Basic Principles Photogrammetry V: Image Convolution & Moving Window: Dr. Gamal Hassan Seedahmed Dept. of Surveying Eng. Faculty of Eng. University of Khartoum Gamal.Seedahmed@gmail.com

Background Convolution is the process of combining the input signal with the effect of the response function (Filter). Input signal here refer to a digital image in terms of pixel values.

Convolution in Digital Images Input image is the signal Response function is a small array called a kernel or a filter Process is termed a Moving Window Product of kernel and image

Kernels Generally, sum of kernel elements equals one or zero Common size is 3x3, but 5x5, 7x7, etc. are also used. (very rare even dimensions)

The 3 Elements of Image Convolution You need an input image. You need a convolution kernel. Then you end up with an output image, which is the result of convolving the kernel with image.

The 3 Elements of Image Convolution (cont’d) Convolution kernel (K) Input image (I) Task: convolve the kernel to the left with the image to the right

The 3 Elements of Image Convolution (cont’d) The practical implementation of convolution requires the centering of the convolution kernel (K) on each image location and compute the sum of the multiplication of the kernel with local image neighborhood. Convolution kernel Centered at image Location (1,1). Center of the kernel

The 3 Elements of Image Convolution (cont’d) 1 2 3 4 5 6 7 8 9 Convolution kernel Centered at image Location (1,1). Center of the kernel Centering the kernel at the boundary of the image creates a problem. In the example above five of the elements of kernel are sitting outside the image {1,2,3,4,7}.

The 3 Elements of Image Convolution (cont’d) 1 2 3 4 5 6 7 8 9 One of the techniques of dealing with the boundary problem is to ignore the boundary. Then the starting image location for the Convolution is (2,2) (indicated above by a blue square).

The 3 Elements of Image Convolution: How to compute the convolution of an image with a kernel Input image Output image Now let us compute the result of the convolution at output image on Location (2,2):

The 3 Elements of Image Convolution: How to compute the convolution of an image with a kernel Output image Observe that we put zeros around the boundary of the image. This is known as padding by zeros to avoid the boundary problem.

The 3 Elements of Image Convolution: How to compute the convolution of an image with a kernel Input image Output image Now let us compute the result of the convolution at output image on Location (2,3):

The 3 Elements of Image Convolution: How to compute the convolution of an image with a kernel Input image Output image Now let us compute the result of the convolution at output image on Location (2,4):

The 3 Elements of Image Convolution: How to compute the convolution of an image with a kernel Input image Output image Then we need to repeat the process for the rest of the image. The practical implementation of convolution is very simple..

Average Kernel (Mean Filter) The kernel above is known as the average kernel of the Mean Filter. This is a 3 x 3 mean filter.

Gaussian Kernel Please very this. Question: Where is the number 16 come from? It came from the summation of the kernel elements. Please very this.

Vertical Edge Filter

Horizontal Edge Filter