Image Processing

Bi-dimensional digital images are represented by pixels disposed on rows and columns in a matrix form. As shown in figure 1, digital processing of an image means obtaining a new image (output image) by applying an (linear or nonlinear) operator on the original image (input image).

Figure 1. Digital image processing.

Convolution filters are linear operators, by which the value of a pixel in the output image is calculated as a linear combination between the pixels of same coordinates in the input image and the values of its neighbors. The convolution operator is usually in the form of a square matrix of odd size (3 × 3, 5 × 5, etc..), also known as convolution mask. This operator will be centered, in turn, on each pixel in the input image. Convolution matrix elements represent the weighting coefficients of the window pixels, used for calculating the value of one pixel in the output image. All pixels covered, at a given time, by the convolution mask are considered in the computation of the value of a pixel in the output image. The output pixel value is sum of the products between the corresponding values in the convolution matrix and the input image. In other words, to calculate the pixel at the coordinates (x,y) in the output image, only the pixels in the input image which can be seen through a window size equal to the convolution matrix and centered on pixel coordinates (x,y) are taken into account.

For example we consider the image in figure.2.a) and the generalized operator of convolution of size 3 × 3 in figure.2.b)

Figure 2.

a) 5×5 image, b) generalized 3×3 convolution operator

In order to determine the pixel value at coordinates (x,y) in the output image, we will take into account pixels of window size 3 × 3 centered on the coordinates of the input image. Coordinates of the 9 pixels that appear in the window are:

(x+1,y-1) (x+1,y) (x+1,y+1)

(x,y-1) (x,y) (x,y+1)

(x-1,y-1) (x-1,y) (x-1,y+1)

Considering in(col,lin) and out(col,lin) the pixel values at coordinates (col,lin) in the input and out put image, respectively, we can write the following formula for computing the output pixel value at coordinates (x,y):

For example, the result at coordinates (1,2) obtained by applying the convolution operator in figure 2.b. on the image in figure 2.a is:

Border effect and ways to reducing it

As shown before, when computing the convolution of an image with a linear filter, for one pixel computation, besides the current pixel value, its neighboring pixels values are considered also. A special situation occurs when the processed pixels are located on the edges of the image, or when using operators larger than 3 × 3, the pixels in a broad band near the image edges are involved.

As noted in figure 3, where a 3 × 3 operator is used to process a pixel on the edge of an image, in the processing window occur 6 pixels from the image, and 3 pixels outside the image, while in the corners, just 4 pixels of the image enter the window and 5 from the outside.