La quête de représentations optimales pour l'analyse d'images ou la vision par ordinateur est confrontée à la variété de contenu des données bidimensionnelles (images, maillages, graphes). De nombreux travaux se sont attelés aux tâches de séparation de zones régulières, de contours, de textures géométriques, de bruits, à la recherche d'un compromis entre complexité et efficacité de représentation. La combinaison d'aspects multi-échelles à la sélectivité fréquentielle et directionnelle a donné une série de méthodes efficaces, permettant de mieux prendre en compte l'orientation locale des éléments d'intérêt de ces images.
Leur fréquente redondance leur permet d'atteindre des représentations plus parcimonieuses et parfois quasi-invariantes pour certaines transformations usuelles (translation, rotation). Ces méthodes sont la motivation d'un panorama thématique, entre espace-échelle, pyramides, ondelettes en arbre dual, ondelettes directionnelles, curvelets, contourlets et autres shearlets.
A novel approach to Image Fusion using combination of Wavelet Transform and C...IJSRD
Panchromatic furthermore multi-spectral image fusion outstands common methods of high-resolution color image amalgamation. In digital image reconstruction, image fusion is standout pre-processing step that aims increasing hotspot image quality to extricate all suitable information from source images ruining inconsistencies or artifacts. Around the different strategies available for image fusion, Wavelet and Curvelet based algorithms are mostly preferred. Wavelet transform is useful for point singularities while Curvelet transform, as the name describes, is more useful for the analysis of images having curved shape edges. This paper reveals a study of development in the field of image fusion.
Multi modal face recognition using block based curvelet featuresijcga
In this paper, we present multimodal 2D +3D face recognition method using block based curvelet features. The 3D surface of face (Depth Map) is computed from the stereo face images using stereo vision technique. The statistical measures such as mean, standard deviation, variance and entropy are extracted from each block of curvelet subband for both depth and intensity images independently.In order to compute the decision score, the KNN classifier is employed independently for both intensity and depth map. Further, computed decision scoresof intensity and depth map are combined at decision level to improve the face recognition rate. The combination of intensity and depth map is verified experimentally using benchmark face database. The experimental results show that the proposed multimodal method is better than individual modality.
Wavelet Multi-resolution Analysis of High Frequency FX RatesaiQUANT
This document summarizes a research paper on using wavelet multi-resolution analysis to analyze high frequency foreign exchange (FX) rate data. It describes decomposing time series data into trends, seasonal patterns, cycles and irregular components. It then discusses using discrete wavelet transforms to analyze financial time series, extracting features at different time scales. Finally, it presents an algorithm for summarizing and predicting FX rates based on extracted trends and cycles, and evaluates the approach on intraday exchange rate data.
This document proposes a grid-based feature extraction method for offline signature verification. It begins with an introduction and discusses existing techniques and their limitations. It then presents the proposed work, which involves signature acquisition, preprocessing, feature extraction by segmenting the signature image into a grid, and verification. The algorithms, mathematical model, advantages and applications are described. The document concludes that the proposed method requires only low-cost hardware and has a low error rate for signature verification.
This document summarizes a seminar presentation on an image denoising method based on the curvelet transform. The presentation covered:
1) How image noise occurs and traditional denoising methods like linear filters and edge-preserving smoothing.
2) The curvelet transform process including sub-band decomposition, smooth partitioning, renormalization, and ridgelet analysis.
3) An image denoising algorithm that applies wavelet and curvelet transforms, then combines results using quad tree decomposition.
This document discusses image denoising techniques. It begins by defining image denoising as removing unwanted noise from an image to restore the original signal. It then discusses several types of noise like additive Gaussian noise, impulse noise, uniform noise, and periodic noise. For denoising, it covers spatial domain techniques like linear filters (mean, weighted mean), non-linear filters (median filter), and frequency domain techniques that apply a low-pass filter to the Fourier transform of the noisy image. The document provides examples of denoising noisy images using mean and median filters to remove different types of noise.
This document describes a project report submitted by three students for their Bachelor of Engineering degree. The project involves developing a system for classifying brain images using machine learning techniques. It discusses challenges in detecting brain tumors and the need for automated classification methods. It also provides an overview of techniques for image segmentation, clustering, and feature extraction that will be used in the project.
This document presents a proposed methodology for offline signature recognition using global and grid features extracted from signature images. The methodology involves preprocessing signatures, extracting global and grid features using discrete wavelet transforms, training a backpropagation neural network on the features, and classifying signatures based on the trained network. Experimental results show classification accuracy rates ranging from 89-93% for signatures from 10 to 50 individuals. Future work could involve exploring different signature features to potentially improve recognition performance.
A novel approach to Image Fusion using combination of Wavelet Transform and C...IJSRD
Panchromatic furthermore multi-spectral image fusion outstands common methods of high-resolution color image amalgamation. In digital image reconstruction, image fusion is standout pre-processing step that aims increasing hotspot image quality to extricate all suitable information from source images ruining inconsistencies or artifacts. Around the different strategies available for image fusion, Wavelet and Curvelet based algorithms are mostly preferred. Wavelet transform is useful for point singularities while Curvelet transform, as the name describes, is more useful for the analysis of images having curved shape edges. This paper reveals a study of development in the field of image fusion.
Multi modal face recognition using block based curvelet featuresijcga
In this paper, we present multimodal 2D +3D face recognition method using block based curvelet features. The 3D surface of face (Depth Map) is computed from the stereo face images using stereo vision technique. The statistical measures such as mean, standard deviation, variance and entropy are extracted from each block of curvelet subband for both depth and intensity images independently.In order to compute the decision score, the KNN classifier is employed independently for both intensity and depth map. Further, computed decision scoresof intensity and depth map are combined at decision level to improve the face recognition rate. The combination of intensity and depth map is verified experimentally using benchmark face database. The experimental results show that the proposed multimodal method is better than individual modality.
Wavelet Multi-resolution Analysis of High Frequency FX RatesaiQUANT
This document summarizes a research paper on using wavelet multi-resolution analysis to analyze high frequency foreign exchange (FX) rate data. It describes decomposing time series data into trends, seasonal patterns, cycles and irregular components. It then discusses using discrete wavelet transforms to analyze financial time series, extracting features at different time scales. Finally, it presents an algorithm for summarizing and predicting FX rates based on extracted trends and cycles, and evaluates the approach on intraday exchange rate data.
This document proposes a grid-based feature extraction method for offline signature verification. It begins with an introduction and discusses existing techniques and their limitations. It then presents the proposed work, which involves signature acquisition, preprocessing, feature extraction by segmenting the signature image into a grid, and verification. The algorithms, mathematical model, advantages and applications are described. The document concludes that the proposed method requires only low-cost hardware and has a low error rate for signature verification.
This document summarizes a seminar presentation on an image denoising method based on the curvelet transform. The presentation covered:
1) How image noise occurs and traditional denoising methods like linear filters and edge-preserving smoothing.
2) The curvelet transform process including sub-band decomposition, smooth partitioning, renormalization, and ridgelet analysis.
3) An image denoising algorithm that applies wavelet and curvelet transforms, then combines results using quad tree decomposition.
This document discusses image denoising techniques. It begins by defining image denoising as removing unwanted noise from an image to restore the original signal. It then discusses several types of noise like additive Gaussian noise, impulse noise, uniform noise, and periodic noise. For denoising, it covers spatial domain techniques like linear filters (mean, weighted mean), non-linear filters (median filter), and frequency domain techniques that apply a low-pass filter to the Fourier transform of the noisy image. The document provides examples of denoising noisy images using mean and median filters to remove different types of noise.
This document describes a project report submitted by three students for their Bachelor of Engineering degree. The project involves developing a system for classifying brain images using machine learning techniques. It discusses challenges in detecting brain tumors and the need for automated classification methods. It also provides an overview of techniques for image segmentation, clustering, and feature extraction that will be used in the project.
This document presents a proposed methodology for offline signature recognition using global and grid features extracted from signature images. The methodology involves preprocessing signatures, extracting global and grid features using discrete wavelet transforms, training a backpropagation neural network on the features, and classifying signatures based on the trained network. Experimental results show classification accuracy rates ranging from 89-93% for signatures from 10 to 50 individuals. Future work could involve exploring different signature features to potentially improve recognition performance.
Duval l 20130523_lect_nyu-poly_tech-newyork_tutorial-2d-waveletsLaurent Duval
The document provides an overview of developments in multiscale geometric representations over the past 15 years. It begins with motivations from applications in geophysics where directional representations are needed to separate different types of waves. Early approaches included isotropic wavelets but these were not optimal for capturing contours and textures. Later approaches incorporated directionality, geometry, and adaptivity to provide sparser representations, including ridgelets, curvelets, contourlets, and adaptive lifting schemes on meshes and manifolds. The goal was representations that are fast, compact, practical and more informative for image analysis tasks.
Transformations en ondelettes 2D directionnelles - Un panoramaLaurent Duval
La quête de représentations optimales en traitement d'images et vision par ordinateur se heurte à la variété de contenu des données bidimensionnelles. De nombreux travaux se sont cependant attelés aux tâches de séparation de zones régulières, de contours, de textures, à la recherche d'un compromis entre complexité et efficacité de représentation. La prise en compte des aspects multi-échelles, dans le siècle de l'invention des ondelettes, a joué un rôle important en l'analyse d'images. La dernière décennie a ainsi vu apparaître une série de méthodes efficaces, combinant des aspects multi-échelle à des aspects directionnels et fréquentiels, permettant de mieux prendre en compte l'orientation des éléments d'intérêt des images (curvelets, shearlets, contourlets, ridgelets). Leur fréquente redondance leur permet d'obtenir des représentations plus parcimonieuses et parfois quasi-invariantes pour certaines transformations usuelles (translation, rotation). Ces méthodes sont la motivation d'un panorama thématique. Quelques liens avec des outils plus proches de la morphologie mathématique seront evoqués.
Ondelettes, représentations bidimensionnelles, multi-échelles et géométriques...Laurent Duval
La quête de représentations optimales en traitement d'images et vision par ordinateur se heurte à la variété de contenu des données bidimensionnelles. De nombreux travaux se sont cependant attelés aux tâches de séparation de zones régulières, de contours, de textures, à la recherche d'un compromis entre complexité et efficacité de représentation. La prise en compte des aspects multi-échelles, dans le siècle de l'invention des ondelettes, a joué un rôle important en l'analyse d'images. La dernière décennie a ainsi vu apparaître une série de méthodes efficaces, combinant des aspects multi-échelle à des aspects directionnels et fréquentiels, permettant de mieux prendre en compte l'orientation des éléments d'intérêt des images : Activelet, AMlet, Armlet, Bandlet, Barlet, Bathlet, Beamlet, Binlet, Bumplet, Brushlet, Camplet, Caplet, Chirplet, Chordlet, Circlet, Coiflet, Contourlet, Cooklet, Coslet, Craplet, Cubelet, CURElet, Curvelet, Daublet, Directionlet, Dreamlet, Edgelet, ERBlet, FAMlet, FLaglet, Flatlet, Fourierlet, Framelet, Fresnelet, Gaborlet, GAMlet, Gausslet, Graphlet, Grouplet, Haarlet, Haardlet, Heatlet, Hutlet, Hyperbolet, Icalet (Icalette), Interpolet, Lesslet (cf. Morelet), Loglet, Marrlet, MIMOlet, Monowavelet, Morelet, Morphlet, Multiselectivelet, Multiwavelet, Needlet, Noiselet, Ondelette/wavelet, Ondulette, Prewavelet, Phaselet, Planelet, Platelet, Purelet, Quadlet/q-Quadlet, QVlet, Radonlet, RAMlet, Randlet, Ranklet, Ridgelet, Riezlet, Ripplet (original, type-I and II), Scalet, S2let, Seamlet, Seislet, Shadelet, Shapelet, Shearlet, Sinclet, Singlet, Sinlet, Slantlet, Smoothlet, Snakelet, SOHOlet, Sparselet, Spikelet, Splinelet, Starlet, Steerlet, Stokeslet, SURE-let (SURElet), Surfacelet, Surflet, Symlet/Symmlet, S2let, Tetrolet, Treelet, Vaguelette, Wavelet-Vaguelette, Wavelet, Warblet, Warplet, Wedgelet, Xlet/X-let
Galaxy Forum SEA Indonesia 2017 -- Pam Tuan-Anh VNSC/VASTILOAHawaii
Galaxy Forum Southeast Asia 2017 — Jakarta
Saturday 18 February (08:30 – 13:30) @ Skyworld TMII, Jakarta, Indonesia
ILOA is very pleased to have cooperation and participation in organizing this Galaxy Forum Southeast Asia of Among Putro SKYWORLD Indonesia, which is a private space/aerospace, astronomy and related science/technology educational and recreational institution located on a national semi-governmental cultural conservation, education and recreational park called “Taman Mini Indonesia Indah” (Wonderful Indonesian Miniature Park) in the city of Jakarta.
Background:
Galaxy Forum is the primary education and outreach initiative of ILOA, it is an architecture designed to advance 21st Century science, education, enterprise and development around the world.
Galaxy Forums are public events specifically geared towards high school teachers, educators, astronomers of all kinds, students and the general public. Presentations are provided by experts in the fields of astrophysics / galaxy research, space exploration and STEM education, as well as related aspects of culture and traditional knowledge. Interactive panel discussions allow for community participation and integration of local perspectives.
Stats:
More than 70 Galaxy Forums, with over 300 presentations to date.
Held in 26 locations worldwide including Hawaii, Silicon Valley, Canada, China, India, Southeast Asia, Japan, Europe, Africa, Chile, Brazil, Kansas and New York.
Started with Galaxy Forum USA, July 4, 2008 in Silicon Valley, California.
International Lunar Observatory Association (ILOA) is an interglobal enterprise incorporated in Hawaii as a 501(c)(3) non-profit to expand human knowledge of the Cosmos through observation from our Moon and to participate in internationally cooperative lunar base build-out, with Aloha – the spirit of Hawaii.
Discovery of a_probable_4_5_jupiter_mass_exoplanet_to_hd95086_by_direct_imagingSérgio Sacani
The document reports the discovery of a probable 4-5 Jupiter-mass exoplanet orbiting the young star HD 95086. Deep imaging observations using VLT/NaCo detected a faint source at a separation of 56 AU from the star. Follow-up observations over more than a year found the source to be co-moving with the star, suggesting it is bound. Its luminosity corresponds to a predicted mass of 4-5 Jupiter masses, making it the lowest mass exoplanet directly imaged around a star. If confirmed, this discovery could provide insights into giant planet formation and evolution.
The document describes the SuperWASP project, which uses two robotic telescopes (SuperWASP-N and SuperWASP-S) to search for transiting exoplanets. As of February 2015, SuperWASP had discovered 98 exoplanets, making it the most successful ground-based exoplanet survey. It discusses the history of exoplanet detection, details the SuperWASP instrumentation and detection method, and reviews its discoveries and role alongside space-based missions like Kepler.
ILOA Galaxy Forum Canada 2014 - Bernard Foing - Moon South Pole ExplorationILOAHawaii
Galaxy Forum Canada 2014, with the theme “Moon South Pole and Human Missions: Giant Steps into the Galaxy” was held in conjunction with the 65th International Astronautical Congress at the Metro Toronto Convention Centre in Ontario, Canada. Thousands of scientists, engineers and experts from around the world gather to explore the latest achievements, innovations and ambitions of worldwide space agencies, industries and enterprises.
The Moon’s South Pole is as exciting and enriching a new frontier as humans on Mars or trillion dollar asteroids, and much closer in time and space.
The distinguished international, national and independent experts assembled for the event consider how robotic missions 2016-2018 can function as precursors to Human Moon missions in the 2020s. A fusion of astrophysics and astronautics, the ILOA Galaxy Forum will preview upcoming Luna missions and priorities of major spacefaring powers China, India, Russia, USA, Europe and Japan, as well as Canada, Korea and others; and of the remarkable enterprises at the forefront of the commercial Lunar Renaissance.
The International Astronautical Federation (IAF) is an official co-sponsor for Galaxy Forum Canada 2014 and is providing a plenary hall at the IAC venue for the event.
Galaxy Forums are free and open to the general public. More information about the program will be available soon. If you have any questions, please contact info@iloa.org.
The document describes fundamentals of antennas and wave propagation. It provides an introduction to different types of antennas including wire antennas, aperture antennas, reflector antennas, lens antennas, microstrip antennas and array antennas. It discusses the basic radiation mechanism of antennas which involves time-varying currents and charges that produce electromagnetic waves. Key points covered include antenna basics, radiation from oscillating dipoles, and the current and voltage distributions on dipole antennas.
1. The document describes a lecture on antennas and wave propagation. It introduces different types of antennas like wire antennas, aperture antennas, reflector antennas, lens antennas, microstrip antennas, and array antennas.
2. It explains the basic radiation mechanism of antennas which involves time-varying currents and accelerated charges producing electromagnetic waves. A current only radiates if the wire is bent, curved, or the charge is oscillating.
3. Key antenna parameters like radiation resistance, directivity, gain, polarization and reciprocity are also covered briefly. Current and voltage distribution on a half-wave dipole antenna is shown.
This document summarizes research analyzing data from the Voyager spacecraft regarding diffuse Lyman-alpha emission from the Milky Way galaxy. The analysis of Voyager Ultraviolet Spectrograph data from 1993 to 2011 revealed:
1) An excess of Lyman-alpha emission above what could be explained by local sources like the heliospheric glow, coinciding with regions of star formation along the Galactic Plane.
2) The background spectra of the UVS detectors increased constantly over time rather than decreasing as expected, likely due to energetic particles.
3) Comparison of the Lyman-alpha brightness data to a radiative transfer model showed an excess of a few Rayleigh coinciding with H-alpha bright star
An interactive teaching and learning on earth and space educationMank Zein
This document discusses hands-on exercises for teaching astronomy concepts. It introduces CLEA (Contemporary Laboratory Experiences in Astronomy), a project that provides modular laboratory exercises using simulations and real data. The exercises are designed for non-science majors and illustrate modern astronomical techniques and data analysis. Some example CLEA modules measure star properties, study star clusters using the H-R diagram, and determine the speed of light using observations of Jupiter's moons. The goal is to provide an interactive, hands-on experience of what astronomers do through measurement simulations and analysis of real astronomical data sets.
This document presents UBVI and Hα photometry of 17319 stars in the vicinity of the young double cluster h & χ Persei. The key findings are:
1) The two clusters share a common distance modulus of 11.75±0.05 and ages of log age(yr) = 7.1±0.1 based on the photometry.
2) 33 Be stars are detected in the region, with 8 being new detections, using the V −Hα color as a measure of Hα emission strength.
3) A peak in the Be star fraction is found towards the end of the main sequence, which is discussed in the context of evolutionary enhancement of the Be phenomenon
Imaging the dust_sublimation_front_of_a_circumbinary_diskSérgio Sacani
Aims. We present the first near-IR milli-arcsecond-scale image of a post-AGB binary that is surrounded by hot circumbinary dust.
Methods. A very rich interferometric data set in six spectral channels was acquired of IRAS 08544-4431 with the new RAPID camera
on the PIONIER beam combiner at the Very Large Telescope Interferometer (VLTI). A broadband image in the H-band was reconstructed
by combining the data of all spectral channels using the SPARCO method.
Results. We spatially separate all the building blocks of the IRAS 08544-4431 system in our milliarcsecond-resolution image. Our
dissection reveals a dust sublimation front that is strikingly similar to that expected in early-stage protoplanetary disks, as well as an
unexpected flux signal of 4% from the secondary star. The energy output from this companion indicates the presence of a compact
circum-companion accretion disk, which is likely the origin of the fast outflow detected in H.
Conclusions. Our image provides the most detailed view into the heart of a dusty circumstellar disk to date. Our results demonstrate
that binary evolution processes and circumstellar disk evolution can be studied in detail in space and over time.
The document provides information about various telescopes and their capabilities. It begins with an image and description of the Eagle Nebula taken by the Hubble Space Telescope. It then provides details about the Hubble, including its launch date, dimensions, mirror size, weight, orbital parameters, and main scientific instruments. The next generation of large ground and space-based telescopes are mentioned, including the European Extremely Large Telescope with a 37m mirror, the Atacama Large Millimeter Array, the James Webb Space Telescope, and the Large Synoptic Survey Telescope.
Beyond the disk: EUV coronagraphic observations of the Extreme Ultraviolet Im...Sérgio Sacani
Most observations of the solar corona beyond 2 R consist of broadband visible light imagery carried out with coronagraphs.
The associated diagnostics mainly consist of kinematics and derivations of the electron number density. While the measurement of the
properties of emission lines can provide crucial additional diagnostics of the coronal plasma (temperatures, velocities, abundances,
etc.), these types of observations are comparatively rare. In visible wavelengths, observations at these heights are limited to total
eclipses. In the ultraviolet (UV) to extreme UV (EUV) range, very few additional observations have been achieved since the pioneering
results of the Ultraviolet Coronagraph Spectrometer (UVCS).
Aims. One of the objectives of the Full Sun Imager (FSI) channel of the Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter
mission has been to provide very wide field-of-view EUV diagnostics of the morphology and dynamics of the solar atmosphere in
temperature regimes that are typical of the lower transition region and of the corona.
Methods. FSI carries out observations in two narrowbands of the EUV spectrum centered on 17.4 nm and 30.4 nm that are dominated,
respectively, by lines of Fe ix/x (formed in the corona around 1 MK) and by the resonance line of He ii (formed around 80 kK in the
lower transition region). Unlike previous EUV imagers, FSI includes a moveable occulting disk that can be inserted in the optical path
to reduce the amount of instrumental stray light to a minimum.
Results. FSI detects signals at 17.4 nm up to the edge of its field of view (7 R), which is about twice further than was previously
possible. Operation at 30.4 nm are for the moment compromised by an as-yet unidentified source of stray light. Comparisons with
observations by the LASCO and Metis coronagraphs confirm the presence of morphological similarities and differences between the
broadband visible light and EUV emissions, as documented on the basis of prior eclipse and space-based observations.
Conclusions. The very-wide-field observations of FSI out to about 3 and 7 R, without and with the occulting disk, respectively, are
paving the way for future dedicated instruments.
This document provides an overview and context for a study of the symbiotic star SS Leporis using interferometric imaging with the PIONIER instrument on the VLTI. Key points:
- SS Leporis is a long-period interacting binary system composed of an A star accreting material from an evolved M giant companion, presenting an "Algol paradox" where the more evolved star is less massive.
- Previous studies have not fully constrained the system morphology and characteristics. New interferometric observations with PIONIER were obtained to directly probe the inner parts of the system.
- The observations were used to perform aperture synthesis imaging and model the system as a binary surrounded by a circumbinary disc. This provides the
Matthew Penn has over 25 years of experience in solar astronomy. He is currently an Associate Astronomer with tenure at the National Solar Observatory in Tucson, Arizona. Previously, he held positions at the National Solar Observatory, California State University Northridge, and the University of Hawaii. He has received several awards for his research and teaching.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdfMalak Abu Hammad
Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
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Similar to Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
Duval l 20130523_lect_nyu-poly_tech-newyork_tutorial-2d-waveletsLaurent Duval
The document provides an overview of developments in multiscale geometric representations over the past 15 years. It begins with motivations from applications in geophysics where directional representations are needed to separate different types of waves. Early approaches included isotropic wavelets but these were not optimal for capturing contours and textures. Later approaches incorporated directionality, geometry, and adaptivity to provide sparser representations, including ridgelets, curvelets, contourlets, and adaptive lifting schemes on meshes and manifolds. The goal was representations that are fast, compact, practical and more informative for image analysis tasks.
Transformations en ondelettes 2D directionnelles - Un panoramaLaurent Duval
La quête de représentations optimales en traitement d'images et vision par ordinateur se heurte à la variété de contenu des données bidimensionnelles. De nombreux travaux se sont cependant attelés aux tâches de séparation de zones régulières, de contours, de textures, à la recherche d'un compromis entre complexité et efficacité de représentation. La prise en compte des aspects multi-échelles, dans le siècle de l'invention des ondelettes, a joué un rôle important en l'analyse d'images. La dernière décennie a ainsi vu apparaître une série de méthodes efficaces, combinant des aspects multi-échelle à des aspects directionnels et fréquentiels, permettant de mieux prendre en compte l'orientation des éléments d'intérêt des images (curvelets, shearlets, contourlets, ridgelets). Leur fréquente redondance leur permet d'obtenir des représentations plus parcimonieuses et parfois quasi-invariantes pour certaines transformations usuelles (translation, rotation). Ces méthodes sont la motivation d'un panorama thématique. Quelques liens avec des outils plus proches de la morphologie mathématique seront evoqués.
Ondelettes, représentations bidimensionnelles, multi-échelles et géométriques...Laurent Duval
La quête de représentations optimales en traitement d'images et vision par ordinateur se heurte à la variété de contenu des données bidimensionnelles. De nombreux travaux se sont cependant attelés aux tâches de séparation de zones régulières, de contours, de textures, à la recherche d'un compromis entre complexité et efficacité de représentation. La prise en compte des aspects multi-échelles, dans le siècle de l'invention des ondelettes, a joué un rôle important en l'analyse d'images. La dernière décennie a ainsi vu apparaître une série de méthodes efficaces, combinant des aspects multi-échelle à des aspects directionnels et fréquentiels, permettant de mieux prendre en compte l'orientation des éléments d'intérêt des images : Activelet, AMlet, Armlet, Bandlet, Barlet, Bathlet, Beamlet, Binlet, Bumplet, Brushlet, Camplet, Caplet, Chirplet, Chordlet, Circlet, Coiflet, Contourlet, Cooklet, Coslet, Craplet, Cubelet, CURElet, Curvelet, Daublet, Directionlet, Dreamlet, Edgelet, ERBlet, FAMlet, FLaglet, Flatlet, Fourierlet, Framelet, Fresnelet, Gaborlet, GAMlet, Gausslet, Graphlet, Grouplet, Haarlet, Haardlet, Heatlet, Hutlet, Hyperbolet, Icalet (Icalette), Interpolet, Lesslet (cf. Morelet), Loglet, Marrlet, MIMOlet, Monowavelet, Morelet, Morphlet, Multiselectivelet, Multiwavelet, Needlet, Noiselet, Ondelette/wavelet, Ondulette, Prewavelet, Phaselet, Planelet, Platelet, Purelet, Quadlet/q-Quadlet, QVlet, Radonlet, RAMlet, Randlet, Ranklet, Ridgelet, Riezlet, Ripplet (original, type-I and II), Scalet, S2let, Seamlet, Seislet, Shadelet, Shapelet, Shearlet, Sinclet, Singlet, Sinlet, Slantlet, Smoothlet, Snakelet, SOHOlet, Sparselet, Spikelet, Splinelet, Starlet, Steerlet, Stokeslet, SURE-let (SURElet), Surfacelet, Surflet, Symlet/Symmlet, S2let, Tetrolet, Treelet, Vaguelette, Wavelet-Vaguelette, Wavelet, Warblet, Warplet, Wedgelet, Xlet/X-let
Galaxy Forum SEA Indonesia 2017 -- Pam Tuan-Anh VNSC/VASTILOAHawaii
Galaxy Forum Southeast Asia 2017 — Jakarta
Saturday 18 February (08:30 – 13:30) @ Skyworld TMII, Jakarta, Indonesia
ILOA is very pleased to have cooperation and participation in organizing this Galaxy Forum Southeast Asia of Among Putro SKYWORLD Indonesia, which is a private space/aerospace, astronomy and related science/technology educational and recreational institution located on a national semi-governmental cultural conservation, education and recreational park called “Taman Mini Indonesia Indah” (Wonderful Indonesian Miniature Park) in the city of Jakarta.
Background:
Galaxy Forum is the primary education and outreach initiative of ILOA, it is an architecture designed to advance 21st Century science, education, enterprise and development around the world.
Galaxy Forums are public events specifically geared towards high school teachers, educators, astronomers of all kinds, students and the general public. Presentations are provided by experts in the fields of astrophysics / galaxy research, space exploration and STEM education, as well as related aspects of culture and traditional knowledge. Interactive panel discussions allow for community participation and integration of local perspectives.
Stats:
More than 70 Galaxy Forums, with over 300 presentations to date.
Held in 26 locations worldwide including Hawaii, Silicon Valley, Canada, China, India, Southeast Asia, Japan, Europe, Africa, Chile, Brazil, Kansas and New York.
Started with Galaxy Forum USA, July 4, 2008 in Silicon Valley, California.
International Lunar Observatory Association (ILOA) is an interglobal enterprise incorporated in Hawaii as a 501(c)(3) non-profit to expand human knowledge of the Cosmos through observation from our Moon and to participate in internationally cooperative lunar base build-out, with Aloha – the spirit of Hawaii.
Discovery of a_probable_4_5_jupiter_mass_exoplanet_to_hd95086_by_direct_imagingSérgio Sacani
The document reports the discovery of a probable 4-5 Jupiter-mass exoplanet orbiting the young star HD 95086. Deep imaging observations using VLT/NaCo detected a faint source at a separation of 56 AU from the star. Follow-up observations over more than a year found the source to be co-moving with the star, suggesting it is bound. Its luminosity corresponds to a predicted mass of 4-5 Jupiter masses, making it the lowest mass exoplanet directly imaged around a star. If confirmed, this discovery could provide insights into giant planet formation and evolution.
The document describes the SuperWASP project, which uses two robotic telescopes (SuperWASP-N and SuperWASP-S) to search for transiting exoplanets. As of February 2015, SuperWASP had discovered 98 exoplanets, making it the most successful ground-based exoplanet survey. It discusses the history of exoplanet detection, details the SuperWASP instrumentation and detection method, and reviews its discoveries and role alongside space-based missions like Kepler.
ILOA Galaxy Forum Canada 2014 - Bernard Foing - Moon South Pole ExplorationILOAHawaii
Galaxy Forum Canada 2014, with the theme “Moon South Pole and Human Missions: Giant Steps into the Galaxy” was held in conjunction with the 65th International Astronautical Congress at the Metro Toronto Convention Centre in Ontario, Canada. Thousands of scientists, engineers and experts from around the world gather to explore the latest achievements, innovations and ambitions of worldwide space agencies, industries and enterprises.
The Moon’s South Pole is as exciting and enriching a new frontier as humans on Mars or trillion dollar asteroids, and much closer in time and space.
The distinguished international, national and independent experts assembled for the event consider how robotic missions 2016-2018 can function as precursors to Human Moon missions in the 2020s. A fusion of astrophysics and astronautics, the ILOA Galaxy Forum will preview upcoming Luna missions and priorities of major spacefaring powers China, India, Russia, USA, Europe and Japan, as well as Canada, Korea and others; and of the remarkable enterprises at the forefront of the commercial Lunar Renaissance.
The International Astronautical Federation (IAF) is an official co-sponsor for Galaxy Forum Canada 2014 and is providing a plenary hall at the IAC venue for the event.
Galaxy Forums are free and open to the general public. More information about the program will be available soon. If you have any questions, please contact info@iloa.org.
The document describes fundamentals of antennas and wave propagation. It provides an introduction to different types of antennas including wire antennas, aperture antennas, reflector antennas, lens antennas, microstrip antennas and array antennas. It discusses the basic radiation mechanism of antennas which involves time-varying currents and charges that produce electromagnetic waves. Key points covered include antenna basics, radiation from oscillating dipoles, and the current and voltage distributions on dipole antennas.
1. The document describes a lecture on antennas and wave propagation. It introduces different types of antennas like wire antennas, aperture antennas, reflector antennas, lens antennas, microstrip antennas, and array antennas.
2. It explains the basic radiation mechanism of antennas which involves time-varying currents and accelerated charges producing electromagnetic waves. A current only radiates if the wire is bent, curved, or the charge is oscillating.
3. Key antenna parameters like radiation resistance, directivity, gain, polarization and reciprocity are also covered briefly. Current and voltage distribution on a half-wave dipole antenna is shown.
This document summarizes research analyzing data from the Voyager spacecraft regarding diffuse Lyman-alpha emission from the Milky Way galaxy. The analysis of Voyager Ultraviolet Spectrograph data from 1993 to 2011 revealed:
1) An excess of Lyman-alpha emission above what could be explained by local sources like the heliospheric glow, coinciding with regions of star formation along the Galactic Plane.
2) The background spectra of the UVS detectors increased constantly over time rather than decreasing as expected, likely due to energetic particles.
3) Comparison of the Lyman-alpha brightness data to a radiative transfer model showed an excess of a few Rayleigh coinciding with H-alpha bright star
An interactive teaching and learning on earth and space educationMank Zein
This document discusses hands-on exercises for teaching astronomy concepts. It introduces CLEA (Contemporary Laboratory Experiences in Astronomy), a project that provides modular laboratory exercises using simulations and real data. The exercises are designed for non-science majors and illustrate modern astronomical techniques and data analysis. Some example CLEA modules measure star properties, study star clusters using the H-R diagram, and determine the speed of light using observations of Jupiter's moons. The goal is to provide an interactive, hands-on experience of what astronomers do through measurement simulations and analysis of real astronomical data sets.
This document presents UBVI and Hα photometry of 17319 stars in the vicinity of the young double cluster h & χ Persei. The key findings are:
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2) 33 Be stars are detected in the region, with 8 being new detections, using the V −Hα color as a measure of Hα emission strength.
3) A peak in the Be star fraction is found towards the end of the main sequence, which is discussed in the context of evolutionary enhancement of the Be phenomenon
Imaging the dust_sublimation_front_of_a_circumbinary_diskSérgio Sacani
Aims. We present the first near-IR milli-arcsecond-scale image of a post-AGB binary that is surrounded by hot circumbinary dust.
Methods. A very rich interferometric data set in six spectral channels was acquired of IRAS 08544-4431 with the new RAPID camera
on the PIONIER beam combiner at the Very Large Telescope Interferometer (VLTI). A broadband image in the H-band was reconstructed
by combining the data of all spectral channels using the SPARCO method.
Results. We spatially separate all the building blocks of the IRAS 08544-4431 system in our milliarcsecond-resolution image. Our
dissection reveals a dust sublimation front that is strikingly similar to that expected in early-stage protoplanetary disks, as well as an
unexpected flux signal of 4% from the secondary star. The energy output from this companion indicates the presence of a compact
circum-companion accretion disk, which is likely the origin of the fast outflow detected in H.
Conclusions. Our image provides the most detailed view into the heart of a dusty circumstellar disk to date. Our results demonstrate
that binary evolution processes and circumstellar disk evolution can be studied in detail in space and over time.
The document provides information about various telescopes and their capabilities. It begins with an image and description of the Eagle Nebula taken by the Hubble Space Telescope. It then provides details about the Hubble, including its launch date, dimensions, mirror size, weight, orbital parameters, and main scientific instruments. The next generation of large ground and space-based telescopes are mentioned, including the European Extremely Large Telescope with a 37m mirror, the Atacama Large Millimeter Array, the James Webb Space Telescope, and the Large Synoptic Survey Telescope.
Beyond the disk: EUV coronagraphic observations of the Extreme Ultraviolet Im...Sérgio Sacani
Most observations of the solar corona beyond 2 R consist of broadband visible light imagery carried out with coronagraphs.
The associated diagnostics mainly consist of kinematics and derivations of the electron number density. While the measurement of the
properties of emission lines can provide crucial additional diagnostics of the coronal plasma (temperatures, velocities, abundances,
etc.), these types of observations are comparatively rare. In visible wavelengths, observations at these heights are limited to total
eclipses. In the ultraviolet (UV) to extreme UV (EUV) range, very few additional observations have been achieved since the pioneering
results of the Ultraviolet Coronagraph Spectrometer (UVCS).
Aims. One of the objectives of the Full Sun Imager (FSI) channel of the Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter
mission has been to provide very wide field-of-view EUV diagnostics of the morphology and dynamics of the solar atmosphere in
temperature regimes that are typical of the lower transition region and of the corona.
Methods. FSI carries out observations in two narrowbands of the EUV spectrum centered on 17.4 nm and 30.4 nm that are dominated,
respectively, by lines of Fe ix/x (formed in the corona around 1 MK) and by the resonance line of He ii (formed around 80 kK in the
lower transition region). Unlike previous EUV imagers, FSI includes a moveable occulting disk that can be inserted in the optical path
to reduce the amount of instrumental stray light to a minimum.
Results. FSI detects signals at 17.4 nm up to the edge of its field of view (7 R), which is about twice further than was previously
possible. Operation at 30.4 nm are for the moment compromised by an as-yet unidentified source of stray light. Comparisons with
observations by the LASCO and Metis coronagraphs confirm the presence of morphological similarities and differences between the
broadband visible light and EUV emissions, as documented on the basis of prior eclipse and space-based observations.
Conclusions. The very-wide-field observations of FSI out to about 3 and 7 R, without and with the occulting disk, respectively, are
paving the way for future dedicated instruments.
This document provides an overview and context for a study of the symbiotic star SS Leporis using interferometric imaging with the PIONIER instrument on the VLTI. Key points:
- SS Leporis is a long-period interacting binary system composed of an A star accreting material from an evolved M giant companion, presenting an "Algol paradox" where the more evolved star is less massive.
- Previous studies have not fully constrained the system morphology and characteristics. New interferometric observations with PIONIER were obtained to directly probe the inner parts of the system.
- The observations were used to perform aperture synthesis imaging and model the system as a binary surrounded by a circumbinary disc. This provides the
Matthew Penn has over 25 years of experience in solar astronomy. He is currently an Associate Astronomer with tenure at the National Solar Observatory in Tucson, Arizona. Previously, he held positions at the National Solar Observatory, California State University Northridge, and the University of Hawaii. He has received several awards for his research and teaching.
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Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
1. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
Curvelets, contourlets, shearlets, *lets, etc.:
multiscale analysis and directional wavelets for
images
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré
UCL, IFPEN, AMU, Dauphine
21/11/2013
Séminaire Cristolien d’Analyse Multifractale
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
2. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
Wavelets for the eye
Artlets: painting wavelets (Hokusai/A. Unser)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
3. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
Wavelets for 1D signals
1D scaling functions and wavelets
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
4. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
Wavelets for 2D images
2D scaling functions and wavelets
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
5. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
1D signals
1D and 2D data appear quite different, even under simple:
◮
time shift
◮
scale change
◮
amplitude drift
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
6. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
1D signals
Figure : 1D and 2D → 1D related signals
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
7. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
2D images
Figure : 1D → 2D and 2D related images
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
8. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
1D signals & 2D images
Only time shift/scale change/amplitude drift between:
◮
John F. Kennedy Moon Speech (Rice Stadium, 12/09/1962)
◮
A Man on the Moon: Buzz Aldrin (Apollo 11, 21/07/196)
Two motivations: JFK + a Rice wavelet toolbox
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
9. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
9/28
1.5D signals: motivations for 2D directional "wavelets"
Figure : Geophysics: seismic data recording (surface and body waves)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
10. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
9/28
1.5D signals: motivations for 2D directional "wavelets"
100
Time (smpl)
200
300
400
500
600
700
0
50
100
150
200
250
300
Offset (traces)
Figure : Geophysics: surface wave removal (before)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
11. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
9/28
1.5D signals: motivations for 2D directional "wavelets"
100
Time (smpl)
200
300
400
500
600
700
0
(b)
50
100
150
200
250
300
Offset (traces)
Figure : Geophysics: surface wave removal (after)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
12. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
9/28
1.5D signals: motivations for 2D directional "wavelets"
Issues in geophysics:
◮ different types of waves on seismic "images"
◮
appear hyperbolic [layers], linear [noise] (and parabolic)
◮
not the standard “mid-amplitude random noise problem”
◮
no contours enclosing textures, more the converse
◮
kind of halfway between signals and images (1.5D)
◮
yet local, directional, frequency-limited, scale-dependent
structures to separate
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
13. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
10/28
Agenda
◮
To survey 15 years of improvements in 2D wavelets
◮
◮
◮
◮
◮
◮
spatial, directional, frequency selectivity increased
sparser representations of contours and textures
from fixed to adaptive, from low to high redundancy
generally fast, practical, compact (or sparse?), informative
1D/2D, discrete/continuous hybridization
Outline
◮
◮
introduction + early days ( 1998)
fixed: oriented & geometrical (selected):
◮
◮
◮
± separable (Hilbert/dual-tree wavelet)
isotropic non-separable (Morlet-Gabor)
anisotropic scaling (ridgelet, curvelet, contourlet, shearlet)
◮
(hidden bonuses):
◮
conclusions
◮
adaptive, lifting, meshes, spheres, manifolds, graphs
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
14. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
11/28
In just one slide
Figure : A standard, “dyadic”, separable wavelet decomposition
Where do we go from here? 15 years, 300+ refs in 30 minutes
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
15. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels (but...):
Figure : Image block as a (canonical) linear combination of pixels
◮
suffices for (simple) data and (basic) manipulation
◮
very limited in higher level understanding tasks
◮
◮
◮
counting, enhancement, filtering
looking for other (meaningful) linear combinations
what about
67 + 93 + 52 + 97, 67 + 93 − 52 − 97
67 − 93 + 52 − 97, 67 − 93 − 52 + 97?
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
16. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels (but...):
A review in an active research field:
◮ (partly) inspired by:
◮
◮
◮
◮
◮
motivated by first successes (JPEG 2000 compression)
aimed either at pragmatic or heuristic purposes:
◮
◮
early vision observations [Marr et al.]
sparse coding: wavelet-like oriented filters and receptive fields
of simple cells (visual cortex) [Olshausen et al.]
a widespread belief in sparsity
known formation model or unknown information content
developed through a legion of *-lets (and relatives)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
17. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, wavelets are legion
Room(let) for improvement:
Activelet, AMlet, Armlet, Bandlet, Barlet, Bathlet, Beamlet, Binlet, Bumplet, Brushlet,
Caplet, Camplet, Chirplet, Chordlet, Circlet, Coiflet, Contourlet, Cooklet, Craplet,
Cubelet, CURElet, Curvelet, Daublet, Directionlet, Dreamlet, Edgelet, FAMlet, FLaglet,
Flatlet, Fourierlet, Framelet, Fresnelet, Gaborlet, GAMlet, Gausslet, Graphlet, Grouplet,
Haarlet, Haardlet, Heatlet, Hutlet, Hyperbolet, Icalet (Icalette), Interpolet, Loglet,
Marrlet, MIMOlet, Monowavelet, Morelet, Morphlet, Multiselectivelet, Multiwavelet,
Needlet, Noiselet, Ondelette, Ondulette, Prewavelet, Phaselet, Planelet, Platelet, Purelet,
QVlet, Radonlet, RAMlet, Randlet, Ranklet, Ridgelet, Riezlet, Ripplet (original, type-I
and II), Scalet, S2let, Seamlet, Seislet, Shadelet, Shapelet, Shearlet, Sinclet, Singlet,
Slantlet, Smoothlet, Snakelet, SOHOlet, Sparselet, Spikelet, Splinelet, Starlet, Steerlet,
Stockeslet, SURE-let (SURElet), Surfacelet, Surflet, Symmlet, S2let, Tetrolet, Treelet,
Vaguelette, Wavelet-Vaguelette, Wavelet, Warblet, Warplet, Wedgelet, Xlet, not
mentioning all those not in -let!
Now, some reasons behind this quantity
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
18. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, but altogether different
Figure : Different kinds of images
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
19. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, but altogether different
Figure : Different kinds of images
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
20. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, but might be described by models
“Template” image decomposition models:
◮
edge cartoon + texture [Meyer-2001]:
inf E (u) =
u
Ω
|∇u| + λ v
∗, f
=u+v
edge cartoon + texture + noise [Aujol-Chambolle-2005]:
w
1
v
f −u −v −w
+B∗
+
inf F (u, v , w ) = J(u) + J ∗
u,v ,w
µ
λ
2α
◮
◮
L2
heuristically: piecewise-smooth + contours + geometrical
textures + noise (or unmodeled)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
21. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, but resolution/scale helps with models
Coarse-to-fine and fine-to-coarse relationships
Figure : Notion of sufficient resolution [Chabat et al., 2004]
◮
◮
discrete 80’s wavelets were “not bad” for: piecewise-smooth
(moments) + contours (gradient-behavior) + geometrical
textures (oscillations) + noise (orthogonality)
yet, not enough with noise, complicated images (poor sparsity
decay)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
22. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
12/28
Images are pixels, but decay with regularity
Compressibility vs regularity: MSE with M-term approximation
◮ 1D
◮
◮
2D
◮
◮
◮
◮
piecewise C α → O(M −2α )
C α → O(M −α ) (standard wavelets)
piecewise C α /C α → O(M −1 ) (standard wavelets)
piecewise C 2 /C 2 → O(M −2 ) (triangulations)
Notes:
◮
◮
◮
very imprecise statements, many deeper results
piecewise C 2 /C 2 → O(M −2 f (M)) w/ directional wavelets?
do much better with other regularities (α = 2, BV)?
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
23. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
13/28
Images are pixels, but sometimes deceiving
Figure : Real world image and illusions
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
24. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
13/28
Images are pixels, but sometimes deceiving
Figure : Real world image and illusions
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
25. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
13/28
Images are pixels, but sometimes deceiving
Figure : Real world image and illusions
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
26. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
14/28
Images are pixels, but resolution/scale helps
To catch important "objects" in their context
◮
◮
use scales, pyramidal or multiresolution schemes,
combine w/ different description/detection/modeling:
◮
smooth curve or polynomial fit, oriented regularized derivatives
(Sobel, structure tensor), discrete (lines) geometry, parametric
curve detectors (e.g. Hough transform), mathematical
morphology, empirical mode decomposition, local frequency
estimators, Hilbert and Riesz (analytic and monogenic),
quaternions, Clifford algebras, optical flow, smoothed random
models, generalized Gaussian mixtures, warping operators, etc.
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
27. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
15/28
Images are pixels, and need efficient descriptions
Depend on application, with sparsity priors:
◮ compression, denoising, enhancement, inpainting, restoration,
contour detection, texture analysis, fusion, super-resolution,
registration, segmentation, reconstruction, source separation,
image decomposition, MDC, learning, etc.
4
10
3
Magnitude
10
2
10
1
10
0
10
−1
10
100
200
300
400
500
Index
600
700
800
900
1000
Figure : Image (contours/textures) and decaying singular values
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
28. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
16/28
Images are pixels: a guiding thread (GT)
Figure : Memorial plaque in honor of A. Haar and F. Riesz: A szegedi
matematikai iskola világhírű megalapítói, courtesy Prof. K. Szatmáry
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
29. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Fourier approach: critical, orthogonal
Figure : GT luminance component amplitude spectrum (log-scale)
Fast, compact, practical but not quite informative (not local)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
30. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Scale-space approach: (highly)-redundant, more local
Figure : GT with Gaussian scale-space decomposition
Gaussian filters and heat diffusion interpretation
Varying persistence of features across scales ⇒ redundancy
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
31. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Pyramid-like approach: (less)-redundant, more local
Figure : GT with Gaussian pyramid decomposition
Varying persistence of features across scales + subsampling
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
32. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Differences in scale-space with subsampling
Figure : GT with Laplacian pyramid decomposition
Laplacian pyramid: complete, reduced redundancy, enhances image
singularities, low-activity regions/small coefficients, algorithmic
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
33. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Isotropic wavelets (more axiomatic)
Consider
Wavelet ψ ∈ L2 (R2 ) such that ψ(x) = ψrad ( x ), with x = (x1 , x2 ),
for some radial function ψrad : R+ → R (with adm. conditions).
Decomposition and reconstruction
For ψ(b,a) (x) =
tion:
1
x−b
a ψ( a ),
f (x) =
if cψ =
(2π)2
2π
cψ
Wf (b, a) = ψ(b,a) , f
+∞
0
R2
2
ˆ
R2 |ψ(k)| / k
2
with reconstruc-
Wf (b, a) ψ(b,a) (x) d2 b
da
a3
d2 k < ∞.
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
34. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Wavelets as multiscale edge detectors: many more potential
wavelet shapes (difference of Gaussians, Cauchy, etc.)
Figure : Example: Marr wavelet as a singularity detector
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
35. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
Definition
The family B is a frame if there exist two constants 0 < µ♭
such that for all f
µ♭ f
2
m
| ψm , f |2
µ♯ f
µ♯ < ∞
2
Possibility of discrete orthogonal bases with O(N) speed. In 2D:
Definition
Separable orthogonal wavelets: dyadic scalings and translations
ψm (x) = 2−j ψ k (2−j x − n) of three tensor-product 2-D wavelets
ψ V (x) = ψ(x1 )ϕ(x2 ), ψ H (x) = ϕ(x1 )ψ(x2 ), ψ D (x) = ψ(x1 )ψ(x2 )
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
36. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
1D scaling functions ψ(x1 ) and wavelets ϕ(x2 )
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
37. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
17/28
Guiding thread (GT): early days
So, back to orthogonality with the discrete wavelet transform: fast,
compact and informative, but... is it sufficient (singularities, noise,
shifts, rotations)?
Figure : Discrete wavelet transform of GT
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
38. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
18/28
Oriented, ± separable
To tackle orthogonal DWT limitations
◮
1D, orthogonality, realness, symmetry, finite support (Haar)
Approaches used for simple designs (& more involved as well)
◮
relaxing properties: IIR, biorthogonal, complex
◮
M-adic MRAs with M integer > 2 or M = p/q
◮
hyperbolic, alternative tilings, less isotropic decompositions
◮
with pyramidal-scheme: steerable Marr-like pyramids
◮
relaxing critical sampling with oversampled filter banks
◮
complexity: (fractional/directional) Hilbert, Riesz, phaselets,
monogenic, hypercomplex, quaternions, Clifford algebras
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
39. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
19/28
Oriented, ± separable
Illustration of a combination of Hilbert pairs and M-band MRA
H{f }(ω) = −ı sign(ω)f (ω)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−4
−3
−2
−1
0
1
2
3
Figure : Hilbert pair 1
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
40. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
19/28
Oriented, ± separable
Illustration of a combination of Hilbert pairs and M-band MRA
H{f }(ω) = −ı sign(ω)f (ω)
1
0.5
0
−0.5
−4
−3
−2
−1
0
1
2
3
Figure : Hilbert pair 2
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
41. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
19/28
Oriented, ± separable
Illustration of a combination of Hilbert pairs and M-band MRA
H{f }(ω) = −ı sign(ω)f (ω)
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−4
−3
−2
−1
0
1
2
3
4
Figure : Hilbert pair 3
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
42. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
19/28
Oriented, ± separable
Illustration of a combination of Hilbert pairs and M-band MRA
H{f }(ω) = −ı sign(ω)f (ω)
3
2
1
0
−1
−2
−4
−3
−2
−1
0
1
2
3
Figure : Hilbert pair 4
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
43. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
19/28
Oriented, ± separable
Illustration of a combination of Hilbert pairs and M-band MRA
H{f }(ω) = −ı sign(ω)f (ω)
Compute two wavelet trees in parallel, wavelets forming Hilbert
pairs, and combine, either with standard 2-band or 4-band
Figure : Dual-tree wavelet atoms and frequency partinioning
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
44. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT for horizontal subband(s): dyadic, 2-band and 4-band DTT
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
45. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT (reminder)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
46. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT for horizontal subband(s) (reminder)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
47. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT for horizontal subband(s): 2-band, real-valued wavelet
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
48. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT for horizontal subband(s): 2-band dual-tree wavelet
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
49. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
20/28
Oriented, ± separable
Figure : GT for horizontal subband(s): 4-band dual-tree wavelet
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
50. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
21/28
Directional, non-separable
Non-separable decomposition schemes, directly n-D
◮
non-diagonal subsampling operators & windows
◮
non-rectangular lattices (quincunx, skewed)
◮
non-MRA directional filter banks
◮
steerable pyramids
◮
M-band non-redundant directional discrete wavelets
served as building blocks for:
◮
◮
◮
contourlets, surfacelets
first generation curvelets with (pseudo-)polar FFT, loglets,
directionlets, digital ridgelets, tetrolets
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
51. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
21/28
Directional, non-separable
Directional wavelets and frames with actions of rotation or
similitude groups
ψ(b,a,θ) (x) =
1
a
−1
ψ( 1 Rθ x − b) ,
a
where Rθ stands for the 2 × 2 rotation matrix
Wf (b, a, θ) = ψ(b,a,θ) , f
inverted through
−1
f (x) = cψ
0
∞
da
a3
2π
dθ
0
d2 b
R2
Wf (b, a, θ) ψ(b,a,θ) (x)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
52. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
21/28
Directional, non-separable
Directional wavelets and frames:
◮
examples: Conical-Cauchy wavelet, Morlet-Gabor frames
Figure : Morlet Wavelet (real part) and Fourier representation
◮
possibility to decompose and reconstruct an image from a
discretized set of parameters; often (too) isotropic
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
53. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Ridgelets: 1-D wavelet and Radon transform Rf (θ, t)
Rf (b, a, θ) =
ψ(b,a,θ) (x) f (x) d2 x =
Rf (θ, t) a−1/2 ψ((t−b)/a) dt
Figure : Ridgelet atom and GT decomposition
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
54. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Curvelet transform: continuous and frame
◮
curvelet atom: scale s, orient. θ ∈ [0, π), pos. y ∈ [0, 1]2 :
−1
ψs,y ,θ (x) = ψs (Rθ (x − y ))
ψs (x) ≈ s −3/4 ψ(s −1/2 x1 , s −1 x2 ) parabolic stretch; (w ≃
Near-optimal decay: C 2 in C 2 : O(n−2 log3 n)
◮
√
l)
tight frame: ψm (x) = ψ2j ,θℓ ,x n (x) where m = (j, n, ℓ) with
sampling locations:
θℓ = ℓπ2⌊j/2⌋−1 ∈ [0, π) and x n = Rθℓ (2j/2 n1 , 2j n2 ) ∈ [0, 1]2
◮
related transforms: shearlets, type-I ripplets
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
55. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Curvelet transform: continuous and frame
Figure : A curvelet atom and the wegde-like frequency support
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
56. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Curvelet transform: continuous and frame
Figure : GT curvelet decomposition
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
57. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Contourlets: Laplacian pyramid + directional filter banks
Figure : Contourlet atom and frequency tiling
from close to critical to highly oversampled
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
58. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Contourlets: Laplacian pyramid + directional filter banks
Figure : Contourlet GT (flexible) decomposition
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
59. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Shearlets
Figure : Shearlet atom in space and frequency, and frequency tiling
Do they have it all?
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
60. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
22/28
Directional, anisotropic scaling
Additional transforms
◮
previously mentioned transforms are better suited for edge
representation
◮
oscillating textures may require more appropriate transforms
examples:
◮
◮
◮
◮
◮
wavelet and local cosine packets
best packets in Gabor frames
brushlets [Meyer, 1997; Borup, 2005]
wave atoms [Demanet, 2007]
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
61. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
23/28
Lifting representations
Lifting scheme is an unifying framework
◮ to design adaptive biorthogonal wavelets
◮ use of spatially varying local interpolations
◮ at each scale j, aj−1 are split into ao and d o
j
j
◮ wavelet coefficients dj and coarse scale coefficients aj : apply
λ
λ
(linear) operators Pj j and Uj j parameterized by λj
λ
dj = djo − Pj j ajo
λ
and aj = ajo + Uj j dj
It also
◮ guarantees perfect reconstruction for arbitrary filters
◮ adapts to non-linear filters, morphological operations
◮ can be used on non-translation invariant grids to build
wavelets on surfaces
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
62. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
23/28
Lifting representations
λ
dj = djo − Pj j ajo
n = m − 2j−1
λ
and aj = ajo + Uj j dj
m + 2j−1
m
aj−1 [n]
aj−1 [m]
ao [n]
j
do [m]
j
Gj−1
Lazy
1
−
2
Predict
1
4
Update
−
dj [m]
G j ∪ Cj =
∪
1
2
1
4
aj [n]
Figure : Predict and update lifting steps; MaxMin lifting of GT
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
63. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
23/28
Lifting representations
Extensions and related works
◮ adaptive predictions:
◮
◮
◮
◮
◮
possibility to design the set of parameter λ = {λj }j to adapt
the transform to the geometry of the image
λj is called an association field, since it links a coefficient of ajo
to a few neighboring coefficients in djo
each association is optimized to reduce the magnitude of
wavelet coefficients dj , and should thus follow the geometric
structures in the image
may shorten wavelet filters near the edges
grouplets: association fields combined to maintain
orthogonality
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
64. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
65. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
66. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
67. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
68. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
69. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
24/28
Images are colors, not monochrome!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
70. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
25/28
One result among many others
Context: multivariate Stein-based denoining of a multi-spectral
satellite image
Different spectral bands
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
71. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
26/28
One result among many others
Context: multivariate Stein-based denoining of a multi-spectral
satellite image
Form left to right: original, noisy, denoised
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
72. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
26/28
One result among many others
Context: multivariate Stein-based denoining of a multi-spectral
satellite image
Form left to right: original, noisy, denoised
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
73. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
26/28
One result among many others
Context: multivariate Stein-based denoining of a multi-spectral
satellite image
Form left to right: original, noisy, denoised
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
74. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
27/28
What else? Images are not (all) flat
Many multiscale designs have been transported, adapted to:
◮
meshes
◮
spheres
◮
two-sheeted hyperboloid and
paraboloid
◮
2-manifolds (case dependent)
◮
big deal: data on graphs
see 300+ reference list!
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
75. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
28/28
Conclusion: on a (frustrating) panorama
Take-away messages anyway?
If you only have a hammer, every problem looks like a nail
◮ Is there a "best" geometric and multiscale transform?
◮
no: intricate data/transform/processing relationships
◮
maybe: many candidates, progresses awaited:
◮
◮
◮
more needed on asymptotics, optimization, models
“so ℓ2 ”! Low-rank (ℓ0 /ℓ1 ), math. morph. (+, × vs max, +)
yes: those you handle best, or (my) on wishlist
◮
mild redundancy, invariance, manageable correlation, fast
decay, tunable frequency decomposition, complex or more
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images
76. Motivations
Intro.
Early days
Oriented & geometrical
Far away from the plane
End
28/28
Conclusion: on a (frustrating) panorama
Postponed references & toolboxes
◮
A Panorama on Multiscale Geometric Representations, Intertwining
Spatial, Directional and Frequency Selectivity
Signal Processing, Dec. 2011
Toolboxes, images, and names
http://www.sciencedirect.com/science/article/pii/S0165168411001356
http://www.laurent-duval.eu/siva-panorama-multiscale-geometric-representations.html
http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html
Cymatiophilic/leptostatonymomaniac acknowledgments to:
◮
the many *-lets (last picks: Speclets/Gabor shearlets)
Laurent Jacques, Laurent Duval†, Caroline Chaux, Gabriel Peyré:
UCL, IFPEN, AMU, Dauphine
Curvelets, contourlets, shearlets, *lets, etc.: multiscale analysis and directional wavelets for images