Signal and image processing - simple Matlab tutorial

Overview:

Description
Signal processing summary
Image processing summary
More in depth signal processing
More in depth image processing
Results

Description

This is a very short and simple demo/tutorial for signal and image processing with Matlab without using specific high-level libraries

Signal processing:

Execute script signal_filter.m

We start from a clean sinusoid
Add some Gaussian noise and few other sinusoids ad different frequencies to mess the signal up a little bit.
Perform a low pass filtering
Perform a moving average filtering
Finally perform a moving average filtering on the previously low-pass filtered signal
plot the signals to see the results

Image processing:

Execute script image_filter.m

Read the input image lena_noise.png -> it is a grayscale image with Gaussian noise
Create kernels/filters:
- Laplacian filtering: edges enphasizing
- Directional filtering
- High-pass filtering: sharpening
Perform convolution between the image and the filters
Show the better-looking output image

More in depth explanation

More in depth signal processing:

Low-pass filter:

LTI filter --> system equation: $\Sigma_{i=0}^n a_i v[k-i] = \Sigma_{i=0}^n b_i u[k-i]$

For filters, it is useful to design in the frequency domain --> Z Transform

$Y(z) = H(z)X(z)$

Transfer function H(z): a filter's behavior can be entirely described by its transfer function

$H(z) = \frac{b_0+\Sigma_{j=1}^n b_j z^{-j}}{a_0+\Sigma_{k=1}^m a_k z^{-k}}$

In Matlab we have a signal with white noise and some high frequency disturbing components, so I used butter to get the transfer function of a low-pass filter to cut off those waves

Output:

$\Sigma_{k=0}^m a_k y[i-k] = \Sigma_{j=0}^n b_J x[i-j]$

$y[i] = \frac{(\Sigma_{j=0}^n b_j x[i-j])-(\Sigma_{k=1}^m a_k y[i-k])}{a_0}$

The i-th value of the output needs the previous output values, how many depends on the degree of the transfer function.
In out case the degree is 8, so I forced the first 7 values of the output to zero.
Collateral effect: this introduced a slight phase shifting

Moving average:

We have a window of dimension d, make it slide through the signal and every point of it is approximated to the average of the adjacent d values.
It’s important to choose a correct window dimension: