University of Michigan - Ann Arbor

Edge detection is a basic image processing technique that helps to find the boundaries of objects in an image by detecting the sharp changes of brightness. It has been widely used in many fields, such as finding fingerprints of tumors and self-driving cars.

One of the applications of the convolution operation in CNN is edge detection. Tuning the parameters of the filter matrix (kernel) allows detecting distinct kinds of boundaries in the image, such as vertical and horizontal edges.

Boundary (Edge) Detection Using Convolution Operation

We have a 6 x 6 image. On its left half we have all tens and on the right half, all zeros. If you plot this as an image, it might look like the one below it, where the left half shows brighter pixels and the right half depict darker pixels. We use a shade of gray to denote zeros. In this image, there is clearly a very strong vertical edge as it transitions from white to black or white to darker color. When you convolve this with the 3 by 3 filter, in the middle, with brighter pixels on the left, mid tone zeroes in the middle, and darker on the right, you'd get the matrix on the right. If we plot the right matrix's image, we will find a lighter region in the middle that corresponds to a vertical edge.

Convolution-based Vertical Edge Detection

If there is a lot of noise in an image, it is hard to detect the edges of an object directly. As a solution, we first preprocess using Gaussian filters to blur the image. After the Gaussian blurring, the edge detection will be more effective.

Preprocessing Before Boundary (Edge) Detection Using Convolution Operation

Image gradient is a value that describes changes in images. It defines as follows:
$\nabla I =(\frac{\partial I}{\partial x},\frac{\partial I}{\partial y})$
It can be approximated by discrete derivatives:
$d_0 = [1, -1], f \circ d_0 = f[n] - f[n-1]$
$d_1 = [1, 0, -1]/2, f \circ d_1 =\frac{f[n+1] - f[n-1]}{2}$
In practice, we can use a filter such as $[1, -1]$, $[1, 0 ,-1]$, $[1, -1]^T$ and convolute it with our image.


Math Behind Boundary (Edge) Detection Using Convolution Operation

We can apply different filters to detect edges in different directions.

Visualization of Boundary (Edge) Detection Using Convolution Operation

Convolution Edge Detection in Positive vs. Negative OR Light to Dark vs. Dark to Light Edge Transitions

For detecting edges in some complicated images, it is proposed to use back-propagation when training the Deep learning model to find the optimal weights for the filter (kernel), rather than handpicking the weights. The objective is to find. the optimal values for these weights as parameters so that it appropriately detects the edges.
It may even learn to detect edges that are at 45 degrees or 73 degrees.

Setting Convolution-based Edge Detection Filter (Kernel) Weights Using Backpropagation

For vertical edge detection, we usually use the following kernels:
- **Sobel Filter**: gives more weight to the pixels in the central rows, which may make it more robust.
- **Scharr Filter**

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