Lecture

Image Processing II: Wavelet Transform and Multiresolution Analysis

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In course

MICRO-512: Image processing II

Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in Jupyter Notebooks; application to real-world examples in industrial vision and biomedical imaging. ...

Description

This lecture covers the motivation behind multiresolution analysis in image processing, including adapting resolution strategies and the compact representation provided by wavelet transforms. It explains the Haar transform, multiscale signal representation, and the connection between wavelets and image pyramids. The lecture also delves into the two-scale relation sequence, wavelet decomposition, and the error pyramid. Additionally, it discusses the Haar expansion, filterbank formulation, and the connection between wavelets and image processing, as well as the basis functions and expansion coefficients in 2D. The presentation concludes with a summary of continuous/discrete image representation, interpolation, B-splines, multiscale image representation, and the concise representation of residues using wavelets.

Instructors (4)

Michaël Unser

Expertise Image Processing, Medical Imaging, Biological Imaging, Wavelets, Splines, Multiresolution Michael Unser is Full Professor with EPFL’s school of Engineering and the academic director of EPFL's Center for Imaging, Lausanne, Switzerland. His primary areas of investigation are signal processing, biomedical imaging and applied functional analysis. He is internationally recognized for his research contributions to sampling theory, wavelets, the use of splines for image processing, stochastic processes,

Daniel Sage

Expertise PublicationsPublications Coordinates Biomedical Imaging GroupAddressEPFL STI-IMT-LIB, BM 4.135, Station 17, CH-1015 Lausanne, SwitzerlandTel.+41 21 693 51 89Fax.+41 21 693 68 10Emaildaniel.sage@epfl.chWeb pagehttp://bigwww.epfl.ch/sage/OfficeBM 4.135 Teaching & PhD Past EPFL PhD Students as codirector Daniel Andreas Schmitter, Anaïs Badoual, Laurène Donati Courses Bioimage informatics BIO-410 The course provides a comprehensive overview of methods, algorithms, and computer tools us

Dimitri Nestor Alice Van De Ville

Expertise Brain and spinal cord imaging, signal and image processing, network science, dynamic systems Teaching & PhD PhD Students Alexandre Cionca, Chun Hei Michael Chan, Flavia Petruso, Bassam El Rawas, Nikolaos Molochidis, Christophe Pascal Jérémie Chênes, Fabrice Jean Guibert, Alessio Siviglia, Ekansh Sareen, Beatrice Lugli Past EPFL PhD Students Djano Kandaswamy, Fikret Isik Karahanoglu, Nora Leonardi, Zafer Dogan, Jeffrey Adam Kasten, Naghmeh Ghazaleh, Rotem Roza Kopel, Daniela Maria Zöller, Thomas

Michael Stefan Daniel Liebling

Teaching & PhD Past EPFL PhD Students as codirector Christian Jaques, Olivia Mariani, Adrian Shajkofci, François Thierry M Marelli Courses Fundamentals of machine learning EE-311 This course provides a general overview of machine learning, covering the main algorithms, theoretical formalisms, and experimental protocols.

Official source

Ontological neighbourhood

Information engineering

Signal processing: Time–frequency analysis

Related concepts (26)

Morlet wavelet

In mathematics, the Morlet wavelet (or Gabor wavelet) is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision. Wavelet#History In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution.

Continuous wavelet transform

In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. The continuous wavelet transform of a function at a scale (a>0) and translational value is expressed by the following integral where is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate.

Gabor wavelet

Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized.

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The Great Pyramid of Giza is the largest Egyptian pyramid and served as the tomb of pharaoh Khufu, who ruled during the Fourth Dynasty of the Old Kingdom. Built in the early 26th century BC, over a period of about 27 years, the pyramid is the oldest of the Seven Wonders of the Ancient World, and the only wonder that has remained largely intact. It is the most famous monument of the Giza pyramid complex, which is part of the UNESCO World Heritage Site "Memphis and its Necropolis".

Continuous wavelet

In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency. Most of the continuous wavelets are used for both wavelet decomposition and composition transforms. That is they are the continuous counterpart of orthogonal wavelets. The following continuous wavelets have been invented for various applications: Poisson wavelet Morlet wavelet Modified Morlet wavelet Mexican ha

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