In the numerical analysis and in the functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transformations, one key advantage it has over Fourier transforms is the temporal resolution: it captures both frequency and location information (time location).
The discrete wavelet transform has a large number of applications in science, engineering, mathematics and computer science. Most notably, it is used for signal coding, to represent a discrete signal in a more redundant way, often as a precondition for data compression. Practical applications can also be found in the processing of acceleration signals for the analysis of the gait, image processing, in digital communications and many others.
It is shown that the discrete wave transformation (discrete in scale and displacement, and continuous over time) is successfully implemented as an analog filter bank in biomedical signal processing for the design of low power pacemakers and also in wireless bandwidth communications Ultralarge (UWB).
Wavelets are often used to replace two-dimensional signals, such as images. The following example provides three steps for eliminating unwanted white Gaussian noise from the noisy image shown. Matlab was used to import and filter the image.
The first step is to choose a type of wavelet, and a level N of decomposition. In this case, biorthogonal waves 3.5 were selected with an N level of 10. Biorthogonal waves are commonly used in image processing to detect and filter white Gaussian noise due to their high contrast intensity values Pixel neighbors. Using this wavelets, a wavelet transformation is performed on the two-dimensional image.
After decomposition of the image file, the next step is to determine the threshold values for each level from 1 to N. The Birgé-Massart strategy is a fairly common method for selecting these thresholds. Using this process individual thresholds for N = 10 levels are made. The application of these thresholds is most of the actual filtering of the signal.
The final step is to reconstruct the image from the modified levels. This is achieved using an inverse wavelet transform. The resulting image, with white Gaussian noise, is shown below the original image. When filtering any form of data, it is important to quantify the signal-to-noise ratio of the result. In this case, the SNR of the noisy image compared to the original was 30.4958%, and the SNR of the recessed image is 32.5525%. The resulting improvement in wavelet filtering is a SNR gain of 2.0567%.
It is important to keep in mind that the choice of other wavelets, levels and threshold strategies can result in different types of filtering. In this example, white Gaussian noise was chosen to be eliminated. Although, with different thresholds, it could have been so easily amplified.
