Representing images mathematically

As mentioned in my previous post, digital images are often represented in the form of a matrix. In this post, we only consider the gray-scale images and the color images will be discussed later. If we consider p and q  the location of pixel in the image and X the matrix which is represented, then X(p,q) i.e. the value of matrix in the row p and in the column q, represents the gray level value of the pixel.

Figure 1: Representing a gray-scale image as a matrix for image processing applications

  

Operation on images:

It is very important in the following discussions of our blog to know the so-called image operator. Operator is a function which applies to an image and can represent many modifications done on real images such as degradation caused by the noise, missing pixels, blurring or many possible filters applied to images. Since we are dealing with matrices many image operators can be modeled by matrix element-wise operators. However, in more complicated applications, it is sometimes better to represent the image matrix discussed above as a vector. The ordering can arbitrarily be lexicographical, column-wise, row-wise, etc. In the figure below, row-wise vectorization of the above image is shown.

Figure 2: vectored representation of the image in figure 1.