Images formulation

An image can be described as a 2D function, f(x,y), where (x,y) are the spatial coordinates and the value of f at any point, (x,y), is proportional to the brightness or gray levels of the image. Additionally, when both (x,y) and brightness values of f are all finite discrete quantities, the image is called a digital image. Therefore, f(x,y) takes the following values:

• x ∈ [0, h-1], where h is the height of the image
• y ∈ [0, w-1], where w is the width of the image
• f(x,y) ∈ [0, L-1], where L = 256 (for an 8-bit image)

A color image can be represented in the same way, but we need to define three functions to represent the red, green, and blue values, respectively. Each of these three individual functions follows the same formulation as the f(x,y) function that was defined for grayscale images. We will denote these three functions subindex R, G and B for the three formulations (for the color images) as fR(x,y), fG(x,y), and fB(x,y).

A black and white image follows the same approximation in the way that only one function is required to represent the image. However, one key point is that f(x,y) can only take two values. Usually, these values are 0 (black) and 255 (white).

The following screenshot shows the three types of images (a color image, a grayscale image, and a black and white image):