\; r^5-840\; r^4+756\; r^3-280\; r^2+35\; r$

$R\_\{5,2\}\; =\; 165\; r^5-360\; r^4+252\; r^3-56\; r^2$

$R\_\{5,3\}\; =\; 55\; r^5-90\; r^4+36\; r^3$

$R\_\{5,4\}\; =\; 11\; r^5-10\; r^4$

$R\_\{5,5\}\; =\; r^5$

The pseudo-Zernike Moments (PZM) of order $n$ and repetition $l$ are defined as

:$$

A_{nl}=\frac{n+1}{\pi}\int_0^{2\pi}\int_0^1 [V_{nl}(r\cos\theta,r\sin\theta)]^*

f(r\cos\theta,r\sin\theta)r\,dr\,d\theta,

where $n\; =\; 0,\; \backslash ldots\; \backslash infty$, and $l$ takes on positive and negative integer

values subject to $|l|\backslash leq\; n$.

The image function can be reconstructed by expansion of the pseudo-Zernike coefficients on the unit disk as

:$$

f(x,y) = \sum_{n=0}^{\infty}\sum_{l=-n}^{+n}A_{nl}V_{nl}(x,y).

Pseudo-Zernike moments are derived from conventional Zernike moments and shown

to be more robust and less sensitive to image noise than the Zernike moments.{{cite journal

|first1=C.-H.

|last1=Teh

|last2=Chin

|first2=R.

|title=On image analysis by the methods of moments

|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence

|volume=10

|issue=4

|year=1988

|pages=496–513

|doi=10.1109/34.3913

}}

• Zernike polynomials
• Image moment

{{Reflist}}

• {{cite journal

|last1=Belkasim

|first1=S.

|last2=Ahmadi

|first2=M.

|last3=Shridhar

|first3=M.

|title=Efficient algorithm for the fast computation of zernike moments

|journal=Journal of the Franklin Institute

|year=1996

|volume=333

|issue=4

|pages=577–581

|doi=10.1016/0016-0032(96)00017-8

}}

• {{cite journal

|last1=Haddadnia

|first1=J.

|last2=Ahmadi

|first2=M.

|last3=Faez

|first3=K.

|title=An efficient feature extraction method with pseudo-zernike moment in rbf neural network-based human face recognition system

|journal=EURASIP Journal on Applied Signal Processing

|year=2003

|pages=890–901

|doi=10.1155/S1110865703305128

|bibcode=2003EJASP2003..146H

|volume=2003

|issue=9

|doi-access=free

}}

• {{cite conference

|author= T.-W. Lin

|author2=Y.-F. Chou

|title=A comparative study of zernike moments

|conference=Proceedings of the IEEE/WIC International Conference on Web Intelligence

|year=2003

|pages=516–519

|isbn=0-7695-1932-6

|doi=10.1109/WI.2003.1241255

}}

• {{ cite journal

|first1=C.-W.

|last1=Chong

|first2=P.

|last2=Raveendran

|first3=R.

|last3=Mukundan

|title=The scale invariants of pseudo-Zernike moments

|journal=Pattern Anal. Applic.

|year=2003

|volume=6

|pages=176–184

|doi=10.1007/s10044-002-0183-5

|issue=3

|url=http://shdl.mmu.edu.my/2525/1/The%20scale%20invariants%20of%20pseudo-Zernike%20moments.pdf

}}

• {{cite journal

|journal=Int. J. Pattern Recogn. Artif. Int.

|year=2003

|volume=17

|issue=6

|pages=1011–1023

|doi= 10.1142/S0218001403002769

|url=http://ir.canterbury.ac.nz/bitstream/10092/448/1/12584534_ivcnz01.pdf

|first1=Chee-Way

|last1=Chong

|first2=R.

|last2=Mukundan

|first3=P.

|last3=Raveendran

|title=An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments

|hdl=10092/448

|hdl-access=free

}}

• {{cite web

|first1=Jamie

|last1=Shutler

|url=http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/SHUTLER3/node11.html |title=Complex Zernike Moments

|year=1992

}}

{{DEFAULTSORT:Pseudo-Zernike Polynomials}}

Category:Orthogonal polynomials