Pointwise Shape-Adaptive DCT for

denoising and image reconstruction:

denoising, deblocking and deblurring 

for grayscale and color images

Need for adaptation: Adaptive block-size?

• In regions where the image exhibits some uniform behaviour, a larger block-size corresponds to a lower MASE. On the contrary, in the vicinity of edges, an increase in the block-size typically produces an increase in the MASEbecause of the less sparse representation of the signal. 

A denoising algorithm based on 2D separable DCT with an adaptive block-size was proposed by Öktem et al. in 2001. The “classical” LPA-ICI was used to select the adaptive block-size in a pointwise manner.

•( Öktem, H., V. Katkovnik, K. Egiazarian, and J. Astola, “Local adaptive transform based image de-noising with varying window size”, Proc. IEEE Int. Conf. Image Process., ICIP 2001, Thessaloniki, Greece, pp. 273-276, Oct. 2001 )However, edges can appear at any orientation and complex geometries in the image can impose strong limitations on the adaptivity based on square blocks.Clearly, an improved approach should go beyond the traditional block DCT, in order to better adapt to the various arbitrarily oriented structures that may be present in the image.

• Two problems

• If a block with adaptive size is to be replaced by a more versatile adaptive-shape region, two problems immediately arise. 

• 1. How to efficiently transform the data within a region which is not a square?

• 2. How to find these adaptive-shape regions from the noisy observations?

Two solutions

• 1. Shape-Adaptive DCT (Sikora & Makai, 1995). A 2D (orthonormal) transform based on 1D DCTs that can be applied on arbitrarily-shaped regions. Good energy-compaction, low complexity, part of MPEG4 standard, can be thought as a generalization of the 2D block DCT. 

• 2. Anisotropic LPA-ICI (our team, 2004). Defines, for every point in the image, and adaptive starshaped neighborhood in which the underlying signal is uniform (in polynomial sense). Very accurate, allows adaptation to the finest structures in the image. Based on convolutions, it is among the fastest pointwise-adaptive methods for robust anisotropic filtering.

Shape-Adaptive DCT  

Anisotropic LPA-ICI

www.cs.tut.fi/~lasipKatkovnik, V., A. Foi, K. Egiazarian, and J. Astola, “Directional varying scale approximations for anisotropic signal processing”, Proc. of XII European Signal Process. Conf., EUSIPCO 2004, pp. 101-104, 2004. 

Foi, A., V. Katkovnik, K. Egiazarian, and J. Astola, “A novel anisotropic local polynomial estimator based on directional multiscale optimizations”, Proc. of the 6th IMA Int. Conf. Math. in Signal Processing, Cirencester (UK), pp. 79-82, 2004.

Pointwise Anisotropic LPA-ICI + SA-DCT approach

For every point in the image an adaptive-shape region is defined by the Anisotropic LPA-ICI

Pointwise Anisotropic LPA-ICI + SA-DCT approach

The noisy data restricted to the adaptive-shape region is extracted to be filtered in SA-DCT domain

Pointwise Anisotropic LPA-ICI + SA-DCT approach

The reconstructed data after the filtering in SA-DCT domain (hard-thresholding)

Noisy Denoised

Fast implementation of the LPA-ICI anisotropic neighborhoods

The anisotropic neighborhood is constructed as the polygonal hull of the adaptive-scale kernels' supports.

One-dimensional “linewise” kernels are used.

Fast implementation of the LPA-ICI anisotropic neighborhoods

Thresholding in SA-DCT domain

Thresholding in SA-DCT domain: Local Estimate

Thresholding in SA-DCT domain: Global Estimate

• All the local estimates are averaged together using adaptive weights that depend on their local variances and on the size of the corresponding adaptive-shape regions.