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This paper introduces an image denoising procedure based on a 2D scale-mixing complex-valued wavelet transform. Both the minimal (unitary) and redundant (maximum overlap) versions of the transform are used. The covariance structure of white noise in wavelet domain is established. Estimation is performed via empirical Bayesian techniques, including versions that preserve the phase of the complex-valued wavelet coefficients and those that do not. The new procedure exhibits excellent quantitative and visual performance, which is demonstrated by simulation on standard test images.

Breast cancer is the second leading cause of death in women in the United States. Mammography is currently the most eective method for detecting breast cancer early; however, radiological inter- pretation of mammogram images is a challenging task. Many medical images demonstrate a certain degree of self-similarity over a range of scales. This scaling can help us to describe and classify mammograms. In this work, we generalize the scale-mixing wavelet spectra to the complex wavelet domain. In this domain, we estimate Hurst parameter and image phase and use them as discriminatory descriptors to clas- sify mammographic images to benign and malignant. The proposed methodology is tested on a set of images from the University of South Florida Digital Database for Screening Mammography (DDSM). Keywords: Scaling; Complex Wavelets; Self-similarity; 2-D Wavelet Scale-Mixing Spectra.

SUMMARYIn this paper, we consider some approaches to spatio-temporal modeling of environmental data obtained from an heterogeneous network.Besides discussing modeling details for spatio-temporal dynamics and calibration of different instruments, we consider crossvalidation issues and extensions to the monitoring network assessment based on sensitivity analysis.We then consider a case study based on the heterogeneous network related to fine particulate matter (PM 10 )coming from the Po Valley, North Italy.

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Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

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