Directionlets: Anisotropic Multidirectional representation with separable filtering

In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis functions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N/sup -1.55/), which, while slower than the optimal rate O(N/sup -2/), is much better than O(N/sup -1/) achieved with wavelets, but at similar complexity.

Directionlets

Vladan Velisavljevic

Efficient representation of geometrical information in images is very important in many image processing areas, including compression, denoising and feature extraction. However, the design of transforms that can capture these geometrical features and represent them with a sparse description is very challenging. Recently, the separable wavelet transform achieved a great success providing a computationally simple tool and allowing for a sparse representation of images. However, in spite of the success, the efficiency of the representation is limited by the spatial isotropy of the wavelet basis functions built in the horizontal and vertical directions as well as the lack of directionality. One-dimensional discontinuities in images (edges and contours), which are very important elements in visual perception, intersect with too many wavelet basis functions leading to a non-sparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, more flexible multi-directional and anisotropic transforms are required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic multi-directional wavelet transform. The transform retains the separable filtering, subsampling and simplicity of computations and filter design from the standard two-dimensional wavelet transform, unlike in the case of some other existing directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding anisotropic basis functions, which we call directionlets, have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for non-linear approximation of images, achieving the decay of mean-square error O(N-1.55), which, while slower than the optimal rate O(N-2), is much better than O(N-1) achieved with wavelets, but at similar complexity. Owing to critical sampling, directionlets can easily be applied to image compression since it is possible to use Lagrange optimization as opposed to the case of overcomplete expansions. The compression algorithms based on directionlets outperform the methods based on the standard wavelet transform achieving better numerical results and visual quality of the reconstructed images. Moreover, we have adapted image denoising algorithms to be used in conjunction with an undecimated version of directionlets obtaining results that are competitive with the current state-of-the-art image denoising methods while having lower computational complexity.

EPFL2005

Sparse Image Representation by Directionlets

Pier Luigi Dragotti, Vladan Velisavljevic, Martin Vetterli

Despite the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency and sparsity of its representation are limited by the spatial symmetry and separability of its basis functions built in the horizontal and vertical directions. One-dimensional discontinuities in images (edges or contours), which are important elements in visual perception, intersect too many wavelet basis functions and lead to a non-sparse representation. To capture efficiently these elongated structures characterized by geometrical regularity along different directions (not only the horizontal and vertical), a more complex multidirectional (M-DIR) and asymmetric transform is required. We present a lattice-based perfect reconstruction and critically sampled asymmetric M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional (2D) WT, unlike the case for some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding asymmetric basis functions, called direction-lets, have directional vanishing moments along any two directions with rational slopes, which allows for a sparser representation of elongated and oriented features. As a consequence of the improved sparsity, directionlets provide an efficient tool for nonlinear approximation of images, significantly outperforming the standard 2D WT. Furthermore, directionlets combined with wavelet-based image compression methods lead to a gain in performance in terms of both the mean square error and visual quality, especially at low bit-rate compression, while retaining the same complexity. Finally, a shift-invariant non-subsampled version of directionlets is successfully implemented in image interpolation, where critical sampling is not a key requirement.

Elsevier2010

Space-frequency quantization for image compression with directionlets

Vladan Velisavljevic, Martin Vetterli

The standard separable 2-D wavelet transform (WT) has recently achieved a great success in image processing because it provides a sparse representation of smooth images. However, it fails to efficiently capture 1-D discontinuities, like edges or contours. These features, being elongated and characterized by geometrical regularity along different directions, intersect and generate many large magnitude wavelet coefficients. Since contours are very important elements in the visual perception of images, to provide a good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional features. In our previous work, we proposed a construction of critically sampled perfect reconstruction transforms with directional vanishing moments imposed in the corresponding basis functions along different directions, called directionlets. In this paper, we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method outperforms the standard SFQ in a rate-distortion sense, both in terms of mean-square error and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm.

2007

Directionlets

Vladan Velisavljevic

EPFL2005