Wavelet Transforms in Image Processing (I)
You dig deeper, and it gets more and more complicated, and you get confused, and it’s tricky, and its hard, but it's beautiful.
-Brian Cox
This will be an introductory blog on how we use wavelet transforms in image processing. If you all like it I would love to continue this series.
Wavelet transforms are an excellent tool used in different real-life phenomena. One of the most common is signal and image processing. It is a technique of changing an array of N numbers from their actual numerical values to various N wavelet coefficients. Wavelet functions are compact, they measure variations around a small region of the data array. Wavelets transform to have a “localized” nature which allows them to easily pick out features in data such as discrete objects, noises, etc. This powerful mathematical technique can be used in analyzing signals and images. Using DWT, the signal is decomposed into different frequency sub-bands.
An image is represented as a signal, especially a 2-dimensional signal, the pixel values represent the amplitude. For digital signals, amplitude is typically measured in the number of bits or levels used to represent the signal. The amplitude can be measured as maximum or minimum values of the signals,relative to the range of the values represented by the digital signal format. In signal processing, amplitude refers to the strength or magnitude of a signal. It represents the maximum deviation of a signal from its mean or average values.
Before we dive into the fundamental theories its better to be aware of the basic notions one might come across.
While analyzing the signals we may use Fourier transforms but it decomposes the function into analyzing functions, into sines and cosines. Now in this case there is a slight trade off of information. It is overall very difficult to have both frequency and time resolution so we compromise to get the best of both worlds.
Now the unit of the transformation is termed as wavelets. Wavelet is a shortlined oscillation localized in time.
To be a proper wavelet, some criteria must be fulfilled:
- Zero mean: This is also called the admissibility condition. The total area must be equal i.e area of positive peaks = area of negative troughs.
- It should have finite energy.
Some familiar wavelets are Haar, Daubechies, morlet, Shanon, etc.
Now one might ask why we had to switch to Wavelets when Fourier transforms worked so fine. The Fourier transform converts the time domain to the frequency domain, in the problem and hence solves it. But the Wavelet transform takes both domains into consideration and finds the final outcome. Hence we get precision.
Fourier transforms work for only stationary signals and can give information only for localized signal regions in time.
So Short Time Fourier Series came into the picture. It was developed to overcome the problems by Fourier Transform. It managed to address some of the problems but not all. In this, we take a window function of fixed length and moved it along the signal from start to end, and take a Fourier Transform at each stationary section. But the window function is finite so the frequency resolution decreases. The fixed length of the window means time and frequency resolutions are fixed for the entire length of the signal.
Keeping all the factors into consideration we choose wavelet transformation, especially for image processing. In my next blog, I will introduce you all to the important formulae and the types of wavelet transforms. Till then, bye!
