Image calculations, sometimes termed image math, refer to arithmetic or logic operations on a pixel-by-pixel basis. Image values can either be manipulated with the same constant value for all pixels or with the values of corresponding pixels in a second image. Any mathematical expression is possible, and some operations are very useful in practical applications. […]

# Biomedical Image Analysis

## Binary Image Processing (Biomedical Image Analysis)

The segmentation results discussed in this topic generally lead to binary images, considered to be masks for the segmented features. The information contained in the image values is lost, but the spatial relationship between the mask and the features is retained. Both mask and masked image can be subjected to an analysis of texture (topic […]

## Biomedical Examples (Biomedical Image Analysis)

In this topic, basic tools for a complete image analysis chain are introduced. These include, in order, image preprocessing (image enhancement or restoration, usually by filtering), image segmentation, some post segmentation enhancement, and finally, quantitative analysis. One very typical example was presented by de Reuille et al.,5 who acquired confocal sections of the shoot apical […]

## Image Processing in The Frequency Domain (Biomedical Image Analysis)

In previous topics the image was introduced as a spatial arrangement of discrete values: image values that represent a physical metric. Neighboring pixels relate to each other in a defined spatial relationship. The human eye is very adept at recognizing spatial relationships, such as repeat patterns, irregularities (noise), edges, or contiguous features. For image processing […]

## The Fourier Transform Part 1 (Biomedical Image Analysis)

French mathematician Joseph Fourier found that any 2^–periodic signal f (t) can be represented by an infinite sum of sine and cosine terms according to When a signal f (t) is given, the coefficients ak and bk can be determined using Fourier analysis of the signal f (t): where ak and bk are called […]

## The Fourier Transform Part 2 (Biomedical Image Analysis)

Abrupt changes of image intensity, such as lines or edges, have many frequency components. The Fourier transform of a step (such as the transition from 1 to 0) exhibits a broad spectrum with a (sin w)/w characteristic. If the intensity change is more gradual, its frequency components drop off more rapidly toward higher frequencies. Figure […]

## Fourier-Based Filtering (Biomedical Image Analysis)

In Section 3.1, the possibility of using the Fourier transform to remove periodic components from images was introduced briefly. However, frequency-domain filtering is far more comprehensive. The foundation of frequency-domain filtering is the convolution theorem, which stipulates that a convolution in the spatial domain corresponds to a multiplication in the Fourier domain. Computational effort of […]

## Other Integral Transforms: The Discrete Cosine Transform and The Hartley Transform (Biomedical Image Analysis)

Other integral transforms exist that have properties similar to those of the Fourier transform but are real-valued. Most notably, these are the discrete cosine transform26 and the discrete Hartley transform.4,16 For both of them, a fast recursive formulation similar to the FFT exists.5 Like the Fourier transform, both transforms decompose the image into its frequency […]

## Biomedical Examples (Biomedical Image Analysis)

Image processing in the Fourier domain is a fundamental tool not only in the process of manipulating and enhancing images but also in image formation. Most computerized imaging modalities rely on the Fourier transform at some point of the image formation process. The following examples of computed tomography and magnetic resonance image formation highlight the […]

## The Wavelet Transform and Wavelet-Based Filtering (Biomedical Image Analysis)

The wavelet transform belongs to the class of integral transforms, such as the Fourier transform. Whereas the Fourier transform uses sine and cosine functions as basis functions, the wavelet transform uses special functions with finite support, termed wavelets. The most fundamental difference between wavelets and the Fourier transform is the scalability of wavelets. Whereas the […]