Javier Portilla

Image denoising using scale mixtures of Gaussians in the wavelet domain

We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error.

A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients

We present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform. The model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations, orientations, and scales. We develop an efficient algorithm for synthesizing random images subject to these constraints, by iteratively projecting onto the set of images satisfying each constraint, and we use this to test the perceptual validity of the model. In particular, we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set. We also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures.

Texture characterization via joint statistics of wavelet coefficient magnitudes

We present a parametric statistical characterization of texture images in the context of an overcomplete complex wavelet frame. The characterization consists of the local autocorrelation of the coefficients in each subband, the local autocorrelation of the coefficent magnitudes, and the cross-correlation of coefficient magnitudes at all orientations and adjacent spatial scales. We develop an efficient algorithm for sampling from an implicit probability density conforming to these statistics, and demonstrate its effectiveness in synthesizing artificial and natural texture images.

Efficient spatial-domain implementation of a multiscale image representation based on Gabor functions

Gabor schemes of multiscale image representation are useful in many computer vision applications. However, the classic Gabor expansion is computationally expensive due to the lack of orthogonality of Gabor functions. Some alternative schemes, based on the application of a bank of Gabor filters, have important advantages such as computational efficiency and robustness, at the cost of redundancy and lack of completeness. In a previous work we proposed a quasicomplete Gabor transform, suitable for fast implementations in either space or frequency domains. Reconstruction was achieved by simply adding together the even Gabor channels. We develop an optimized spatial-domain implementation, using one-dimensional 11-tap filter masks, that is faster and more flexible than Fourier implementations. The reconstruction method is improved by applying fixed and independent weights to the Gabor channels before adding them together. Finally, we analyze and implement, in the spatial domain, two ways to incorporate a high-pass residual, which permits a visually complete representation of the image.

Adaptive Wiener denoising using a Gaussian scale mixture model in the wavelet domain

We describe a statistical model for images decomposed in an overcomplete wavelet pyramid. Each coefficient of the pyramid is modeled as the product of two independent random variables: an element of a Gaussian random field, and a hidden multiplier with a marginal log-normal prior. The latter modulates the local variance of the coefficients. We assume subband coefficients are contaminated with additive Gaussian noise of known covariance, and compute a MAP estimate of each multiplier variable based on observation of a local neighborhood of coefficients. Conditioned on this multiplier, we then estimate the subband coefficients with a local Wiener estimator. Unlike previous approaches, we (a) empirically motivate our choice for the prior on the multiplier; (b) use the full covariance of signal and noise in the estimation; (c) include adjacent scales in the conditioning neighborhood. To our knowledge, the results are the best in the literature, both visually and in terms of squared error.

Image Restoration Using Space-Variant Gaussian Scale Mixtures in Overcomplete Pyramids

In recent years, Bayes least squares-Gaussian scale mixtures (BLS-GSM) has emerged as one of the most powerful methods for image restoration. Its strength relies on providing a simple and, yet, very effective local statistical description of oriented pyramid coefficient neighborhoods via a GSM vector. This can be viewed as a fine adaptation of the model to the signal variance at each scale, orientation, and spatial location. Here, we present an enhancement of the model by introducing a coarser adaptation level, where a larger neighborhood is used to estimate the local signal covariance within every subband. We formulate our model as a BLS estimator using space-variant GSM. The model can be also applied to image deconvolution, by first performing a global blur compensation, and then doing local adaptive denoising. We demonstrate through simulations that the proposed method, besides being model-based and noniterative, it is also robust and efficient. Its performance, measured visually and in L2-norm terms, is significantly higher than the original BLS-GSM method, both for denoising and deconvolution.

Image restoration through l0 analysis-based sparse optimization in tight frames

Sparse optimization in overcomplete frames has been widely applied in recent years to ill-conditioned inverse problems. In particular, analysis-based sparse optimization consists of achieving a certain trade-off between fidelity to the observation and sparsity in a given linear representation, typically measured by some ℓp quasi-norm. Whereas most popular choice for p is 1 (convex optimization case), there is an increasing evidence on both the computational feasibility and higher performance potential of non-convex approaches (0 ≤ p ≪ 1). The extreme p = 0 case is especial, because analysis coefficients of typical images obtained using typical pyramidal frames are not strictly sparse, but rather compressible. Here we model the analysis coefficients as a strictly sparse vector plus a Gaussian correction term. This statistical formulation allows for an elegant iterated marginal optimization. We also show that it provides state-of-the-art performance, in a least-squares error sense, in standard deconvolution tests.

Image restoration using Gaussian scale mixtures in the wavelet domain

A statistical model for images decomposed in an overcomplete wavelet pyramid is described. Each neighborhood of pyramid coefficients is modeled as the product of a Gaussian vector of known covariance, and an independent hidden positive scalar random variable. We propose an efficient Bayesian estimator for the pyramid coefficients of an image degraded by linear distortion (e.g., blur) and additive Gaussian noise. We demonstrate the quality of our results in simulations over a wide range of blur and noise levels.

Full blind denoising through noise covariance estimation using Gaussian scale mixtures in the wavelet domain

We describe an efficient generalized expectation maximization algorithm for estimating the spectral features of a noise source corrupting an observed image. We use a statistical model for images decomposed in an overcomplete oriented pyramid. Each neighborhood of clean pyramid coefficients is modeled as a Gaussian scale mixture, whereas the noise is assumed Gaussian. Combining this GEM technique with a previous Bayesian denoise estimator, we obtain a full blind denoising algorithm, able to deal with homogeneous, Gaussian or mesokurtotic, noise sources of arbitrary covariance. Results demonstrate the high performance of the method for a wide range of corruption sources.

Image denoising via adjustment of wavelet coefficient magnitude correlation

We describe a novel method of removing additive white noise of known variance from photographic images. The method is based on a characterization of the statistical properties of natural images represented in a complex wavelet decomposition. Specifically, we decompose the noisy image into wavelet subbands, estimate the autocorrelation of both the noise-free raw coefficients and their magnitudes within each subband, impose these statistics by projecting onto the space of images having the desired autocorrelations, and reconstruct an image from the modified wavelet coefficients. This process is applied repeatedly, and can be accelerated to produce optimal results in only a few iterations. Denoising results compare favorably to three reference methods, both perceptually and in terms of mean squared error.