Image denoising and quality measurements by using filtering and wavelet based techniques

An image is often corrupted by noise in its acquisition and transmission by various kinds of noises. Image denoising using thresholding methods find appropriate values (threshold) which separates noise values to actual image values without affecting the significant features of the image. Wavelet transform represents image energy in compact form and representation helps in determining threshold between noisy features and important image feature. In this paper we have worked with denoising of salt–pepper and Gaussian noise. The work is organized in four steps as follows: (1) image is denoised by filtering method, (2) image is denoised by wavelet based techniques using thresholding, (3) hard thresholding and filtering method applied simultaneously on noisy image, (4) results of PSNR (peak signal to noise ratio) and MSE (mean square error) are calculated by comparing all cases.     

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising--Part I: Theory.

We study the robust estimation of missing regions in images and video using adaptive, sparse reconstructions. Our primary application is on missing regions of pixels containing textures, edges, and other image features that are not readily handled by prevalent estimation and recovery algorithms. We assume that we are given a linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero. We adaptively determine these small magnitude coefficients through thresholding, establish sparsity constraints, and estimate missing regions in images using information surrounding these regions. Unlike prevalent algorithms, our approach does not necessitate any complex preconditioning, segmentation, or edge detection steps, and it can be written as a sequence of denoising operations. We show that the region types we can effectively estimate in a mean-squared error sense are those for which the given transform provides a close approximation using sparse nonlinear approximants. We show the nature of the constructed estimators and how these estimators relate to the utilized transform and its sparsity over regions of interest. The developed estimation framework is general, and can readily be applied to other nonstationary signals with a suitable choice of linear transforms. Part I discusses fundamental issues, and Part II is devoted to adaptive algorithms with extensive simulation examples that demonstrate the power of the proposed techniques.     

Fractal-wavelet image denoising revisited.

The essence of fractal image denoising is to predict the fractal code of a noiseless image from its noisy observation. From the predicted fractal code, one can generate an estimate of the original image. We show how well fractal-wavelet denoising predicts parent wavelet subtress of the noiseless image. The performance of various fractal-wavelet denoising schemes (e.g., fixed partitioning, quadtree partitioning) is compared to that of some standard wavelet thresholding methods. We also examine the use of cycle spinning in fractal-based image denoising for the purpose enhancing the denoised estimates. Our experimental results show that these fractal-based image denoising methods are quite competitive with standard wavelet thresholding methods for image denoising. Finally, we compare the performance of the pixel- and wavelet-based fractal denoising schemes.     

一种小波域与空域相结合的图像滤波方法

利用小波阈值去噪方法和传统空间域Lee 滤波的特点, 提出了一种图像去噪的的组合滤波方案。首先在小波域对图像阈值去噪, 得到预去噪图像; 再在空间域上利用自适应Wiener 滤波器进一步提高恢复图像的精度。为了保证小波域和空间域两种算法之间的匹配, 对预去噪图像中残留噪声的分布进行了研究, 对其噪声方差估计做了改进, 提出了一种估计噪声方差的近似最优公式。仿真实验表明, 与单独的在小波域或空域去噪相比, 该方法的均方误差和信噪比指标均得到了改善。     

Multivariate statistical modeling for image denoising using wavelet transforms

Recently a variety of efficient image denoising methods using wavelet transforms have been proposed by many researchers. In this paper, we derive the general estimation rule in the wavelet domain to obtain the denoised coefficients from the noisy image based on the multivariate statistical theory. The multivariate distributions of the original clean image can be estimated empirically from a sample image set. We define a parametric multivariate generalized Gaussian distribution (MGGD) model which closely fits the sample distribution. Multivariate model makes it possible to exploit the dependency between the estimated wavelet coefficients and their neighbours or other coefficients in different subbands. Also it can be shown that some of the existing methods based on statistical modeling are subsets of our multivariate approach. Our method could achieve high quality image denoising. Among the existing image denoising methods using the same type of wavelet (Daubechies 8) filter, our results produce the highest peak signal-to-noise ratio (PSNR).     

Linear, Worst-Case Estimators for Denoising Quantization Noise in Transform Coded Images

Transform-coded images exhibit distortions that fall outside of the assumptions of traditional denoising techniques. In this paper, we use tools from robust signal processing to construct linear, worst-case estimators for the denoising of transform compressed images. We show that while standard denoising is fundamentally determined by statistical models for images alone, the distortions induced by transform coding are heavily dependent on the structure of the transform used. Our method, thus, uses simple models for the image and for the quantization error, with the latter capturing the transform dependency. Based on these models, we derive optimal, linear estimators of the original image that are optimal in the mean-squared error sense for the worst-case cross correlation between the original and the quantization error. Our construction is transform agnostic and is applicable to transforms from block discrete cosine transforms to wavelets. Furthermore, our approach is applicable to different types of image statistics and can also serve as an optimization tool for the design of transforms/quantizers. Through the interaction of the source and quantizer models, our work provides useful insights and is instrumental in identifying and removing quantization artifacts from general signals coded with general transforms. As we decouple the modeling and processing steps, we allow for the construction of many different types of estimators depending on the desired sophistication and available computational complexity. In the low end of this spectrum, our lookup table based estimator, which can be deployed in low complexity environments, provides competitive PSNR values with some of the best results in the literature.     

Wavelet-domain approximation and compression of piecewise smooth images.

The wavelet transform provides a sparse representation for smooth images, enabling efficient approximation and compression using techniques such as zerotrees. Unfortunately, this sparsity does not extend to piecewise smooth images, where edge discontinuities separating smooth regions persist along smooth contours. This lack of sparsity hampers the efficiency of wavelet-based approximation and compression. On the class of images containing smooth C2 regions separated by edges along smooth C2 contours, for example, the asymptotic rate-distortion (R-D) performance of zerotree-based wavelet coding is limited to D(R) (< or = 1/R, well below the optimal rate of 1/R2. In this paper, we develop a geometric modeling framework for wavelets that addresses this shortcoming. The framework can be interpreted either as 1) an extension to the "zerotree model" for wavelet coefficients that explicitly accounts for edge structure at fine scales, or as 2) a new atomic representation that synthesizes images using a sparse combination of wavelets and wedgeprints--anisotropic atoms that are adapted to edge singularities. Our approach enables a new type of quadtree pruning for piecewise smooth images, using zerotrees in uniformly smooth regions and wedgeprints in regions containing geometry. Using this framework, we develop a prototype image coder that has near-optimal asymptotic R-D performance D(R) < or = (log R)2 /R2 for piecewise smooth C2/C2 images. In addition, we extend the algorithm to compress natural images, exploring the practical problems that arise and attaining promising results in terms of mean-square error and visual quality.     

An unsupervised fusion network for boosting denoising performance

While many efforts have been devoted to addressing image denoising and achieve continuously improving results during the past few decades, it is fair to say that no a stand-alone method is consistently better than others. Nonetheless, many existing denoising methods, each having a different denoising capability, can yield various but complementary denoised images with respect to specific local areas. To effectively exploit the complementarity and diversity among the denoised images obtained with different denoisers, in this work we fuse them to produce an overall better result, which is fundamental to achieve robust and competitive denoising performance especially for complex scenes. A framework called deep fusion network (DFNet) is proposed to generate a consistent estimation about the final denoised image, taking advantage of the complementarity of denoisers and suppressing the bias. Specifically, given a noisy image, we first exploit a set of representative image denoisers to denoise it respectively, and obtain the corresponding initial denoised images. Then these initial denoised images are concatenated and fed into the proposed DFNet, and the proposed DFNet seeks to adjust its network parameters to produce the fused image (as the final denoised image) with an unsupervised training strategy through minimizing the carefully designed loss function. The experimental results show that our approach outperforms the stand-alone methods as well as the ones using combination strategy by large margin both in objective and subjective evaluations. Compared to the those methods that are relatively close to our strategy, the proposed DFNet is extensible and parameter free, which means it can cope with a variable number of different denoisers and avoid the manual intervention during the fusion process. The proposed DFNet has greater flexibility and better practicality.     

基于差分进化的插值置换混叠图像盲分离

针对插值置换混叠图像提出了一套完整的盲分离方案。根据插值图像像素点之间存在的相关性,通过对置换混叠图像分块进行有限差分,检测插值图像与原图像经过有限差分后的值之间的差距,设定适当的阈值,将其分开。为获得最优阈值,用差分进化算法进行优化,获得最优阈值。根据阈值将图像二值化,进而将置换图像分离出来。实验结果表明,该方法比阈值法的鲁棒性强,对经过不同插值方式的置换混叠图像进行盲分离都有较好的效果。     

The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a; trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.     

基于过完备表示的图像去噪算法

提出一种基于过完备线形变换集合的图像去除白高斯噪音算法.对于每个变换使用阈值技术进行图像去噪,图像恢复使用加权平均来融合去噪结果.采用稀疏集中度来计算图像稀疏分解的测度,而权重依赖该结果.对于图像稀疏区域,拥有较大的权.该方法不需要设计复杂的变换系数统计模型,与使用复杂的有向变换和图像统计模型方法相比,取得了较高的去噪性能.该方法简洁高效,尤其是对包含奇异特征的图像得到了较好的去噪结果.实验结果证实了所提方法的有效性.     

基于不可分离MRA的快速小波去噪图像重建

提出了一种基于不可分离MRA的小波去噪重建算法.算法将投影数据进行二通道的小波分解,从而直接得到小波的近似系数和细节系数,对这些小波系数进行基于小波的阀值化去噪处理,再经过逆小波变换就得到了最终的重建图像.算法降低了复杂度,与可分离MRA重建算法比较速度更快,并且可以去噪.     

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising--Part II: Adaptive algorithms.

We combine the main ideas introduced in Part I with adaptive techniques to arrive at a powerful algorithm that estimates missing data in nonstationary signals. The proposed approach operates automatically based on a chosen linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero. Unlike prevalent algorithms, our method does not necessitate any complex preconditioning, segmentation, or edge detection steps, and it can be written as a progression of denoising operations. We show that constructing estimates based on nonlinear approximants is fundamentally a nonconvex problem and we propose a progressive algorithm that is designed to deal with this issue directly. The algorithm is applied to images through an extensive set of simulation examples, primarily on missing regions containing textures, edges, and other image features that are not readily handled by established estimation and recovery methods. We discuss the properties required of good transforms, and in conjunction, show the types of regions over which well-known transforms provide good predictors. We further discuss extensions of the algorithm where the utilized transforms are also chosen adaptively, where unpredictable signal components in the progressions are identified and not predicted, and where the prediction scenario is more general.     

基于非局部均值滤波的小波红外图像去噪

为避免小波去噪时阈值的缺陷和非局部均值滤波去噪时计算的复杂性和更有效地去除红外图像中的噪声,提出了一种采用非局部均值滤波的小波图像去噪方法.对含噪图像进行多层小波分解,采用新的贝叶斯估计阈值对高频系数进行阈值化处理,以消除高频噪声;在部分低层子带上进行非局部均值处理以进一步消除噪声.实验结果表明,与通常的小波阈值去噪和非局部均值去噪相比,该方法能很好地去除红外图像中的噪声,获得更高的信噪比(Signal To Noise Ratio,SNR)和更小的均方误差(MeanSquared Error,MSE),而且该方法计算相对简单,能达到很好的视觉效果.     

Modified Fuzzy Linear Discriminant Analysis for Threshold Selection

Otsu’s thresholding method is a popular and efficient method for image segmentation. However, its performance is greatly affected by noise and the population size of object and background. In this paper, a novel thresholding method is proposed based on modified fuzzy linear discriminant analysis (MFLDA). MFLDA is an extension of linear discriminant analysis to fuzzy domain, where the between-class variance is modified as the distance between the centers of background and object. The optimal threshold is selected such that the MFLDA criterion is maximized. Some images are used to test the performance of the proposed thresholding method and results reveal that the proposed method is less affected by noise, the population size of objects and background, and better segmentation results are obtained than Otsu’s method and other classical thresholding methods.     

Noise reduction using multiscale bilateral decomposition for digital color images

In this paper, we propose a noise reduction algorithm for digital color images using a nonlinear image decomposition approach. Most existing noise reduction methods do not adequately consider spatial correlation of color noise in digital color images. Color noise components in color images captured by digital cameras are observed as irregular grains with various sizes and shapes, which are spatially randomly distributed. We use a modified multiscale bilateral decomposition to effectively separate signal and mixed-type noise components, in which a noisy input image is decomposed into a base layer and several detail layers. A base layer contains strong edges, and most of noise components are contained in detail layers. Noise components in detail layers are reduced by an adaptive thresholding function. We obtain a denoised image by combining a base layer and noise-reduced detail layers. Experimental results show the effectiveness of the proposed algorithm, in terms of both the peak signal-to-noise ratio and visual quality.     

Quantitative characterization of electron micrograph image usingfractal feature

In this investigation, texture analysis was carried out on electron micrograph images. Fractal dimensions and spatial grey level co-occurrence matrices statistics were estimated on each homogeneous region of interest, The fractal model has the advantages that the fractal dimension correlates to the roughness of the surface and is stable over transformations of scale and linear transforms of intensity. It can be calculated using three different methods. The first method estimates fractal dimension based on the average intensity difference of pixel pairs. In the second method, fractal dimension is determined from the Fourier transformed domain. Finally, fractal dimension can be estimated using reticular cell counting approach. Moreover, automatic image segmentation was performed using fractal dimensions, spatial grey level co-occurrence matrices statistics, and grey level thresholding. Each image was segmented into a number of regions corresponding to distinctly different morphologies: heterochromatin, euchromatin, and background. Fractal dimensions and spatial grey level co-occurrence matrices statistics were found to be able to characterize and segment electron micrograph images     

Design of linear equalizers optimized for the structural similarity index.

We propose an algorithm for designing linear equalizers that maximize the structural similarity (SSIM) index between the reference and restored signals. The SSIM index has enjoyed considerable application in the evaluation of image processing algorithms. Algorithms, however, have not been designed yet to explicitly optimize for this measure. The design of such an algorithm is nontrivial due to the nonconvex nature of the distortion measure. In this paper, we reformulate the nonconvex problem as a quasi-convex optimization problem, which admits a tractable solution. We compute the optimal solution in near closed form, with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error (MMSE) solution, independent of the number of filter taps. To demonstrate the usefulness of the proposed algorithm, it is applied to restore images that have been blurred and corrupted with additive white gaussian noise. As a special case, we consider blur-free image denoising. In each case, its performance is compared to a locally adaptive linear MSE-optimal filter. We show that the images denoised and restored using the SSIM-optimal filter have higher SSIM index, and superior perceptual quality than those restored using the MSE-optimal adaptive linear filter. Through these results, we demonstrate that a) designing image processing algorithms, and, in particular, denoising and restoration-type algorithms, can yield significant gains over existing (in particular, linear MMSE-based) algorithms by optimizing them for perceptual distortion measures, and b) these gains may be obtained without significant increase in the computational complexity of the algorithm.     

Sparse molecular image representation

Sparsity-based models have proven to be very effective in most image processing applications. The notion of sparsity has recently been extended to structured sparsity models where not only the number of components but also their support is important. This paper goes one step further and proposes a new model where signals are composed of a small number of molecules, which are each linear combinations of a few elementary functions in a dictionary. Our model takes into account the energy on the signal components in addition to their support. We study our prior in detail and propose a novel algorithm for sparse coding that permits the appearance of signal dependent versions of the molecules. Our experiments prove the benefits of the new image model in various restoration tasks and confirm the effectiveness of priors that extend sparsity in flexible ways especially in case of inverse problems with low quality data.