crazybiocomputing: Convolution

In Image Processing, one of the most important operation is the convolution because it is used everywhere for designing filters, mathematical morphology operators, analyzing particles, etc. Here is a small introduction...

1. Principle of Convolution

The Convolution operation corresponds to the shift of a small image called a convolution mask (or kernel) on top of an input image. The output image is reconstructed pixel by pixel by doing some calculations within the convolution mask (Fig.1). See Wikipedia for more information [Link].

Fig. 1: Principle of convolution. The input and output images are in blue and green, respectively. The 3x3 convolution mask in gray moves over the input image allowing the reconstruction pixel by pixel of the output image (in green). (Source: https://github.com/vdumoulin/conv_arithmetic).

2. Kernel Shapes

The convolution masks (or kernels) may be of various shapes. In ImageJ, the convention used to define the kernel shapes is based on its radius which is not very common. To get the correspondence between the shape you are looking for and the radius, an IJ utility located in Process > Filters > Show Circular Masks is available (Fig.2).

Fig.2: Kernel shapes in ImageJ (from radius 0.5 to 49.5)

Most of the time, we are using very small kernels and here is the correspondence between shape and radius for the most classical ones (Fig. 3).

Fig.3:Classical kernel shapes. A) 3x3 cross (radius=0.5). B) 3x3 square (radius = 1.0). C) 5x5 losange (radius = 1.5). D) 5x5 octagon (radius = 2.0).

By default, in ImageJ, the filters have a radius of 2.0 (5x5 octagon).

3. Managing Edges

For the image edges, the convolution mask cover outside parts of the image as shown in Fig. 1. For these outside pixels, there is no value. Thus, various strategies were defined to set these missing values.

Fig.4: Sample image based on the ImageJ clown image.

4.1. Crop the output image

As shown in Fig.1, the output edge pixels have incorrect values because, the computation in the convolution mask is based on outside undefined pixels. Thus, the simplest strategy is simply to remove those edge pixels of the output filtered image by cropping it.

4.2. Padding the image with values

Another simple strategy is to surround the image with pixels having the same values. The choice of this pixel value may vary. The simplest one is to put zero pixels like in the function Image > Adjust > Canvas Size and check the Zero fill.