For applications in signal and image processing, that is, applications on discrete data, a discrete wavelet transform must exist. For a discrete wavelet transform it is not necessary to translate and scale the mother wavelet continuously. Rather, it is possible to rewrite Equation (4.7) to reflect discrete translation and scaling steps j and k: where […]

# Biomedical Image Analysis

## Two-Dimensional Discrete Wavelet Transform (Biomedical Image Analysis)

Up to this point, we have covered the one-dimensional discrete wavelet transform. In image processing, a multidimensional wavelet transform is desired. Fortunately, the wavelet transform is a linear operation. Therefore, analogous to the Fourier transform, the two-dimensional wavelet transform can be performed by first computing the row-by-row one-dimensional wavelet transform in the horizontal direction, followed […]

## Wavelet-Based Filtering Part 1 (Biomedical Image Analysis)

We discussed the decomposition of a signal (or image) into a lowpass- and highpass-filtered component. Wavelet-based filters can be envisioned as algorithms where wavelet decomposition takes place, followed by an attenuation of either the lowpass-filtered component or the highpass-filtered component. An inverse wavelet transform then restores the filtered image. The underlying principle can be seen […]

## Wavelet-Based Filtering Part 2 (Biomedical Image Analysis)

Wavelet-Based Highpass Filtering The complementary operation to wavelet-based lowpass filtering is wavelet-based highpass filtering, a process analogous to highpass filtering in the frequency domain. To create a wavelet-based highpass filter, the wavelet coefficients of the lowest detail level, that is, the lowpass output of the last subband stage (the Xk), need to be attenuated. Although […]

## Comparison of Frequency-Domain Analysis To Wavelet Analysis (Biomedical Image Analysis)

Both the Fourier transform and the wavelet transform provide linear decompositions of a signal f (t) into coefficients ak such that andare the basis functions. In the case of the Fourier transform, theare complex oscillationsThese oscillations are continuous from which implies that the function f (t) is aperiodic signal that stretches fromThe wavelet transform, on […]

## Biomedical Examples (Biomedical Image Analysis)

The importance of the wavelet transform for biomedical image processing was realized almost immediately after the key concepts became widely known. Two reviews18,20 cover most of the fundamental concepts and applications. In addition, the application of wavelets in medical imaging has led to development of powerful wavelet toolboxes that can be downloaded freely from the […]

## Adaptive Filtering (Biomedical Image Analysis)

Conventional (nonadaptive) filters were introduced in Section 2.3. These are operators that act equally on all areas of an image. Conversely, adaptive filters change their behavior with the properties of the local neighborhood of the image to be filtered. One example of an adaptive filter is local contrast enhancement by histogram equalization. Histogram equalization is […]

## Adaptive Noise Reduction Part 1 (Biomedical Image Analysis)

Noise reduction is one of the most important image processing steps, particularly in biomedical image processing. Noise is broadly distributed over the frequency spectrum of the image. The conventional approach to reducing noise is the application of a blurring filter (e.g., Gaussian smoothing or lowpass filtering in the frequency domain). Unfortunately, degradation of high-frequency componentsâ€”edges […]

## Adaptive Noise Reduction Part 2 (Biomedical Image Analysis)

Anisotropic Diffusion Lowpass Filter Perona and Malik introduced a very powerful adaptive filter that operates by numerically simulating anisotropic diffusion.28 Image intensity values can be thought of as local concentrations caught inside the pixel. If pixel boundaries are thought of as semipermeable, the pixel intensity would diffuse over time into neighboring pixels of lower intensity […]

## Adaptive Filters In The Frequency Domain: Adaptive Wiener Filters (Biomedical Image Analysis)

Frequency-domain filtering was introduced in topic 3. One particularly important filter is the Wiener filter which finds its main application in image restoration (Section 3.2.3). Let us recall the general equation of the Wiener filter in the frequency domain: Here, H(u,v) is the Fourier transform of the point-spread function of the degradation process h(x,y) and […]