Natural image modeling using complex wavelets Andre Jalobeanu a , Laure Blanc-Feraud and Josiane Zerubia b a USRA/RIACS, NASA Ames Research Center, Moffett Field, CA, USA b CNRS/INRIA/UNSA, INRIA Sophia Antipolis, France ABSTRACT 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 behavior of these parameters. This model has been successfully applied to denoising and deconvolution, for real images and simulations provided by the French Space Agency. Keywords: Complex wavelets, image modeling, hierarchical Bayesian inference, denoising, deblurring 1. INTRODUCTION There are two topics presented here, the modeling of natural remote sensing images, and their application to solve ill-posed inverse problems. The images we deal with are highly complex and we do not attempt to model them completely. They are defined over high-dimensional spaces which are difficult to approach. Through experimental study, it is possible to project these images onto lower-dimensional spaces, making them more convenient to handle. The modeling process can be seen as finding the best projection, in the sense that it provides us with a good understanding of the observed phenomena, and therefore a good representation. However, this projection also has to provide the shortest description, so that the results of the projection are accessible (it is not realistic to model each pixel individually). Furthermore, we should not forget our final goal, which is denoising in the present work. Thus the model we construct is well suited for the problem to solve. In Sect. 2, we first recall how to build a complex wavelet transform. In Sect. 3 we study the properties of natural images, leading to a few general consequences on image modeling. After explaining why we choose to use complex wavelets in Sect. 5, we show how to build a statistical model of the subbands. Finally, in Sect. 7, we propose a new hierarchical multiscale model, combining global parameters with the subband model to form a general image model. In Sect. 8, this is applied to noise removal, and we show how to build complex wavelet packets to also deal with blur, then we give the details and results of the new multiscale deblurring technique. 2. THE COMPLEX WAVELET TRANSFORM 2.1. Implementation To build a complex wavelet transform (CWT), Kingsbury 1 has developed a quad-tree algorithm, by noting that an approximate shift invariance can be obtained with a real biorthogonal transform by doubling the sampling rate at each scale. This is achieved by computing 4 parallel wavelet transforms, which are differently subsampled. Thus, the redundancy is limited to 4, compared to real shift invariant transforms. At level j = 1, the CWT it is simply a non-decimated wavelet transform (using a pair of odd-length filters h o and g o ) whose coefficients are re-ordered into 4 interleaved images by using their parity. This defines the 4 trees T =A, B, C and D. If a and d denote approximation and detail coefficients (a 0 ≡ X, the input image), we have: Tree T A B C D (a 1 T ) x,y (a 0 ⋆h o h o ) 2x,2y (a 0 ⋆h o h o ) 2x,2y+1 (a 0 ⋆h o h o ) 2x+1,2y (a 0 ⋆h o h o ) 2x+1,2y+1 (d 1,1 T ) x,y (a 0 ⋆g o h o ) 2x,2y (a 0 ⋆g o h o ) 2x,2y+1 (a 0 ⋆g o h o ) 2x+1,2y (a 0 ⋆g o h o ) 2x+1,2y+1 (d 1,2 T ) x,y (a 0 ⋆h o g o ) 2x,2y (a 0 ⋆h o g o ) 2x,2y+1 (a 0 ⋆h o g o ) 2x+1,2y (a 0 ⋆h o g o ) 2x+1,2y+1 (d 1,3 T ) x,y (a 0 ⋆g o g o ) 2x,2y (a 0 ⋆g o g o ) 2x,2y+1 (a 0 ⋆g o g o ) 2x+1,2y (a 0 ⋆g o g o ) 2x+1,2y+1 E-mail: ajalobea@riacs.edu, blancf@sophia.inria.fr, zerubia@sophia.inria.fr