André Jalobeanu

Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method

Abstract The satellite image deconvolution problem is ill-posed and must be regularized. Herein, we use an edge-preserving regularization model using a ϕ function, involving two hyperparameters. Our goal is to estimate the optimal parameters in order to automatically reconstruct images. We propose to use the maximum-likelihood estimator (MLE), applied to the observed image. We need sampling from prior and posterior distributions. Since the convolution prevents use of standard samplers, we have developed a modified Geman–Yang algorithm, using an auxiliary variable and a cosine transform. We present a Markov chain Monte Carlo maximum-likelihood (MCMCML) technique which is able to simultaneously achieve the estimation and the reconstruction.

An adaptive Gaussian model for satellite image deblurring

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Since real satellite data show spatially variant characteristics, we propose here to use an inhomogeneous model. We use the maximum likelihood estimator (MLE) to estimate its parameters and we show that the MLE computed on the corrupted image is not suitable for image deconvolution because it is not robust to noise. We then show that the estimation is correct only if it is made from the original image. Since this image is unknown, we need to compute an approximation of sufficiently good quality to provide useful estimation results. Such an approximation is provided by a wavelet-based deconvolution algorithm. Thus, a hybrid method is first used to estimate the space-variant parameters from this image and then to compute the regularized solution. The obtained results on high resolution satellite images simultaneously exhibit sharp edges, correctly restored textures, and a high SNR in homogeneous areas, since the proposed technique adapts to the local characteristics of the data.

Satellite image deconvolution using complex wavelet packets

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem. Donoho (1994) has proposed to deconvolve the image without regularization and to denoise the result in a wavelet basis by thresholding the transformed coefficients. We have developed a new filtering method, consisting of using a complex wavelet packet basis. Herein, the thresholding functions associated to the proposed method are automatically estimated. The estimation is performed within a Bayesian framework, by modeling the subbands using generalized Gaussian distributions, and by applying the maximum a posteriori (MAP) estimator on each coefficient. Compared to real wavelet-packet-based algorithms, the proposed method is shift invariant, provides good directionality properties and remains of complexity O(N).

Image deconvolution using hidden Markov tree modeling of complex wavelet packets

In this paper, we propose to use a hidden Markov tree modeling of the complex wavelet packet transform, to capture the inter-scale dependencies of natural images. First, the observed image, blurred and noisy, is deconvolved without regularization. Then its transform is denoised within a Bayesian framework using the proposed model, whose parameters are estimated by an EM technique. The total complexity of this new deblurring algorithm remains O(N).

Estimation of blur and noise parameters in remote sensing

In this paper we propose a new algorithm to estimate the parameters of the noise related to the sensor and the impulse response of the optical system, from a blurred and noisy satellite or aerial image. The noise is supposed to be white, Gaussian and stationary. The blurring kernel has a parametric form and is modeled in such a way as to take into account the physics of the system (the atmosphere, the optics and the sensor). The observed scene is described by a fractal model, taking into account the scale invariance properties of natural images. The estimation is performed automatically by maximizing a marginalized likelihood, which is achieved by a deterministic algorithm whose complexity is limited to O (N), where N is the number of pixels.

Estimation of adaptive parameters for satellite image deconvolution

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Since real satellite data show spatially variant characteristics, we propose to use an inhomogeneous model. We use the maximum likelihood estimator (MLE) to estimate its parameters. We demonstrate that the MLE computed on the corrupted image is not suitable for image deconvolution, because it is not robust to noise. Then we show that the estimation is correct only if it is made from the original image. As this image is unknown, we need to compute an approximation of sufficiently good quality to provide useful estimation results. Such an approximation is provided by a wavelet-based deconvolution algorithm. Thus, an hybrid method is first used to estimate the space-variant parameters from this image and second to compute the regularized solution. The obtained results on high resolution satellite images simultaneously exhibit sharp edges, correctly restored textures and a high SNR in homogeneous areas, since the proposed technique adapts to the local characteristics of the data.

Natural image modeling using complex wavelets

We propose to model satellite and aerial images using a probabilistic approach. We show how the properties of these images, such as scale invariance, rotational invariance and spatial adaptivity lead to a new general model which aims to describe a broad range of natural images. The complex wavelet transform initially proposed by Kingsbury is a simple way of taking into account all these characteristics. We build a statistical model around this transform, by defining an adaptive Gaussian model with interscale dependencies, global parameters, and hyperpriors controlling the behaviour of these parameters. This model has been successfully applied to denoising and deconvolution, for real images and simulations provided by the French Space Agency.

Hyperparameter Estimation for Satellite Image Restoration by a MCMCML Method

Satellite images can be corrupted by an optical blur and electronic noise. Blurring is modeled by convolution, with a known linear operator H, and the noise is supposed to be additive, white and Gaussian, with a known variance. The recovery problem is ill-posed and therefore must be regularized. Herein, we use a regularization model which introduces a function, avoiding noise amplification while preserving image discontinuities (i.e. edges) of the restored image. This model involves two hyperparameters. Our goal is to estimate the optimal parameters in order to reconstruct images automatically. In this paper, we propose to use the Maximum Likelihood estimator, applied to the observed image. To evaluate the derivatives of this criterion, we must estimate expectations by sampling (samples are extracted from a Markov chain). These samples are images whose probability takes into account the convolution operator. Thus, it is very difficult to obtain them directly by using a standard sampler. We have developed a new algorithm for sampling, using an auxiliary variable based on Geman-Yang algorithm, and a cosine transform. We also present a new reconstruction method based on this sampling algorithm. We detail the Markov Chain Monte Carlo Maximum Likelihood (MCMCML) algorithm which ables to simultaneously estimate the parameters, and to reconstruct the corrupted image.

Satellite and aerial image deconvolution using an EM method with complex wavelets

In this paper we present a new deconvolution method, able to deal with noninvertible blurring functions. To avoid noise amplification, a prior model of the image to be reconstructed is used within a Bayesian framework. We use a spatially adaptive prior defined with a complex wavelet transform in order to preserve shift invariance and to better restore variously oriented features. The unknown image is estimated by an EM technique, whose E step is a Landweber update iteration, and the M step consists of denoising the image, which is achieved by wavelet coefficient thresholding. The new algorithm has been applied to high resolution satellite and aerial data, showing better performance than existing techniques when the blurring process is not invertible, like motion blur for instance.