Image processing and nonparametric regression
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1 Image processing and nonparametric regression Rencontres R BoRdeaux 2012 B. Thieurmel Collaborators : P.A. Cornillon, N. Hengartner, E. Matzner-Løber, B. Wolhberg 2 Juillet 2012 Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 1 / 20
2 Preamble Definition : noisy image = original image + noise Our study : gaussian noise Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 2 / 20
3 Preamble image : matrix pixel one pixel represent a value of grey level or color level is spatially defined by its coordinates (i, j) is surrounded with 8 neighbors (vertical, horizontal and diagonal) numerical measure : Peak Signal to Noise Ratio ( PSNR) PSNR = 10 log10( d 2 MSE ) d maximum possible pixel value of the image MSE mean squared error between the original image and the treated image quality of reconstruction, in decibel (db) well reconstructed image 30 PSNR 40 Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 3 / 20
4 As a regression problem Notations : y ij the grey level at pixel (i, j) Y the grey level of the pixel to be denoised X R 10 (or more) the vector of explanatory variables Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 4 / 20
5 As a regression problem Non parametric regression model : (X i, Y i ) R d R pairs of observations Y = m(x ) + ε Ŷ = S λ Y Using method Iterative Bias Reduction (IBR) : developed by Cornillon, Hengartner et Matzner-Løber competes with classic techniques (MARS, GAM) estimation without constraints of the regression function m R package ibr available on cran Problem : n = smoothing matrix very big Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 5 / 20
6 As a regression problem Non parametric regression model : (X i, Y i ) R d R pairs of observations Y = m(x ) + ε Ŷ = S λ Y Using method Iterative Bias Reduction (IBR) : developed by Cornillon, Hengartner et Matzner-Løber competes with classic techniques (MARS, GAM) estimation without constraints of the regression function m R package ibr available on cran Problem : n = smoothing matrix very big Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 5 / 20
7 As a regression problem Non parametric regression model : (X i, Y i ) R d R pairs of observations Y = m(x ) + ε Ŷ = S λ Y Using method Iterative Bias Reduction (IBR) : developed by Cornillon, Hengartner et Matzner-Løber competes with classic techniques (MARS, GAM) estimation without constraints of the regression function m R package ibr available on cran Problem : n = smoothing matrix very big Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 5 / 20
8 Image partitioning Resolution : Image partitioning in small-sized regions Our proposal : free itself from the choice of the size of regions have a certain freedom of shape data dependent regions use of CART(Breiman et al., 1984) and more exactly regression trees explain grey level Y by coordinates (i, j) Modifications of package rpart (T.M. Therneau et al., 2002) : application for high-dimension data control maximum and minimum region size Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 6 / 20
9 Image partitioning Resolution : Image partitioning in small-sized regions Our proposal : free itself from the choice of the size of regions have a certain freedom of shape data dependent regions use of CART(Breiman et al., 1984) and more exactly regression trees explain grey level Y by coordinates (i, j) Modifications of package rpart (T.M. Therneau et al., 2002) : application for high-dimension data control maximum and minimum region size Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 6 / 20
10 Image partitioning Resolution : Image partitioning in small-sized regions Our proposal : free itself from the choice of the size of regions have a certain freedom of shape data dependent regions use of CART(Breiman et al., 1984) and more exactly regression trees explain grey level Y by coordinates (i, j) Modifications of package rpart (T.M. Therneau et al., 2002) : application for high-dimension data control maximum and minimum region size Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 6 / 20
11 Image partitioning Evolution of partitioning via rpart : Characteristics : rectangular regions detection of the horizontal and/or vertical structures of the image Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 7 / 20
12 Regression model for each region : Non parametric denoising Y i = m(x i ) + ε i, i = 1,..., n, with X R 10 the vector of explanatory variables, and n the size of the region Question : Influence of the size of regions? Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 8 / 20
13 Non parametric denoising maximum size of regions : 700 pixels number of regions : 686 region no overlapping visibility of the outlines of regions PSNR = db Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 9 / 20
14 Non parametric denoising Rotation : iteration of image partitioning and image denoising via IBR rotation of plan IJ between every iteration prediction of a same pixel with different regions unique prediction : mean of predicted values Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 10 / 20
15 Non parametric denoising Question : Influence of the number of rotations? Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 11 / 20
16 Non parametric denoising maximum size of regions : 300 pixels number of rotations : 3 PSNR = db db Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 12 / 20
17 BM3D vs IBR Presentation : developed by Dabov et al., 2007 state of art algorithm : standard deviation of the noise in parameter strong influence on the result unknown noise, default standard deviation = 25 source Matlab code available Comparison BM3D / IBR : several images with various noise 3 results : 1 BM3D with good standard deviation of the noise 2 BM3D with default standard deviation of the noise 3 IBR, 3 rotations, maximum size = 700 Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 13 / 20
18 BM3D vs IBR Presentation : developed by Dabov et al., 2007 state of art algorithm : standard deviation of the noise in parameter strong influence on the result unknown noise, default standard deviation = 25 source Matlab code available Comparison BM3D / IBR : several images with various noise 3 results : 1 BM3D with good standard deviation of the noise 2 BM3D with default standard deviation of the noise 3 IBR, 3 rotations, maximum size = 700 Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 13 / 20
19 BM3D vs IBR BM3D with good standard deviation of the noise (red) : IBR (blue) less successful of more or less 2 db the differences increase with the level of noise BM3D with default standard deviation of the noise (green) : IBR better for a low level of noise Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 14 / 20
20 Conclusion Performance : Image denoising by means of the method IBR already gives satisfactory results for its entrance to the vast field of image processing Current research : more freedom of shape in the creation of regions implementation of connections between similar regions Others applications : use of the rpart partitioning for high-dimension data in regression application in chronological series Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 15 / 20
21 Inpainting Preamble Objective : reconstruct the missing parts of an image by means of the iterative bias reduction Picture data base : black and white images and color images accent put on the structural reconstruction Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 16 / 20
22 Inpainting Image processing Process : 1 pixels treated one by one, from extremities to the center 2 definition of a region formed by the d available neighbors in a neighborhood of 4 pixels 3 data base : K nearest neighbors n most similar regions Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 17 / 20
23 Inpainting Method and Results color images : formed by three different images (Red, Green, Blue) treatment separated by image Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 18 / 20
24 Inpainting Method and Results Conclusion : Good structural reconstruction No impression of blur Current research : use of the structural information of the image in the choice of the filling order Others applications : treatment of missing data, tested at the moment for chronological series Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 19 / 20
25 Image processing and nonparametric regression Rencontres R BoRdeaux 2012 B. Thieurmel Collaborators : P.A. Cornillon, N. Hengartner, E. Matzner-Løber, B. Wolhberg 2 Juillet 2012 Rencontres R BoRdeaux 2012 B. Thieurmel Image processing and nonparametric regression 20 / 20
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