Wavelets | Trees (data structures)
Embedded Zerotrees of Wavelet transforms (EZW) is a lossy image compression algorithm. At low bit rates, i.e. high compression ratios, most of the coefficients produced by a subband transform (such as the wavelet transform)will be zero, or very close to zero. This occurs because "real world" images tend to contain mostly low frequency information (highly correlated). However where high frequency information does occur (such as edges in the image) this is particularly important in terms of human perception of the image quality, and thus must be represented accurately in any high quality coding scheme. By considering the transformed coefficients as a tree (or trees) with the lowest frequency coefficients at the root node and with the children of each tree node being the spatially related coefficients in the next higher frequency subband, there is a high probability that one or more subtrees will consist entirely of coefficients which are zero or nearly zero, such subtrees are called zerotrees. Due to this, we use the terms node and coefficient interchangeably, and when we refer to the children of a coefficient, we mean the child coefficients of the node in the tree where that coefficient is located. We use children to refer to directly connected nodes lower in the tree and descendants to refer to all nodes which are below a particular node in the tree, even if not directly connected. In zerotree based image compression scheme such as EZW and SPIHT, the intent is to use the statistical properties of the trees in order to efficiently code the locations of the significant coefficients. Since most of the coefficients will be zero or close to zero, the spatial locations of the significant coefficients make up a large portion of the total size of a typical compressed image. A coefficient (likewise a tree) is considered significant if its magnitude (or magnitudes of a node and all its descendants in the case of a tree) is above a particular threshold. By starting with a threshold which is close to the maximum coefficient magnitudes and iteratively decreasing the threshold, it is possible to create a compressed representation of an image which progressively adds finer detail. Due to the structure of the trees, it is very likely that if a coefficient in a particular frequency band is insignificant, then all its descendants (the spatially related higher frequency band coefficients) will also be insignificant. EZW uses four symbols to represent (a) a zerotree root, (b) an isolated zero (a coefficient which is insignificant, but which has significant descendants), (c) a significant positive coefficient and (d) a significant negative coefficient. The symbols may be thus represented by two binary bits. The compression algorithm consistsof a number of iterations through a dominant pass and a subordinate pass, the threshold is updated (reduced by a factor of two) after each iteration. The dominant pass encodes the significance of the coefficients which have not yet been found significant in earlier iterations, by scanning the trees and emitting one of the four symbols. The children of a coefficient are only scanned if the coefficient was found to be significant, or if the coefficient was an isolated zero. The subordinate pass emits one bit (the most significant bit of each coefficient not so far emitted) for each coefficient which has been found significant in the previous significance passes. The subordinate pass is therefore similar to bit-plane coding. There are several important features to note. Firstly, it is possible to stop the compression algorithm at any time and obtain an approximation of the original image, the greater the number of bits received, the better the image. Secondly, due to the way in which the compression algorithm is structured as a series of decisions, the same algorithm can be run at the decoder to reconstruct the coefficients, but with the decisions being taken according to the incoming bit stream. In practical implementations, it would be usual to use an entropy code such as arithmetic code to further improve the performance of the dominant pass. Bits from the subordinate pass are usually random enough that entropy coding provides no further coding gain. The coding performance of EZW has since been exceeded by SPIHT and its many derivatives. (Wikipedia).
Wavelets: a mathematical microscope
Wavelet transform is an invaluable tool in signal processing, which has applications in a variety of fields - from hydrodynamics to neuroscience. This revolutionary method allows us to uncover structures, which are present in the signal but are hidden behind the noise. The key feature of w
From playlist Fourier
Poles and Zeros of z-Transforms
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Definition of poles and zeros for z-transforms that are a ratio of polynomials in z. Examples.
From playlist The z-Transform
Emmanuel Candès: Wavelets, sparsity and its consequences
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Inverting the Z transform and Z transform of systems
I move from signals to systems in describing discrete systems in the z domain
From playlist Discrete
Introduction to the z-Transform
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces the definition of the z-transform, the complex plane, and the relationship between the z-transform and the discrete-time Fourier transfor
From playlist The z-Transform
The Two-Dimensional Discrete Fourier Transform
The two-dimensional discrete Fourier transform (DFT) is the natural extension of the one-dimensional DFT and describes two-dimensional signals like images as a weighted sum of two dimensional sinusoids. Two-dimensional sinusoids have a horizontal frequency component and a vertical frequen
From playlist Fourier
Compositional Structure of Classical Integral Transforms
The recently implemented fractional order integro-differentiation operator, FractionalD, is a particular case of more general integral transforms. The majority of classical integral transforms are representable as compositions of only two transforms: the modified direct and inverse Laplace
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Understanding Wavelets, Part 2: Types of Wavelet Transforms
Explore the workings of wavelet transforms in detail. •Try Wavelet Toolbox: https://goo.gl/m0ms9d •Ready to Buy: https://goo.gl/sMfoDr You will also learn important applications of using wavelet transforms with MATLAB®. Video Transcript: In the previous session, we discussed wavelet co
From playlist Understanding Wavelets
An introduction to the wavelet transform (and how to draw with them!)
The wavelet transform allows to change our point of view on a signal. The important information is condensed in a smaller space, allowing to easily compress or filter the signal. A lot of approximations are made in this video, like a lot of missing √2 factors. This choice was made to keep
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Stéphane Mallat: "Deep Generative Networks as Inverse Problems"
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From playlist New Deep Learning Techniques 2018
CS25 I Stanford Seminar - Audio Research: Transformers for Applications in Audio, Speech and Music
Transformers have touched many fields of research and audio and music is no different. This talk will present 3 of my papers as a case study done, on how we can leverage powerfulness of Transformers, with that of representation learning, signal processing and clustering. For the first part
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Stéphane Mallat - Apprentissage par invariants en grande dimension
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From playlist Huawei-IHÉS Workshop on Mathematical Sciences
Dirichlet Eta Function - Integral Representation
Today, we use an integral to derive one of the integral representations for the Dirichlet eta function. This representation is very similar to the Riemann zeta function, which explains why their respective infinite series definition is quite similar (with the eta function being an alte rna
From playlist Integrals
Stéphane Mallat: High dimensional learning from images to physics
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From playlist 30 years of wavelets
Autoencoder Image Generation with Multiscale Sparse (...) - Mallat - Workshop 3 - CEB T1 2019
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Hans Feichtinger: Wavelet theory, coorbit spaces and ramifications
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
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Uniform p-adic wave front sets and zero loci of function ...- R.Cluckers - Workshop 2 - CEB T1 2018
Raf Cluckers (CNRS – Université de Lille & KU Leuven) / 08.03.2018 Uniform p-adic wave front sets and zero loci of functions of C exp-class. I will recall some concrete parts of the course on motivic integration given at the IHP by Halupczok, and use it to define distributions of Cexp cl
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An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale (Paper Explained)
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From playlist Papers Explained
Hans Feichtinger: Fourier Analysis via the Banach Gelfand Triple
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From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
How to apply Fourier transforms to solve differential equations
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From playlist Partial differential equations