Convolution Principles

Building the Cantata Workspace

At the time this lesson was prepared, both the linear and circular convolution operators in Khoros had small problems. So to fully reproduce linear and circular convolution with the Khoros Linear operator, the original images have to be zero extended for linear convolution or periodically extended for the circular convolution. Also, in both cases the correct portion of the image has to be extracted.

In this experiment the image $DIP/data/draw-small.viff is used. It is a 32 x 32 image with floating point pixels with values 0.0 and 1.0.

1. Create the zero extended image

Select the image $DIP/data/draw-small.viff and zero extended it to size 64 x 64, so that the input image appears in the center. Use operator Pad.

2. Create the periodic image

Select the image $DIP/data/draw-small.viff and create a 4 x 4 mosaic using the operator Inset twice. First to put two images side by side and the second to put this two-images top by bottom.

3. Create the Zoom Display Procedure

As the extended or periodic image is small (64x64), create a procedure to display it zoomed by 4 using the operators: Expand and Display Image. Call this procedure Zoom Display. You can see how to create a procedure in cantata.

4. Building the kernels

shifted h(2,2) = 1
Use the operator Impulse.
3x3 average
Use the operator Constant.
Sobel X
Use the file $DIP/data/sobel_x.viff and operator User defined. You can view this ASCII file using the File Viewer operator.

5. Performing the convolution

Use operator Linear Operator to perform the convolution.

Extracting the correct portion of the resultant image

Use the operator Extract to select the correct portion of the image.

For linear convolution, you need to extract an image 2 pixels wider and 2 pixels higher that the original since the kernel dimensions are 3x3.

For periodic convolution, you need to extract the bottom left image, since the kernel origin is at (0,0)

Exercises

Explain how the position of the extracted image is dependent on the kernel origin.
Create an encapsulate workspace operator to fully implement the linear and circular convolution for any kind of kernel.

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