An improved analysis of Optical Coherence Tomography (OCT) images of the retina is of essential importance for the correct diagnosis of retinal abnormalities. Unfortunately, OCT images suffer from noise arising from different sources. In particular, speckle noise caused by the scattering of light waves strongly degrades the quality of OCT image acquisitions. In this paper, we employ a Modified Morphological Component Analysis (MMCA) to provide a new method that separates the image into components that contain different features as texture, piecewise smooth parts, and singularities along curves. Each image component is computed as a sparse representation in a suitable dictionary. To create these dictionaries, we use non-data-adaptive multi-scale ( X -let) transforms which have been shown to be well suitable to extract the special OCT image features. In this way, we reach two goals at once. On the one hand, we achieve strongly improved denoising results by applying adaptive local thresholding techniques separately to each image component. The denoising performance outperforms other state-of-the-art denoising algorithms regarding the PSNR as well as no-reference image quality assessments. On the other hand, we obtain a decomposition of the OCT images in well-interpretable image components that can be exploited for further image processing tasks, such as classification.

Optical Coherence Tomography (OCT) images, as well as a majority of medical images, are imposed to speckle noise while capturing. Since the quality of these images is crucial for detecting any abnormalities, we develop an improved denoising algorithm that is particularly appropriate for OCT images. The essential idea is to combine two non-data-adaptive transform-based denoising methods that are capable to preserve different important structures appearing in OCT images while providing a very good denoising performance. Based on our numerical experiments, the most appropriate non-data-adaptive transforms for denoising and feature extraction are the Discrete Cosine Transform (DCT) (capturing local patterns) and the Dual-Tree Complex Wavelet Transform (DTCWT) (capturing piecewise smooth image features). These two transforms are combined using the Dual Basis Pursuit Denoising (DBPD) algorithm. Further improvement of the denoising procedure is achieved by total variation (TV) regularization and by employing an iterative algorithm based on DBPD.

This paper presents a new approach for reconstruction of disconnected digital lines (DDLs) based on a constrained regularization model which ensures connectivity of the digital lines (DLs) in the discrete image plane. The first step in this approach is to determine the order of given pixels of the DDL. To determine connectivity of pixels, we use the usual 8-neighbor connectivity in discrete images. For any neighboring pixels of the DDL that are not connected, we determine a number of new pixel values that need to be reconstructed between these pixels. Next, the integer-valued x - and y -coordinates of the location of the pixels of the DDLs are segregated into two 1D signal vectors. Then the x - and y -coordinates of the missing pixels of the DDLs are estimated using a new constrained regularization. While the solution of this constrained minimization problem provides real values for the x - and y -coordinates of pixels positions, the imposed constraint ensures connectivity of the resulting DLs in the image plane after transforming the computed values from [Formula: see text] to [Formula: see text]. The proposed regularization approach forces connected lines with small curvature. The experimental results demonstrate that the proposed technique improves DL intersection detection, as well. Moreover, this technique has a high potential to be used as a fast approach in binary image inpainting particularly overcoming the shortcomings of conventional methods which cause destruction of thin objects and blurring in the recovered regions.

In this paper, we apply compressed sensing (CS) to video compression. CS techniques exploit the observation that one needs much fewer random measurements than given by the Shannon-Nyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial domain or in a transform domain). In the CS framework, we can achieve sensing, compression, and denoising simultaneously. We propose a fast and simple online encoding by the application of pseudorandom downsampling of the 2-D fast Fourier transform to video frames. For offline decoding, we apply a modification of the recently proposed approximate message passing (AMP) algorithm. The AMP method has been derived using the statistical concept of “state evolution,” and it has been shown to considerably accelerate the convergence rate in special CS-decoding applications. We shall prove that the AMP method can be rewritten as a forward-backward splitting algorithm. This new representation enables us to give conditions that ensure convergence of the AMP method and to modify the algorithm in order to achieve higher robustness. The success of reconstruction methods for video decoding also essentially depends on the chosen transform, where sparsity of the video signals is assumed. We propose incorporating the 3-D dual-tree complex wavelet transform that possesses sufficiently good directional selectivity while being computationally less expensive and less redundant than other directional 3-D wavelet transforms.

The easy path wavelet transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image approximation that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure.

Sparse representation of seismic data is a crucial step for seismic forward modeling and seismic processing such as coherent noise separation, imaging, and sparsity-promoting data recovery. In this letter, a new locally adaptive wavelet transform, called easy-path wavelet transform (EPWT), is applied for the sparse representation of seismic data. The EPWT is an adaptive geometric wavelet transform that works along a series of special pathways through the input data and exploits the local correlations of the data. The transform consists of two steps: reorganizing the data following the pathways according to the data values and then applying a 1-D wavelet transform along the pathways. This leads to a very sparse wavelet representation. In comparison to conventional wavelets, the EPWT concentrates most of the energy of signals at smooth scales and needs less significant wavelet coefficients to represent signals. Numerical experiments show that the new method is really superior over the conventional wavelets and curvelets in terms of sparse representation and compression of seismic data.

Denoising is always a challenging problem in natural imaging and geophysical data processing. In this paper, we consider the denoising of texture images using a nonlinear reaction-diffusion equation and directional wavelet frames. In our model, a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures. We derive a digital reaction-diffusion filter that lives on graphs and show convergence of the corresponding iteration process. Experimental results and comparisons show very good performance of the proposed model for texture-preserving denoising.

In this paper, a diffusion-based curvelet shrinkage is proposed for discontinuity-preserving denoising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme. In order to suppress the pseudo-Gibbs and curvelet-like artifacts, the conventional shrinkage results are further processed by a projected total variation diffusion, in which only the insignificant curvelet coefficients or high-frequency part of the signal are changed by use of a constrained projection. Numerical experiments from piecewise-smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components, in comparison to some existing methods.