Image enhancement algorithms: grey scale transformations

Any digital image can be represented mathematically in matrix form. The number of lines in the matrix is the number of lines of the digital image (also called image height), whereas the number of columns in the matrix is the number of columns in the digital image (also called image width). In the following we will consider the simplest case, i.e., the case of grey scale image processing; in that case, the elements of the matrix used to represent our digital image are integer positive values in a finite set, which represent the brightness of the pixels in the digital image. The point processing (also known as pixel processing) of grey scale images is, mathematically speaking, the simplest class of algorithms that can be used to process (modify) a digital image. This class of processing algorithms is used for a long time in the digital image processing field. The point processing mathematical algorithms often constitute the "foundation" for much more complex types of image processing algorithms. Since these operations basically just modify the brightness in a spatial location, independent to the surrounding brightness or to the region in the image, they can be simply seen as modifications of the grey levels in the image; therefore they are also called grey scale transformations and can be implemented directly on the image histogram. The point processing (or pixel processing or grey scale processing) operations is the class of algorithms in which the brightness (grey level) of each individual pixel in the output image depends solely on the grey level of the pixel found in the same spatial position in the input image. The point processing algorithms are generally based on simple mathematical operations. Usually, the processing is described by a linear or non-linear transformation of the grey scale range of the input image, and this transformation determines the new grey scale range in the output image. This linear or non-linear transformation is typically described either analytically or graphically. In the most common representation of grey scale digital images, the brightness Y takes integer values in the range [0; 255]; therefore the grey scale transformation must be defined for 256 discrete values. An important class of point processing operations is the one devoted to image enhancement. The goal of image enhancement is to improve the visual appearance of an image or to enhance particular features in the image, to make the examination of its contents easier for the human observer. In this laboratory, we will examine only some of the image enhancement algorithms based on point processing. In the following, the input and output (processed) images are grey scale images, with the minimum grey level – corresponding to black – represented by the value 0, and the maximum grey level – corresponding to white – denoted by LMax (typically, LMax=255, for an 8 bit per pixel image representation). The input (original) grey scale image (to be processed) is represented by the matrix U[M×N], where M is the number of lines in the image, and N is the number of columns in the image. Let us denote the brightness of any pixel in the input image U by u (regardless of its spatial position in the image), } ,..., 1 , 0 { Max L u ∈ . In a similar way, we represent the processed