Statistical moments - An introduction

Moments are applicable to many different aspects of image processing, ranging from invariant pattern recognition and image encoding to pose estimation. When applied to images, they describe the image content (or distribution) with respect to its axes. They are designed to capture both global and detailed geometric information about the image. Here we are using them to characterise a grey level image so as to extract properties that have analogies in statistics or mechanics. In continuous form an image can be considered as a two-dimensional Cartesian density distribution function $f(x,y)$$f(x,y)$Two dimensional continuous function. With this assumption, the general form of a moment of order $(p+q)$, evaluating over the complete image plane $\xi$ is:

$$M_{pq} = \int \!\int_{\xi} \psi_{pq}(x,y) f(x,y)~dx~dy \hspace{1cm};\hspace{1cm} {p,q = 0,1,2,...,\infty}$$ (1)

The weighting kernel or basis function is $\psi_{pq}$. This produces a weighted description of $f(x,y)$ over the entire plane $\xi$. The basis functions may have a range of useful properties that may be passed onto the moments, producing descriptions which can be invariant under rotation, scale, translation and orientation. To apply this to digital images, Equation 1.1 needs to be expressed in discrete form. The probability density function (of a continuous distribution) is different from that of the probability of a discrete distribution. For simplicity we assume that $\xi$ is divided into square pixels of dimensions $1\times1$, with constant intensity $I$ over each square pixel. So if $P_{xy}$ $P_{xy}$Discrete pixelis a discrete pixel value then:

$$P_{xy} = I(x,y)\Delta A$$ (2)

where $\Delta A$ is the sample or pixel area equal to one. Thus, analysing over the complete discrete image intensity plane produces:

$$M_{pq} = \sum_x \sum_y \psi_{pq}(x,y) P_{xy} \hspace{1cm};\hspace{1cm}{p,q = 0,1,2,...,\infty}$$ (3)

The choice of basis function depends on the application and any desired invariant properties. The choice of basis may introduce constraints including limiting the $x$ and $y$ range, or translating the description and image to polar co-ordinates (eg. mapping it to the unit disc).