# Image Filtering

### Three views of filtering

• Image filters in spatial domain
• Filter is a mathematical operation of a grid of numbers
• Smoothing, sharpening, measuring texture
• Image filters in the frequency domain
• Filtering is a way to modify the frequencies of images
• Denoising, sampling, image compression
• Templates and Image Pyramids
• Filtering is a way to match a template to the image
• Detection, coarse-to-fine registration

### Example

Box filter

• Replaces each pixel with an average of its neighborhood
• Smoothing

Given a 3-by-3 box filter in the graph below

We will be able to find the filtered image, and the result looks like below (right one).

We also have some other popular and useful filters.

Sobel filter

Vertical Sobel filter
Horizontal Sobel filter

Now you may think that a Sobel filter can be used to find the edge in an image. And you are right. I have tried to merge two result images from the vertical and horizontal

### Properties of linear filters

Linearity:

filter(f1 + f2) = filter(f1) + filter(f2)

Shift invariance: same behavior regardless of
pixel location

filter(shift(f)) = shift(filter(f))

Any linear, shift-invariant operator can be
represented as a convolution

### Important filter: Gaussian

Weight contributions of neighboring pixels by nearness

Smoothing with Gaussian filter

Smoothing with Gaussian filter

Smoothing with box filter

Smoothing with box filter

A Gaussian filter can do this better since it keeps "more information" than a box filter by weighting contributions from neighbors.

### Practical matters

How big should the filter be?

• Values at edges should be near zero
• Rule of thumb for Gaussian: set filter half-width to

• the filter window falls off the edge of the image
• need to extrapolate
• methods:
• clip filter (black)
• wrap around
• copy edge
• reflect across edge

What is the size of the output?

### Median filter

• A Median Filter operates over a window by
selecting the median intensity in the window.
• What advantage does a median filter have over
a mean filter? (Check the picture below!)
• Is a median filter a kind of convolution?
Comparison: salt and pepper noise

Have you seen the superior advantage of applying a mean filter?

### Reference:

Thanks for reading. If you find any mistake / typo in this blog, please don't hesitate to let me know, you can reach me by email: jyang7[at]ualberta.ca