The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove fine structures in images. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the “method noise,” defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods is compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of $L^2$ distances of the denoised version to the original image. The fourth and perhaps most powerful evaluation method is, however, the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.

Image Denoising Methods. A New Nonlocal Principle

Texture analysis describes a variety of image-analysis techniques that quantify the variation in surface intensity or patterns, including some that are imperceptible to the human visual system. Texture analysis may be particularly well-suited for lesion segmentation and characterization and for the longitudinal monitoring of disease or recovery. We begin this review by outlining the general procedure for performing texture analysis, identifying some potential pitfalls and strategies for avoiding them. We then provide an overview of some intriguing neuro-MR imaging applications of texture analysis, particularly in the characterization of brain tumors, prediction of seizures in epilepsy, and a host of applications to MS.

http://www.ajnr.org/content/ajnr/31/5/809.full.pdf

Texture analysis: a review of neurologic MR imaging applications.

Diagnostic radiology requires accurate interpretation of complex signals in medical images. Content-based image retrieval (CBIR) techniques could be valuable to radiologists in assessing medical images by identifying similar images in large archives that could assist with decision support. Many advances have occurred in CBIR, and a variety of systems have appeared in nonmedical domains; however, permeation of these methods into radiology has been limited. Our goal in this review is to survey CBIR methods and systems from the perspective of application to radiology and to identify approaches developed in nonmedical applications that could be translated to radiology. Radiology images pose specific challenges compared with images in the consumer domain; they contain varied, rich, and often subtle features that need to be recognized in assessing image similarity. Radiology images also provide rich opportunities for CBIR: rich metadata about image semantics are provided by radiologists, and this information is not yet being used to its fullest advantage in CBIR systems. By integrating pixel-based and metadata-based image feature analysis, substantial advances of CBIR in medicine could ensue, with CBIR systems becoming an important tool in radiology practice.

http://www.cba-research.com/pdfs/cbir_in_radiology_jdi2011.pdf

Content-Based Image Retrieval in Radiology: Current Status and Future Directions

Digital images are matrices of equally spaced pixels, each containing a photon count. This photon count is a stochastic process due to the quantum nature of light. It follows that all images are noisy. Ever since digital images have existed, numerical methods have been proposed to improve the signal-to-noise ratio. Such ‘denoising’ methods require a noise model and an image model. It is relatively easy to obtain a noise model. As will be explained in the present paper, it is even possible to estimate it from a single noisy image.

Secrets of image denoising cuisine

Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.

A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly

The Stockwell transform (ST), recently developed for geophysics, combines features of the Fourier, Gabor and wavelet transforms; it reveals frequency variation over time or space. This valuable information is obtained by Fourier analysis of a small segment of a signal at a time. Localization of the Fourier spectrum is achieved by filtering the signal with frequency-dependent Gaussian scaling windows. This multi-scale time–frequency analysis provides information about which frequencies occur and more importantly when they occur. Furthermore, the Stockwell domain can be directly inferred from the Fourier domain and vice versa. These features make the ST a potentially effective tool to visualize,analyze, and process medical imaging data. The ST has proven useful in noise reduction and tissue texture analysis. Herein, we focus on the theory and effectiveness of the ST for medical imaging. Its effectiveness and comparison with other linear time–frequency transforms, such as the Gabor and wavelet transforms, are discussed and demonstrated using functional magnetic resonance imaging data.

A new local multiscale Fourier analysis for medical imaging

The paper focuses on progressive transmission of CT or MR images, and introduces two general schemes that are built around information embedded in transforms of images. A direct, a priori ordering of 93, parallel, CT slices of a head is obtained by successively finer sweepings of their natural subscript ordering to give a benchmark illustration. By comparison, an ordering of these CT slices simply by their energies is seen to not provide a viable progressive imaging scheme, at least when an overall, 3D skin level rendering is the gauge employed. To investigate progressive imaging that does not obscure internal detail, two techniques based on transform space information are introduced here, and illustrated in detail with a 128?128 MR slice of a head I(x, y). The first uses decreasing size of the moduli of the elements of the Fourier transform F(k x , k y ) of I(x, y). The second, a one parameter generalization, exploits the localization feature of the recent S-transform and also provides a capability for an observer to outline a region of interest within the progressive transmission process. Both transform based methods are effective for the specific illustrations included, and the latter opens important research questions for application in analysis, handling or interpretation of the massive data sets arising in magnetic resonance imaging.

/pdf/progressive-imaging-s-transform-order-1kzdljymr7.pdf

Progressive imaging: S-transform order

Results presented in this paper represent part of an ongoing research programme dedicated to the resolution enhancement of Fourier domain magnetic resonance (MR) data. Here we explore the use of self-similarity methods that may aid in frequency extrapolation of such data. To this end, we present analytical and empirical results demonstrating the self similarity of complex, Fourier domain MR data.

/pdf/self-similarity-of-images-in-the-fourier-domain-with-1ivj6iynis.pdf

M. L. Lauzon

Papers

A new local multiscale Fourier analysis for medical imaging

Progressive imaging: S-transform order

Self-similarity of Images in the Fourier Domain, with Applications to MRI