Final Activity Report Summary - MAMEBIA (Mathematical Methods in Biological Image Analysis)
The aim of the MAMEBIA project is the development of theoretical and concrete mathematical methods to model and analyse biological image data, with an emphasis on complex-valued methods and phase information.
At the moment, the mathematical models and transforms regarded in biological contexts, apart from the Fourier transform, are mostly real-valued. This restriction is often based on the assumption that all biological data is real valued, and that complex-valued methods only increase the needed storage space and computation time of algorithms, but don't contribute to better analysis quality.
But researchers in image analysis become more and more aware that even for real data complex-valued methods yield much better performance. These methods extract phase, which gives directional information for edge detection, and codes local features. Most information of an image is coded in the phase. But its extraction with mathematical transforms and its interpretation is not yet fully understood. This might be the reason why phase information is rarely used for biological image analysis.
The MAMEBIA project aimed to bridge this gap. The team models biological problems, and formulates them in a sound mathematical manner. On this basis, the team develops new, and adapts existing complex-valued transforms to extract the modelled image features. Particularly, harmonic and nonharmonic Fourier transforms, as well as multiresolution approaches, as Gabor analysis and complex continuous and discrete wavelet methods are emphasized. Algorithms and concrete programs are developed and implemented. The team closely cooperates with collaborators from biology to ensure the high quality of the models and to have a significant validation. This close cooperation allows the amalgamation of knowledge and know-how, and ensures that both mathematics and biology benefits strongly from this interdisciplinary research.
A choice of scientific highlight are:
Complex B-Splines, Dirichlet means, statistics and multivariate versions: Complex B-splines are an extension of Schoenberg's cardinal splines to include complex orders. They are indeed piecewise polynomials, with an additional phase and a modulation/scaling factor in the Fourier domain. The frequency components on the negative and positive axis are enhanced by the opposite sign. The complex B-splines are scaling functions for multiresolution analyses. We exhibited relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and proved a generalization of the Hermite-Genocchi formula. This generalised Hermite-Genocchi formula then gave rise to a more general class of complex B-splines that allows for some interesting stochastic interpretation, e.g. for submartingales and Poisson processes. Moreover, the notion of complex B-splines was extended to a multivariate setting by means of ridge functions employing the known geometric relationship between ordinary B-splines and multivariate B-splines. A Fourier domain representation makes them useful for signal and image analysis tasks.
Riesz Wavelets and steerable frames: We considered an extension of the 1D concept of analytical wavelets to nD which is by construction compatible with rotations. This extension, the monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using Riesz transforms and perfectly isotropic wavelets frames. Employing the new concept of Clifford frames, we showed that the monogenic wavelet generates a wavelet frame. Furthermore, this approach yields steerable wavelet frames. Applications in image processing are e.g. descreening of half tone images and contrast enhancement.
Color Filter Arrays (CFAs): We proposed two new types of random patterns with RGB colours, which allow to design CFAs with better spectral properties. With these new CFAs, the demosaicing artefacts appear as incoherent noise, which is less visually disturbing than the Moiré structures characteristic of CFAs with periodic patterns.
At the moment, the mathematical models and transforms regarded in biological contexts, apart from the Fourier transform, are mostly real-valued. This restriction is often based on the assumption that all biological data is real valued, and that complex-valued methods only increase the needed storage space and computation time of algorithms, but don't contribute to better analysis quality.
But researchers in image analysis become more and more aware that even for real data complex-valued methods yield much better performance. These methods extract phase, which gives directional information for edge detection, and codes local features. Most information of an image is coded in the phase. But its extraction with mathematical transforms and its interpretation is not yet fully understood. This might be the reason why phase information is rarely used for biological image analysis.
The MAMEBIA project aimed to bridge this gap. The team models biological problems, and formulates them in a sound mathematical manner. On this basis, the team develops new, and adapts existing complex-valued transforms to extract the modelled image features. Particularly, harmonic and nonharmonic Fourier transforms, as well as multiresolution approaches, as Gabor analysis and complex continuous and discrete wavelet methods are emphasized. Algorithms and concrete programs are developed and implemented. The team closely cooperates with collaborators from biology to ensure the high quality of the models and to have a significant validation. This close cooperation allows the amalgamation of knowledge and know-how, and ensures that both mathematics and biology benefits strongly from this interdisciplinary research.
A choice of scientific highlight are:
Complex B-Splines, Dirichlet means, statistics and multivariate versions: Complex B-splines are an extension of Schoenberg's cardinal splines to include complex orders. They are indeed piecewise polynomials, with an additional phase and a modulation/scaling factor in the Fourier domain. The frequency components on the negative and positive axis are enhanced by the opposite sign. The complex B-splines are scaling functions for multiresolution analyses. We exhibited relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and proved a generalization of the Hermite-Genocchi formula. This generalised Hermite-Genocchi formula then gave rise to a more general class of complex B-splines that allows for some interesting stochastic interpretation, e.g. for submartingales and Poisson processes. Moreover, the notion of complex B-splines was extended to a multivariate setting by means of ridge functions employing the known geometric relationship between ordinary B-splines and multivariate B-splines. A Fourier domain representation makes them useful for signal and image analysis tasks.
Riesz Wavelets and steerable frames: We considered an extension of the 1D concept of analytical wavelets to nD which is by construction compatible with rotations. This extension, the monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using Riesz transforms and perfectly isotropic wavelets frames. Employing the new concept of Clifford frames, we showed that the monogenic wavelet generates a wavelet frame. Furthermore, this approach yields steerable wavelet frames. Applications in image processing are e.g. descreening of half tone images and contrast enhancement.
Color Filter Arrays (CFAs): We proposed two new types of random patterns with RGB colours, which allow to design CFAs with better spectral properties. With these new CFAs, the demosaicing artefacts appear as incoherent noise, which is less visually disturbing than the Moiré structures characteristic of CFAs with periodic patterns.