## Abstract

We propose a class of generalized moment functions (GMFs) that can be used to determine a set of geometric points, namely, generalized centroids (G centroids), within an object. Based on a linear GMF, a mass centroid and its subcentroids can be defined and extracted, which provide information on the location and orientation of an object. Similar to traditional moment functions, GMFs can also be used to describe the global shape of an object, including symmetry and fullness. However, GMFs, along with G centroids and subcentroids, can further serve to construct a feature vector of an object, which is critical for image registration and invariant pattern recognition. One can extract more distinguishing features from the same object by changing the combination of GMFs. We present results that show simulations of pattern recognition from uniform backgrounds.

© 2005 Optical Society of America

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### Equations (21)

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(1)
$${m}_{pq}={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}f(x,y){x}^{p}{y}^{q}\text{d}x\text{d}y,$$
(2)
$${m}_{pq}={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}f(x,y){(x-{x}_{0})}^{p}{(y-{y}_{0})}^{q}\text{d}x\text{d}y.$$
(3)
$${\mathrm{\Phi}}_{3}={({\mathrm{\eta}}_{30}-3{\mathrm{\eta}}_{12})}^{2}+{(3{\mathrm{\eta}}_{21}-{\mathrm{\eta}}_{03})}^{2},$$
(4)
$${m}_{pq}(x,y)=f(x,y)\otimes ({x}^{p},{y}^{q}).$$
(5)
$${m}_{10}=f(x,y)\otimes x,$$
(6)
$${m}_{01}=f(x,y)\otimes y.$$
(7)
$${m}_{10}({x}_{0},y)=0.$$
(8)
$${m}_{01}(x,{y}_{0})=0.$$
(9)
$$\text{Centroid}=\text{location}\{[f(x,y)\otimes (x+iy)]=0{\mid}_{\text{inside}\hspace{0.17em}f(x,y)}\}.$$
(10)
$$\text{Centroid}=\text{location}\{\text{Max}[|f(x,y)\times \otimes {m}_{f}(x,y)\mid \otimes G(x,y)]\},$$
(11)
$${m}_{w}(x,y)=[Sr(x)+iSi(y)]\text{circ}(2r/w),$$
(12)
$$\text{GMF}(x,y)=f(x,y)\otimes {m}_{w}(x,y).$$
(13)
$$Sr(-x)=-Sr(x),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}Si(-y)=-Si(y).$$
(14)
$$\begin{array}{l}\text{GMF}(s,t)=[f(x,y)+n(x,y)]\otimes {m}_{w}(x,y)\\ =f(x,y)\otimes {m}_{w}(x,y)+n(x,y)\otimes {m}_{w}(x,y)\\ =f(x,y)\otimes {m}_{w}(x,y)+\int \int n(x,y)Sr\\ \times \hspace{0.17em}(x-s)\text{d}x\text{d}y+i\int \int n(x,y)Si(y-t)\text{d}x\text{d}y\\ =f(x,y)\otimes {m}_{w}(x,y)+\int Sr(x-s)\\ \times \hspace{0.17em}\left[\int n(x,y)\text{d}y\right]\text{d}x+i\int Si(y-t)\\ \times \hspace{0.17em}\left[\int n(x,y)\text{d}x\right]\text{d}y\\ \cong f(x,y)\otimes {m}_{w}(x,y).\end{array}$$
(15)
$${\mathbf{P}}_{1}={P}_{1}(x,y)-{P}_{m}(x,y),$$
(16)
$${\mathbf{P}}_{2}={P}_{2}(x,y)-{P}_{m}(x,y).$$
(17)
$$\mathbf{P}={C}_{0}({x}_{0},{y}_{0})-{C}_{1}({x}_{1},{y}_{1})={\mathbf{p}}_{01}\hspace{0.17em}\text{exp}(i{\mathrm{\phi}}_{01}),$$
(18)
$$\mathbf{V}=[{\mathbf{P}}_{1},{\mathbf{P}}_{2},\dots {\mathbf{P}}_{n}].$$
(19)
$${D}_{f}=\mid {\mathbf{V}}_{1}-{\mathbf{V}}_{2}\mid .$$
(20)
$$m={{p}_{ij}}^{2}/{M}_{0},$$
(21)
$${M}_{0}={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}f(x,y)\text{d}x\text{d}y.$$