Today, digital images are massively used in all kinds of applications: entertainment, multimedia, meteorology, medical applications. The quality of the acquisition device and the conditions in which the image is taken affect the quality of the resulting image in a drastic way. Depending of these factors, the images can be over or underexposed, blurry or noisy.This thesis focuses on the problem of removing the noise from digital images. The developments proposed in this work aim at improving the denoising results of the statistical filters and provide solutions to use them in the specific case of medical images The statistical filters are essentially mode estimators. As mode separation improves in high dimensional spaces, we propose to adapt the statistical filters to use high-dimensional data. As a result, two new types of filters are created: Patch-based filters and patchwise filters. These filters turns out to be a generalisation of some other patch-based filters proposed in the literature. The statistical filters use the Gaussian kernel as notion of similarity. This kernel is not adapted to the high-dimensional spaces. We propose to use other kernels, constructed in order to avoid the complications linked to the use of high-dimensional data in the statistical filters. In real life applications, such as medical images, the parameters of the filters are chosen in a heuristic way. A new method to model the noise created by a medical scanner is proposed. It estimates the optimal parameters needed to remove the noise from the images taken with that scanner. Some medical images are polluted with a noise that follows a Poisson law. The statistical filters are not designed to filter with this kind of noise. Variance stabilizing transforms are used in order to change the noise on the image to Gaussian noise, and then apply the statistical filters. An integration of the variance stabilizing transforms into the algorithms is proposed.

Mode estimation is extensively studied in statistics. One of the most widely used methods of mode estimation is hill-climbing on a kernel density estimator with gradient ascent or a fixed-point approach. Within this framework, Gaussian kernels proves to be a natural and intuitive option for non-parametric density estimation. This paper shows that in the case of high-dimensional data, mode estimation can be improved by using differently shaped kernels, called flat-top kernels. The improvement are illustrated with an image denoising application, in which pictures are decomposed into small patches, i.e. groups of adjacent pixels, that are vectorized. Noise in the patches can be attenuated by substituting them with the closest mode in the observed distribution of patches. The quality of the denoised picture then depends on the accuracy of mode estimation in a high-dimensional space. Experiments conducted on usual benchmarks in the image processing community show that flat-top kernels outperform the Gaussian one.

Denoising is a cornerstone of image analysis and remains a very active research field. This paper deals with image filters that rely on similarity kernels to compute weighted pixel averages. Whereas similarities have been based on the comparison of isolated pixel values until recently, modern filters extend the paradigm to groups of pixels called patches. Significant quality improvements result from the mere replacement of pixel differences with patch-to-patch comparisons directly into the filter. Our objective is to cast this generalization within the framework of mode estimation. Starting from objective functions that are extended to patches, this leads us to slightly different formulations of filters proposed in the literature, such as the local M-smoothers, bilateral filters, and the nonlocal means. A fast implementation of these new filters relying on separable linear-time convolutions is detailed. Experiments show that this principled approach further improves the denoising quality without increasing the computational complexity.

In the field of medical image analysis, denoising is one of the most important preprocessing steps before medical analysis. The design of an efficient, robust, and computationally effective edge-preserving denoising algorithm is a widely studied, and yet unsolved problem. One of the most efficient edge-preserving denoising algorithms is the bilateral filter, which is an intuitive generalization of the local M-smoother. In this paper, we propose to modify both the bilateral filter and the local M-smoother to use patches of the image instead of single voxels in the denoising process. Using patches instead of single voxels in the filtering process is a way to adapt the filter to the textures, ramps, and edges of the image, and make the filter more discriminant. The filtering performances of the patch-based algorithms are evaluated on a benchmark and a CT phantom image and compared to the bilateral filter and local M-smoother.

Signal denoising proves to be important in many domains such as pattern recognition and image analysis. This paper investigates several refinements of adaptive local filters that rely on local mode finding. These spatial filters are anisotropic and offer the advantage of attenuating noise without smoothing salient signal features such as discontinuities or other sharp transitions. In particular, a bootstrapped procedure is developed and leads to an improvement of the denoising quality without increasing the computational complexity. Experiments with an artificial benchmark allow the quantification of the performance gain.

Denoising is a key step in the processing of medical images. It aims at improving both the interpretability and visual aspect of the images. Yet, designing a robust and efficient denoising tool remains an unsolved challenge and a specific issue concerns the noise model. Many filters typically assume that noise is additive and Gaussian, with uniform variance. In contrast, noise in medical images often has more complex properties. This paper considers images with Poissonian noise and the patch-based bilateral filters, that is, filters that involve a tonal kernel and pair wise comparisons between shifted blocks of the images. The main aim is then to integrate two variance stabilizing transformations that allow the filters to work with Gaussianized noise. The performances of these filters are compared to those of the classical bilateral filter with the same transformations. The experiments include an artificial benchmark as well as a positron emission tomography image.

In the field of image analysis, denoising is an important preprocessing task. The design of an efficient, robust, and computationally effective edge preserving denoising algorithm is a widely studied, and yet unsolved problem. One of the most efficient edge-preserving denoising algorithms is the bilateral filter, which is an intuitive generalization of the local M-smoother. In this paper, we propose to modify both the bilateral filter and the local M-smoother to use patches of the image instead of single pixels in the denoising process. With this modification, the filtering effect becomes more sensitive to the different areas of the image and the filtering results improve. The denoising quality of these patch-based filters is evaluated on test images and compared to the classical bilateral filtering and local M-smoother.

The wavelet transform is a widely used pre-filtering step for subsequent R spike detection by thresholding of the coefficients. The time-frequency decomposition is indeed a powerful tool to analyze non-stationary signals. Still, current methods use consecutive wavelet scales in an a priori restricted range and may therefore lack adaptativity. This paper introduces a supervised learning algorithm which learns the optimal scales for each dataset using the annotations provided by physicians on a small training set. For each record, this method allows a specific set of non consecutive scales to be selected, based on the record’s characteristics. The selected scales are then used for the decomposition of the original long-term ECG signal recording and a hard thresholding rule is applied on the derivative of the wavelet coefficients to label the R spikes. This algorithm has been tested on the MIT-BIH arrhythmia database and obtains an average sensitivity rate of 99.7% and average positive predictivity rate of 99.7%.

One of the most important tasks in automatic annotation of the ECG is the detection of the R spike. The wavelet transform is a widely used tool for R spike detection. The time-frequency decomposition is indeed a powerful tool to analyze non-stationary signals. Still, current methods use consecutive wavelet scales in an a priori restricted range and may therefore lack adaptivity. This paper introduces a supervised learning algorithm which learns the optimal scales for each dataset using the annotations provided by physicians on a small training set. For each record, this method allows a specific set of non consecutive scales to be selected, based on the record characteristics. The selected scales are then used on the original long-term ECG signal recording and a hard thresholding rule is applied on the derivative of the wavelet coefficients to label the R spikes. This algorithm has been tested on the MIT-BIH arrhythmia database and obtains an average sensitivity rate of 99.7% and average positive predictivity rate of 99.7%.

Functional data are often sampled at high frequency which leads to high-dimensional vectors. The curse of dimensionality makes this type of signal difficult to handle with standard data analysis tools. Functional data analysis uses the functional nature of data to project them on a smooth basis. This paper shows how to extend functional Self-Organizing Maps (SOM) to signal windows having different lengths using functional data analysis. This technique may be applied for example on regularly sampled signals, for which the duration of each signal is varying; an example concerns electrocardiography (ECG), where the signal is usually cut according to the variable period between two heart beats.