Editing Digital image processing (section)

=== Filtering ===
Digital filters are used to blur and sharpen digital images. Filtering can be performed by:
* [[Kernel (image processing)#Convolution|convolution]] with specifically designed [[Kernel (image processing)|kernels]] (filter array) in the spatial domain<ref name=":0">{{Cite journal|last1=Zhang|first1=M. Z.|last2=Livingston|first2=A. R.|last3=Asari|first3=V. K.|date=2008|journal=International Journal of Computers and Applications|volume=30|issue=4|pages=298–308|doi=10.1080/1206212x.2008.11441909|title=A High Performance Architecture for Implementation of 2-D Convolution with Quadrant Symmetric Kernels|s2cid=57289814}}</ref>
* masking specific frequency regions in the frequency (Fourier) domain

The following examples show both methods:<ref name="Gonzalez 2008">{{cite book
  |last = Gonzalez
  |first = Rafael
  |title = Digital Image Processing, 3rd
  |publisher = Pearson Hall
  |date = 2008
  |isbn = 978-0-13-168728-8
}}</ref>

{| class="wikitable"
|-
! Filter type
! Kernel or mask
! Example
|-
| '''Original Image'''
| align="center" | <math>
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}
</math>
| [[File:Affine Transformation Original Checkerboard.jpg]]
|-
| '''[[lowpass|Spatial Lowpass]]'''
| align="center" | <math>
\frac{1}{9}\times
\begin{bmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{bmatrix}
</math>
| [[File:Spatial Mean Filter Checkerboard.png]]
|-
| '''[[Edge detection|Spatial Highpass]]'''
| align="center" | <math>
\begin{bmatrix}
0 & -1 & 0 \\
-1 & 4 & -1 \\
0 & -1 & 0
\end{bmatrix}
</math>
 | [[File:Spatial Laplacian Filter Checkerboard.png]]
|-
| '''[[Fast Fourier transform|Fourier Representation]]'''
| Pseudo-code:

image = checkerboard

F = Fourier Transform of image

Show Image: log(1+Absolute Value(F))

| align="center"| [[File:Fourier Space Checkerboard.png]]
|-
| '''Fourier Lowpass'''
| align="center"| [[File:Lowpass Butterworth Checkerboard.png]]
| align="center"| [[File:Lowpass FFT Filtered checkerboard.png]]
|-
| '''Fourier Highpass'''
| align="center"| [[File:Highpass Butterworth Checkerboard.png]]
| align="center"| [[File:Highpass FFT Filtered checkerboard.png]]
|-
|}

==== Image padding in Fourier domain filtering ====

Images are typically padded before being transformed to the Fourier space, the [[highpass filter]]ed images below illustrate the consequences of different padding techniques:

{| class="wikitable"
|-
! Zero padded
! Repeated edge padded
|-
| [[File:Highpass FFT Filtered checkerboard.png]]
| [[File:Highpass FFT Replicate.png]]
|-
|}

Notice that the highpass filter shows extra edges when zero padded compared to the repeated edge padding.

==== Filtering code examples ====

MATLAB example for spatial domain highpass filtering.

<syntaxhighlight lang="matlab">
img=checkerboard(20);                           % generate checkerboard
% **************************  SPATIAL DOMAIN  ***************************
klaplace=[0 -1 0; -1 5 -1;  0 -1 0];             % Laplacian filter kernel
X=conv2(img,klaplace);                          % convolve test img with
                                                % 3x3 Laplacian kernel
figure()
imshow(X,[])                                    % show Laplacian filtered
title('Laplacian Edge Detection')
</syntaxhighlight>