Error Analysis in the Computation of Orthogonal Rotation Invariant Moments

Abstract

Orthogonal rotation invariant moments (ORIMs) are among the best region based shape descriptors. Being orthogonal and complete, they possess minimum information redundancy. The magnitude of moments is invariant to rotation and reflection and with some geometric transformation, they can be made translation and scale invariant. Apart from these characteristics, they are robust to image noise. These characteristics of ORIMs make them suitable for many pattern recognition and image processing applications. Despite these characteristics, the ORIMs suffer from many digitization errors, thus they are incapable of representing subtle details in image, especially at high orders of moments. Among the various errors, the image discretization error, geometric and numerical integration errors are the most prominent ones. This paper investigates the contribution and effects of these errors on the characteristics of ORIMs and performs a comparative analysis of these errors on the accurate computation of the three major ORIMs: Zernike moments (ZMs), Pseudo Zernike moments (PZMs) and orthogonal Fourier-Mellin moments (OFMMs). Detailed experimental analysis reveals some interesting results on the performance of these moments.

Appendix

1.1 A.1 Cubic Interpolation of Image Function

An image consisting of N×N pixels assumes that the pixel values f(i,k) are defined at the center of the pixel which occupies the area

$$\biggl[ x_{i} - \frac{\Delta x}{2},y_{k} - \frac{\Delta y}{2} \biggr] \times\biggl[ x_{i} + \frac{\Delta x}{2},y_{k} + \frac{\Delta y}{2} \biggr] $$

and has the center at (x _i ,y _k ) as shown in Fig. 1(a). The cubic interpolation assumes that the discrete data, which is converted into a continuous function, must be defined at the crossing of the grids which are given the name ‘nodes’ for further reference. Let the coordinates of nodes be denoted by (X _i ,Y _k ), which are given by

$$ \begin{aligned}[c] &X_{i} = \frac{2i - N}{N}, \\ & Y_{k} = \frac{2k - N}{N}, \quad i,k = 0,1,\ldots,N \end{aligned} $$

(32)

The cubic interpolation method requires the discrete values of the function at locations (X _i ,Y _k ), for i,k,=0,1,…,N, whereas the pixels are defined at pixel centers (x _i ,y _k ),i,k=0,1,…,N−1. Thus for an N×N image, there are (N+1)×(N+1) nodes at which the discrete values are required. A simple solution to this problem is to define a discrete function which is obtained by taking the average of pixel values meeting at a given node. Mathematically, we can obtain g(i,k) as follows

$$\begin{aligned} &g(i,k) = \frac{1}{S}\sum_{u = 0}^{1} \sum_{v = 0}^{1} f(i - u,k - v), \\ &\quad i,k = 0,1,2,\ldots, N \end{aligned}$$

(33)

where

$$\begin{aligned} &f(l,m) = 0,\quad\mbox{for}\ l = - 1,l = N,m = - 1,m = N,\quad\mbox{and} \\ & S = \begin{cases} 1,&\mbox{for corner nodes} \\ 2,&\mbox{for edge nodes} \\ 4,&\mbox{for interior nodes} \end{cases} \end{aligned}$$

After deriving g(i,k) the value of the image intensity can be obtained at a point (x,y)∈[x _i ,y _k ]×[x _i+1,y _k+1], i,k=0,1,…,N−1, as given below

$$ g(x,y) = \sum_{l = 0}^{3} \sum _{m = 0}^{3} a_{lm} x^{l}y^{m}, $$

(34)

where

$$ a_{mn} = \frac{1}{4\Delta^{6}}\sum_{r = 0}^{3} \sum_{s = 0}^{3} g(i + r - 1,k + s - 1) u_{rl}(x_{i})u_{sm}(y_{k}) $$

(35)

with

$$\Delta= \frac{2}{N} $$

The expressions u _rl (x _i ) and u _sm (y _k ) are given in [7]. The intensity function g(i,k) is derived for i,k=0,1,…,N as given by Eq. (33). However, Eq. (35) requires values for the extended grid which are determined by the procedure given in [7].