Continuous Wavelet Transform with Arbitrary Scales and O(N) Complexity

Summary The continuous wavelet transform (CWT) is a common signal-processing tool for the analysis of nonstationary signals. We propose here a new B-spline-based method that allows the CWT computation at any scale. A nice property of the algorithm is that the computational cost is independent of the scale value. Its complexity is of the same order as that of the fastest published methods, without being restricted to dyadic or integer scales. The method reduces to the filtering of an auxiliary (pre-integrated) signal with an expanded mask that acts as a kind of modified ‘à trous’ filter.The algorithm is well-suited for a parallel implementation. Résumé La transformée continue en ondelettes (Continuous Wavelet Transform, CWT) est un outil de traitement de signal que l'on utilise volontiers pour analyser des signaux non-stationnaires. Nous proposons ici une nouvelle méthode de calcul de CWT, basée sur les B-splines, valide à toute échelle. Une propriété intéressante de l'algorithme est que son coût de calcul est indépendant de l'échelle. Son ordre de complexité est identique à celui des méthodes les plus rapides de la littérature, sans restriction à des échelles entières ou dyadiques. La méthode se résume à filtrer un signal auxiliaire (préalablement intégré) par un masque étendu, qui agit à la façon d'un filtre ‘à trous’ modifié. L'algorithme se prête facilement à une implémentation parallèle. Zusammenfassung Auf dem Gebiet der Signalverarbeitung ist die stetige Wavelet Transformation (CWT) eine weit verbreitete Methode zur Analyse nicht stationärer Signale. Wir schlagen eine B-spline basierte Methode vor, die die Berechnung der CWT auf einer beliebigen Skala ermöglicht. Ein Vorteil dieser Methode besteht darin, daß der Aufwand für die Berechnung unabhängig von der Skala ist. Der Berechnungsaufwand ist von gleicher Ordnung wie derjenige der schnellsten Methoden, die bisher veröffentlicht wurden. Die Methode ist nicht beschränkt auf dyadische oder ganzzahlige Skalen. Die Berechnungsmethode entspricht der Filterung eines vorher integrierten Hilfssignals mit einer erweiterten Maske, die als eine Art ‘à trous’ Filter dient. Der Algorithmus eignet sich sehr gut für eine parallel Implementation

Fast adaptive elliptical filtering using box splines

We demonstrate that it is possible to filter an image with an elliptic window of varying size, elongation and orientation with a fixed computational cost per pixel. Our method involves the application of a suitable global pre-integrator followed by a pointwise-adaptive localization mesh. We present the basic theory for the 1D case using a B-spline formalism and then appropriately extend it to 2D using radially-uniform box splines. The size and ellipticity of these radially-uniform box splines is adaptively controlled. Moreover, they converge to Gaussians as the order increases. Finally, we present a fast and practical directional filtering algorithm that has the capability of adapting to the local image features.Comment: 9 pages, 1 figur

Fast space-variant elliptical filtering using box splines

The efficient realization of linear space-variant (non-convolution) filters is a challenging computational problem in image processing. In this paper, we demonstrate that it is possible to filter an image with a Gaussian-like elliptic window of varying size, elongation and orientation using a fixed number of computations per pixel. The associated algorithm, which is based on a family of smooth compactly supported piecewise polynomials, the radially-uniform box splines, is realized using pre-integration and local finite-differences. The radially-uniform box splines are constructed through the repeated convolution of a fixed number of box distributions, which have been suitably scaled and distributed radially in an uniform fashion. The attractive features of these box splines are their asymptotic behavior, their simple covariance structure, and their quasi-separability. They converge to Gaussians with the increase of their order, and are used to approximate anisotropic Gaussians of varying covariance simply by controlling the scales of the constituent box distributions. Based on the second feature, we develop a technique for continuously controlling the size, elongation and orientation of these Gaussian-like functions. Finally, the quasi-separable structure, along with a certain scaling property of box distributions, is used to efficiently realize the associated space-variant elliptical filtering, which requires O(1) computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201

Watermarking scheme using slantlet transform and enhanced knight tour algorithm for medical images

Digital watermarking has been employed as an alternative solution to protect the medical healthcare system with a layer of protection applied directly on top of data stored. Medical image that is highly sensitive to the image processing and cannot tolerate any visual degradation has become the focus of digital watermarking. However, since watermarking introduces some changes on medical images, it is a challenge for medical image watermarking to maintain high imperceptibility and robustness at the same time. Research to date has tended to focus on the embedding method instead of the sequence of embedding of the watermarking itself. Also, although watermarking has been introduced into medical images as a layer of protection, it still cannot prevent a knowledgeable hacker from retrieving the watermark. Therefore, this research proposes a robust watermarking scheme with high imperceptibility for medical images to increase the effectiveness of the medical healthcare system in terms of perceptibility, embedding technique, embedding region and embedding sequence of the watermarking scheme. To increase imperceptibility of a watermark, this research introduces Dynamic Visibility Threshold, a new parameter that increases visual quality in terms of imperceptibility. It is a unique number which differs for each host image using descriptive statistics. In addition, two new concepts of embedding region, namely Embeddable zone (EBD) and Non-Embeddable zone (NEBD) to function as a non-parametric decision region to complicate the estimate of the detection function are also proposed. The sequence of embedding is shuffled using enhanced Knight Tour algorithm based on Slantlet Transform to increase the complexity of the watermarking scheme. A significant result from the Peak Signal-to-Noise Ratio (PSNR) evaluation showing approximately 270 dB was obtained, suggesting that this proposed medical image watermarking technique outperforms other contemporary techniques in the same working domain. Based on the experimental result using the standard dataset, all host images are resilient to Salt and Pepper Noise, Speckle Noise, Poisson Noise, Rotation and Sharpen Filter with minimum Bit Error Rate (BER) of 0.0426 and Normalized Cross-Correlation (NCC) value of as high as 1. Since quartile theory is used, this experiment has shown that among all three quartiles, the Third Quartile performs the best in functioning as Dynamic Visibility Threshold (DVT) with 0 for BER and 1 for NCC evaluation

Nondyadic and nonlinear multiresolution image approximations

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes

Análisis y síntesis de señales de audio a través de la Transformada Wavelet continua y compleja: El algoritmo CWAS

En esta Tesis se pretende demostrar que la Transformada Wavelet Continua y Compleja (CCWT) puede ser una herramienta precisa para la obtención de características de alto nivel de la señal de audio, a través de un algoritmo generalista del modelo que se propone de la misma. Se presenta un algoritmo funcional basado en la CCWT, el algoritmo de Síntesis Aditiva por Wavelet Complejas, o CWAS por sus siglas en inglés (Complex Wavelet Additive Synthesis). El desarrollo matemático presentado permite llegar finalmente la obtención de un novedoso modelo de la señal de audio bien posicionado de cara a posibles aplicaciones. Un filtrado pasobanda complejo unitario permite el cálculo de los coeficientes wavelet, en cuyo módulo se indican de forma implícita las bandas del espectro de frecuencia que conforman la zona de influencia de cada componente detectada. La suma de los coeficientes wavelet en las bandas asociadas a cada componente proporciona una función compleja para cada parcial, de amplitud y fase instantáneas altamente coherentes (es decir, muy cercanas al par canónico teórico de la señal). Es precisamente la coherencia en fase la principal ventaja del modelo propuesto. Un simple modelo se síntesis aditiva permite la generación de una señal sintética de características tímbricas y tonales muy similares a la señal original, con la característica añadida de una diferencia punto por punto entre las señales analizada y sintética que resulta despreciable numéricamente para la mayoría de las aplicaciones. El algoritmo CWAS se ha utilizado en síntesis de sonidos, localización de onsets y detección de fundamentales entre otras aplicaciones, con resultados muy prometedores. Del mismo modo, se han hecho grandes progresos de cara a aplicaciones más completas y complejas, como la separación ciega de fuentes monaurales de sonido. El algoritmo CWAS presenta una serie de ventajas e inconvenientes sobre otras técnicas basadas en diferentes Distribuciones Tiempo--Frecuencia, como la STFT. Entre las ventajas, a partir de la coherencia en fase se consiguen resultados en la síntesis de sonido por encima de los arrojados utilizando otras técnicas, lo cual permite abordar con elevadas esperanzas de éxito aplicaciones más ambiciosas. La principal limitación del algoritmo propuesto es el tiempo de procesado, que ha impedido por el momento el empleo de esta técnica en aplicaciones en tiempo real. Sin embargo, se demuestra que el algoritmo CWAS es sensiblemente más rápido que la STFT en condiciones de trabajo equivalentes