[en] We propose the SHAH (SHape-Adaptive Haar) transform for images, which results in an orthonormal, adaptive decomposition of the image into Haar-wavelet-like components, arranged hierarchically according to decreasing importance, whose shapes reflect the features present in the image. The decomposition is as sparse as it can be for piecewise-constant images. It is performed via an stepwise bottom-up algorithm with quadratic computational complexity; however, nearly-linear variants also exist. SHAH is rapidly invertible. We show how to use SHAH for image denoising. Having performed the SHAH transform, the coefficients are hard- or soft-thresholded, and the inverse transform taken. The SHAH image denoising algorithm compares favourably to the state of the art for piecewise-constant images. A clear asset of the methodology is its very general scope: it can be used with any images or more generally with any data that can be represented as graphs or networks.
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Bibliography
J.-P.Antoine, D.Rosca, P.Vandergheynst, (2010), Wavelet Transform on Manifolds:Old and New Approaches, Applied and Computational Harmonic Analysis, 28, 189–202.
F.Arandiga, A.Cohen, R.Donat, N.Dyn, B.Matei, (2008), Approximation of Piecewise Smooth Functions and Images by Edge-Adapted (ENO-EA) Nonlinear Multiresolution Techniques, Applied Computational and Harmonic Analysis, 24, 225–250.
E.J.Candès, D.L.Donoho, (2001), Curvelets and Curvilinear Integrals, Journal of Approximation Theory, 113, 59–90.
R.L.Claypoole, and R.G.Baraniuk, (2000), “A Multiresolution Wedgelet Transform for Image Processing,” in SPIE Technical Conference on Wavelet Applications in Signal Processing VIII, pp. 253–262.
D.L.Donoho, (1999), Wedgelets:Nearly-Minimax Estimation of Edges, Annals of Statistics, 27, 859–897.
P.Fryzlewicz, (2007), Unbalanced Haar Technique for Nonparametric Function Estimation, Journal of the American Statistical Association, 102, 1318–1327.
M.Gavish, B.Nadler, R.R.Coifman, (2010), Multiscale Wavelets on Trees, Graphs and High Dimensional Data:Theory and Applications to Semi Supervised Learning, International Conference on Machine Learning,. 367–374.
D.K.Hammond, P.Vandergheynst, R.Gribonval, (2009), Wavelets on Graphs via Spectral Graph Theory, Applied and Computational Harmonic Analysis, 30, 129–150.
H.Heijmans, J.Goutsias, (2000), Nonlinear Multiresolution Signal Decomposition Schemes—Part II:Morphological Wavelets, IEEE Transactions on Image Processing, 9, 1897–1913.
M.Jansen, G.Nason, B.Silverman, (2009), Multiscale Methods for Data on Graphs and Irregular Multidimensional Situations, Journal of the Royal Statistical Society, Series B, 71, 97–125.
A.Kovac, A.Smith, (2011), Nonparametric Regression on a Graph, Journal of Computational and Graphical Statistics, 20, 432–447.
J.Krommweh, (2010), Tetrolet Transform:A New Adaptive Haar Wavelet Algorithm for Sparse Image Representation, Journal of Visual Communication and Image Representation, 21, 364–374.
A.B.Lee, B.Nadler, L.Wasserman, (2008), Treelets—An Adaptive Multi-Scale Basis for Sparse Unordered Data, Annals of Applied Statistics, 2, 435–471.
E.Le Pennec, S.Mallat, (2005), Sparse Geometrical Image Representation with Bandelets, IEEE Transactions on Image Processing, 14, 423–438.
M.Maggioni, J.C.Bremer, R.R.Coifman, and A.D.Szlam, (2005), “Biorthogonal Diffusion Wavelets for Multiscale Representations on Manifolds and Graphs,” in Proceedings SPIE Wavelet XI (Vol. 5914), eds. M.Papadakis, A.F.Laine, and M.A.Unser.
——— (2009a), A Wavelet Tour of Signal Processing (3rd ed.), New York:Academic Press.
2007, The Haar Wavelet Transform of a Dendrogram, Journal of Classification, 24, 3–32.
P.Narendra, M.Goldberg, (1980), Image Segmentation With Directed Trees, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 185–191.
G.Peyré, (2011), A Review of Adaptive Image Representations, IEEE Journal of Selected Topics in Signal Processing, 5, 896–911.
G.Plonka, (2009), The Easy Path Wavelet Transform:A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data, Multiscale Modeling and Simulation, 7, 1474–1496.
J.Polzehl, V.Spokoiny, (2000), Adaptive Weights Smoothing With Applications to Image Restoration, Journal of the Royal Statistical Society, Series B, 62, 335–354.
A.Singh, R.D.Nowak, A.R.Calderbank, (2010), Detecting Weak But Hierarchically-Structured Patterns in Networks, Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS-10),. 749–756.
W.Sweldens, (1996), The Lifting Scheme:A Custom-Design Construction of Biorthogonal Wavelets, Applied and Computational Harmonic Analysis, 3, 186–200.
A.D.Szlam, M.Maggioni, R.R.Coifman, and J.C.Bremer, (2005), “Diffusion-Driven Multiscale Analysis on Manifolds and Graphs:Top-Down and Bottom-Up Constructions,” in Proceedings SPIE Wavelet XI (Vol. 5914), eds. M.Papadakis, A.F.Laine, and M.A.Unser.
C.Timmermans, R.von Sachs, (2015), A Novel Semi-Distance for Measuring Dissimilarities of Curves With Sharp Local Patterns, Journal of Statistical Planning and Inference, 160, 35–50.
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