Isidora Stankovic

Image reconstruction from a reduced set of pixels using a simplified gradient algorithm

A reconstruction of images in the DCT transformation domain based on the adaptive gradient algorithm is considered in this paper. Two approaches are used in the reconstruction. In the first approach, the image is pre-processed using 8×8 blocks, such that the smallest DCT coefficients are set to zero in order to make the image sparse. The second approach reconstructs the image without the pre-processing step. It has been assumed that the sparsity is an intrinsic property of the analyzed image. An adaptive gradient based algorithm is used to recover a large number of missing pixels in the image. In order to improve the calculation complexity, in this paper we propose an improved version of recently proposed adaptive gradient algorithm, which is now reduced to a single, automatically determined parameter. The previous reconstruction of black and white and colour images is repeated with a significant calculation efficiency improvement.

Denoising of sparse images in impulsive disturbance environment

The paper presents a method for denoising and reconstruction of sparse images based on a gradient-descent algorithm. It is assumed that the original (non-noisy) image is sparse in the two-dimensional Discrete Cosine Transform (2D-DCT) domain. It is also assumed that a number of image pixels is corrupted by a salt and pepper noise. In addition, we assume that there are pixels corrupted by a noise of any value. In this paper we introduce a method to find the positions of the corrupted pixels when the noise is not of the salt and pepper form. The proposed algorithm for noisy pixels detection and reconstruction works blindly. It does not require the knowledge about the positions of corrupted pixels. The only assumption is that the image is sparse and that the noise degrades this property. The advantage of this reconstruction algorithm is that we do not change the uncorrupted pixels in the process of the reconstruction, unlike common reconstruction methods. Corrupted pixels are detected and removed iteratively using the gradient of sparsity measure as a criterion for detection. After the corrupted pixels are detected and removed, the gradient algorithm is employed to reconstruct the image. The algorithm is tested on both grayscale and color images. Additionally, the case when both salt and pepper noise and a random noise, within the pixel values range, are combined is considered. The proposed method can be used without explicitly imposing the image sparsity in a strict sense. Quality of the reconstructed image is measured for different sparsity and noise levels using the structural similarity index, the mean absolute error, mean-square error and peak signal-to-noise ratio and compared to the traditional median filter and recent algorithms, one based on the total-variations reconstruction and a two-stage adaptive algorithm.

Nonsparsity influence on the ISAR recovery from reduced data [Correspondence]

The analysis of inverse synthetic aperture radar (ISAR) image recovery from a reduced set of data presented previously (IEEE Transactions on Aerospace and Electronic Systems, 51, 3 [Jul. 2015], 2093–2106) is extended in this correspondence to an important topic of signal nonsparsity (approximative sparsity). In real cases the ISAR images are noisy and only approximately sparse. Formula for the mean square error in the nonsparse ISAR, reconstructed under the sparsity assumption, is derived. The results are tested on examples and compared with statistical data.

On the reconstruction of nonsparse time-frequency signals with sparsity constraint from a reduced set of samples

Abstract Nonstationary signals, approximately sparse in the joint time-frequency domain, are considered. Reconstruction of such signals with sparsity constraint is analyzed in this paper. The short-time Fourier transform (STFT) and time-frequency representations that can be calculated using the STFT are considered. The formula for error caused by the nonreconstructed coefficients is derived and presented in the form of a theorem. The results are examined statistically on examples.

Iterative denoising of sparse images

The paper examines an application of the gradient-based algorithm to image denoising with noise values being in the range of the available (non-noisy) pixel values. The analyzed image is considered to be sparse in the 2D-DCT domain. The presented algorithm is a generalization of the previous results on denoising images when the noisy pixels can be detected and eliminated using the L-statistics. The algorithm is based on the recently developed technique used for denoising of one-dimensional signals. A significant advantage of the presented algorithm is that it does not use any a priori knowledge about the positions, values or distribution of the noisy pixels. It is assumed that the positions of noisy pixels cannot be determined using the methods like the L-statistics based ones. Hence, the proposed approach reconstructs the pixels values iteratively using the highest gradient as pixel selection criterion, thus performing blind denoising on a pixel-by-pixel basis. The examples with synthetic two-dimensional signal and a test image are presented. Quality of the image reconstruction is measured using the structural similarity index and the mean absolute error (MAE).

Error in the Reconstruction of Nonsparse Images

Sparse signals, assuming a small number of nonzero coefficients in a transformation domain, can be reconstructed from a reduced set of measurements. In practical applications, signals are only approximately sparse. Images are a representative example of such approximately sparse signals in the two-dimensional (2D) discrete cosine transform (DCT) domain. Although a significant amount of image energy is well concentrated in a small number of transform coefficients, other nonzero coefficients appearing in the 2D-DCT domain make the images be only approximately sparse or nonsparse. In the compressive sensing theory, strict sparsity should be assumed. It means that the reconstruction algorithms will not be able to recover small valued coefficients (above the assumed sparsity) of nonsparse signals. In the literature, this kind of reconstruction error is described by appropriate error bound relations. In this paper, an exact relation for the expected reconstruction error is derived and presented in the form of a theorem. In addition to the theoretical proof, the presented theory is validated through numerical simulations.

Decomposition of signals in dispersive channels using dual polynomial fourier transform

The acoustic waves transmitted in a dispersive environments can be quite complex for decomposition and localization. A signal which is transmitted through a dispersive channel is usually non-stationary. Even if a simple signal is transmitted, it can change its characteristics (phase and frequency) during the transmission through an underwater acoustic dispersive communication channel. Commonly, several components with different paths are received. In this paper, we present a method of decomposition of multicomponent acoustic signals using the dual polynomial Fourier transform and time-frequency methods.

Time-frequency signal reconstruction of nonsparse audio signals

In this paper, the reconstruction of non-stationary audio signals is considered. Audio signals are approximately sparse in the joint time-frequency representation domain. The reconstruction is based on a reduced set of samples, and it is considered that the signals are sparse. The short-time Fourier transform (STFT) is considered as the representation domain where the audio signals are sparse. The formula for error caused by the reconstruction of approximately sparse signals under the sparsity assumption is derived. The results are numerically illustrated on three audio signals.

Reconstruction of global Ozone density data using a gradient-descent algorithm

The Ozone density and its decrease in the atmosphere are one of the main concerns of nowadays. The Ozone density is tracked using the Ozone Measurements Instrument (OMI). During tracking and collection of information, missing data occur because of the pathway the instrument is going. Reconstruction of those missing data can be done using the results from the compressive sensing (CS) approach. The application of a CS algorithm based on the gradient-descent method for the Ozone density data reconstruction is presented in this paper. The algorithm is based on varying the missing data to promote sparsity in each frame. The algorithm is modified for a more efficient reconstruction by using the dynamic information about previous frames. The reconstruction results for some recent Ozone data are presented.

Compressive sensing reconstruction of video data based on DCT and gradient-descent method

The reconstruction of video with missing data using a gradient-based compressive sensing algorithm is presented in this paper. The video is assumed to have missing data randomly spread over the frames. Also, it has been assumed that the video frames are sparse in the DCT domain. The value of the gradient is calculated in the sparsity domain and then used to update the pixel values. The efficiency of the video reconstruction using the gradient-descent algorithm is analyzed in terms of the number of available samples and sparsity level, showing better reconstruction performance comparing to the greedy algorithm. Also, it has been shown that the gradient-based algorithm provides high quality reconstruction for real video sequences.