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In this work, a dynamically adaptive curvelet technique has been developed for solving nonlinear Schrödinger equation (NLS). Central finite difference method is used for approximating the one- and two-dimensional differential operators and radial-basis functions (RBFs) are utilized for approximating the differential operators on the sphere. The grid on which the equation is solved, is obtained using curvelets. For test problems 1 and $2$ (1d & 2d problems) considered in this paper, the computational time carried out by the proposed technique is analyzed with the computational time carried out by the finite difference technique. Moreover, the problem on the sphere has been considered, for which the computational time carried out by the RBF collocation technique is analyzed with the computational time carried out by the proposed technique. It is found that the developed technique performs better in terms of computational time, for example, on sphere, computational effort reduces by four times using the proposed method.

The sensors or electronic components are vulnerable to interference in the camera’s imaging process, usually leading to random directional stripes. Therefore, a method of stripe noise removal based on compressed sensing is proposed. First, the measurement matrix of the image with stripe noise is established, which makes the stripe images equivalent to the observation of the original image. Second, the relationships between the corresponding coefficients of adjacent scales are defined. On this basis, the bivariate threshold function is set in the curvelet sparse domain to represent the features of images. Finally, the Landweber iteration algorithm of alternating convex projection and filtering operation is achieved. Furthermore, to accelerate the noise removal at the initial stage of iteration and preserve the image details later, the exponential threshold function is utilized. This method does not need many samples, which is different from the current deep learning method. The experimental results show that the proposed algorithm represents excellent performance in removing the stripes and preserving the texture details. In addition, the PSNR of the denoised image has been dramatically improved compared with similar algorithms.

Image compression is the emerging field to transmit the multimedia products like image, video and audio. Image compression is used to reduce the storage quantity as much as possible. The objective of this paper is to compare the multiscale transform based image compression encoding techniques. The multiscale transforms involved in this paper are wavelet transform, bandelet transform, curvelet transform, ridgelet transform and contourlet transform. Wavelet transform allows good localization both in time and frequency domain. Bandelet transform takes geometric regularity of the natural images to improve the efficiency of representation. Curvelet transform handles curve discontinuities well. Curvelets are the good tool for the analysis and the computation of partial differential equations. Curvelets also have micro local features which make them especially adapted to certain reconstruction problems with missing data. The ridgelet transform is the core idea behind curvelet transform. It is used to represent objects with line singularities. The contourlet transform gets smooth contours and edges at any orientation. It filters noise as well. The Encoding techniques involved in this paper are spatial orientation tree wavelet (STW), set partitioned embedded block (SPECK) and compression with reversible embedded wavelet (CREW). The performance parameters such as peak signal to noise ratio (PSNR), image quality index and structural similarity index (SSIM) are used for the purpose of evaluation. It is found that bandelet transform with all the encoding techniques work well.

Representing the image to be inpainted in an appropriate sparse dictionary, we introduce a novel method for the filling-in of structure and texture in regions of missing image information. In the morphological component analysis (MCA) inpainting approach, a TV penalty is added to better reduce ringing artifacts. However, the incorporation of TV penalty terms leads to PDE schemes that are numerically intensive. Inspired by the works of Daubechies–Teschke and Borup–Nielsen, we replace the TV term by a term. It results in an iterative curvelet thresholding scheme for the structure image inpainting. In the whole inpainting process, an alternative approach is presented to layer inpainting. Experimental results show the performance of the algorithm.

In this paper, a selective curvelet coefficient digital watermarking algorithm is proposed. Traditionally, curvelet watermarks are embedded into all sample frequency bands. However, the study of individual band behavior and the use of single band for watermarking have not been reported. The selective band will provide an addition security feature against any physical tampering. This paper aims to give an intensive study on the robustness of watermarking using selective curvelet coefficients from a single band and to find out the best band for embedding watermark. Wrapping of specially selected Fourier samples is employed to implement Fast Discrete Curvelet Transforms (FDCT) to transform the digital image to the curvelet domain.

In this paper, six robust non-blind watermarking schemes based on curvelet transform are proposed. Single band watermarking method was proposed in Ref. 1. This paper develops the single band watermarking method and adds Human Vision System (HVS) to form six different multi-bands watermarking methods. With the increasing redundancy of watermark, the robustness of the algorithm will be investigated and comparative studies with the single band watermarking will be shown. The experimental results demonstrate that the proposed algorithms have great robustness against various imaging attacks.

Synthetic Aperture Radar (SAR) is an all-weather, day and night satellite imaging technology where the radar is mounted on aircraft and successive pulses of radio waves are transmitted to illuminate the target scene. The signal processing of the recorded backscattered echoes produce SAR images. SAR images contain inherent multiplicative speckle noise which is formed due to the constructive and destructive interference of transmitted signals with the returning signals. Speckle noise appears as granular patterns and makes the image interpretation difficult. Non-local means approaches like Block Matching and 3D filtering (BM3D) are effective scheme for removing speckle noise from SAR images. This method gives good performance for additive noise but is not adaptive to curved edges and discontinuities that occur in SAR images affected by multiplicative noise. This paper proposes a three-step refined algorithm to adapt BM3D for despeckling multiplicative speckle noise. In the proposed scheme curvelet is used to find the transform coefficients and this modification in the transform domain improves the despeckling accuracy of BM3D. Also Wiener filtering is replaced with Importance Sampling Unscented Kalman Filtering (ISUKF) for better adapting to discontinuities in the real SAR image. An improved method of grouping is proposed here based on Manhattan distance which better adapts to constantly changing multiplicative noise statistics. A detailed comparative study is carried out on each step using various well-known performance measures. From the results, it is found that the proposed Curvelet-ISUKF-Manhattan BM3D (CIM-BM3D) method of despeckling has better values for all the performance measure and the results are also visually verified.

In this paper a new ripplet-based shrinkage technique is used to suppress noise from Magnetic Resonance Imaging (MRI). The propitious properties of ripplet transform such as anisotropy, high directionality, good localization, and high-energy compaction make the proposed method efficient and feature preserving when compared to other transforms. Ripplet transform provides efficient representation of edges in images with a higher potential for image processing applications such as image restoration, compression, and de-noising. The proposed method implies a new nonlinear ripplet-based shrinkage technique to extract the spatial and frequency information from MRI corrupted by noise. The choice of this new shrinkage technique is due to its simplicity, versatility, and its efficiency in removing noise from homogenous regions and those regions with singularities, when compared to the existing filtering techniques. Experiments were conducted on several diffusion weighed images and anatomical images. The results show that the proposed de-noising technique shows competitive performance compared to the current state-of-art methods. Qualitative validation was performed based on several quality metrics and profound improvement over existing methods was obtained. Higher values of Peak Signal to Noise Ratio (PSNR), Correlation Coefficient (CC), mean structural similarity index (MSSIM), and lower values of Root Mean Square Error (RMSE) and computational time were obtained for the proposed ripplet-based shrinkage technique when compared to the existing ones.

This paper presents the method of multi-resolution analysis used in 2D image data to extract the curved edge features. The method is based on the combination of multi-resolution decomposition through Wavelet Packet and Prime Ridgelet transform. We call this combination Prime Wavelet Packet Contourlet Transform-PWPC. At each leave of Packet Wavelet Packet Tree, the prime ridgelet transform is applied on the band pass image or packet, which contains the high frequency data. The experiment shows that the PWPC coefficients are good approximations to curved edges. The speed of PWPC is faster than that of the basic Curvelet transform. This transform is very suitable to represent the noisy curved features that often exist in medicine or nano/micro images.