Multiresolution signal decomposition schemes. Part 2: Morphological wavelets

In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens. The aim of this report, which is a sequel to a previous report devoted exclusively to the pyramid transform, is to present an axiomatic framework encompassing most existing linear and nonlinear wavelet decompositions. Furthermore, it introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. This paper discusses one example in considerable detail, the max-lifting scheme, which has the intriguing property that it preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are

Multiresolution signal decomposition schemes

[PNA-R9810] Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This report proposes a general axiomatic pyramid decomposition scheme for signal analysis and synthesis. This scheme comprises the following ingredients: (i) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases towards higher levels; (ii) each step towards a higher level is constituted by an (information-reducing) analysis operator, whereas each step towards a lower level is modeled by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. In this report, several examples are described of linear as well as nonlinear (e.g., morphological) pyramid decomposition schemes. Some of these examples are known from the literature (Laplacian pyramid, morphological granulometries, skeleton decomposition) and some of them are new (morphological Haar pyramid, median pyramid). Furthermore, the report makes a distinction between single-scale and multiscale decomposition schemes (i.e. without or with sample reduction).#[PNA-R9905] In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinea

Adaptive lifting schemes with perfect reconstruction

In this paper, we propose a framework for constructing adaptive wavelet decompositions using the lifting scheme. A major requirement is that perfect reconstruction is possible without any overhead cost. In this paper we restrict ourselves to the update lifting stage. It is assumed that the update filter utilises local gradient information to adapt itself to the signal in the sense that smaller gradients `evoke' stronger update filters. As a result, sharp transitions in a signal will not be smoothed to the same extent as regions which are more homogeneous. The approach taken in this paper differs from other adaptive schemes found in the literature in the sense that that no bookkeeping is required in order to have perfect reconstruction

Fusion of MultiSpectral and Panchromatic Images Based on Morphological Operators

International audienceNonlinear decomposition schemes constitute an alternative to classical approaches for facing the problem of data fusion. In this paper we discuss the application of this methodology to a popular remote sensing application called pansharpening, which consists in the fusion of a low resolution multispectral image and a high resolution panchromatic image. We design a complete pansharpening scheme based on the use of morphological half gradients operators and demonstrate the suitability of this algorithm through the comparison with state of the art approaches. Four datasets acquired by the Pleiades, Worldview-2, Ikonos and Geoeye-1 satellites are employed for the performance assessment, testifying the effectiveness of the proposed approach in producing top-class images with a setting independent of the specific sensor

Combining nonlinear multiresolution system and vector quantization for still image compression

It is popular to use multiresolution systems for image coding and compression. However, general-purpose techniques such as filter banks and wavelets are linear. While these systems are rigorous, nonlinear features in the signals cannot be utilized in a single entity for compression. Linear filters are known to blur the edges. Thus, the low-resolution images are typically blurred, carrying little information. We propose and demonstrate that edge- preserving filters such as median filters can be used in generating a multiresolution system using the Laplacian pyramid. The signals in the detail images are small and localized in the edge areas. Principal component vector quantization (PCVQ) is used to encode the detail images. PCVQ is a tree-structured VQ which allows fast codebook design and encoding/decoding. In encoding, the quantization error at each level is fed back through the pyramid to the previous level so that ultimately all the error is confined to the first level. With simple coding methods, we demonstrate that images with PSNR 33 dB can be obtained at 0.66 bpp without the use of entropy coding. When the rate is decreased to 0.25 bpp, the PSNR of 30 dB can still be achieved. Combined with an earlier result, our work demonstrate that nonlinear filters can be used for multiresolution systems and image coding

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

The Multigrid Image Transform

A second order partial differential operator is applied to an image function. To this end we consider both the Laplacian and a more general elliptic operator. By using a multigrid operator known from the so-called approximation property, we derive a multiresolution decomposition of the image without blurring of edges at coarser levels. We investigate both a linear and a nonlinear variant and compare to some established methods