GIANCourseon

TheoryandApplicationofWaveletsandFramelets

28December2017–04January2018

Overview

As the major tool for multiscale data analysis, wavelets have a wide scope of applications in mathematics,engineering,physics,sciences,andindustries.Forexample,waveletshavebeenadoptedinJPEG-2000standardforimage compression, and wavelet subdivision algorithms have been used in animation movie industry. Being amultidisciplinary researcharea,waveletsand frameletsareveryeffective for representingvarious functionsanddata.Thegreatsuccessofwaveletsandframeletslargelyliesintheirmanydesiredpropertiessuchasmultiscalestructure, sparse representation, efficient approximation schemes, good time-frequency localization, and fastcomputational algorithms. In comparison to traditional wavelets, framelets have the desired properties ofredundancyforrobustnessandflexibilityforadaptivecustomdesign.Thecurrentdevelopmentsonwavelettheoryarefocusingonframeletaspectsandtheirapplicationsinhigh-dimensionaldataanalysis.Forexample,algorithmsusing framelets currently provide the-state-of-the-art results in image processing, and are popular in geometricmodellingprocessing.

Objectives

Introducethealgorithmsandbasictheoryofwaveletsandframeletstostudentsandinterestedresearchers.Thiswillequipthestudentsandresearchersthebasicknowledgeonwavelettheory,teachthemhowtodevelopandimplement their own fast framelet/wavelet transforms, and introduce them to design their own wavelets andframelets for their own purposes. Bring the students and researchers to thewide scope of applications usingwaveletsandframelets.Therearenumerousapplicationsofwaveletsandframeletsandwemainlyconcentrateontheirapplicationstosignal/imageprocessingandgeometricmodelling.Throughoutthelecturesandtutorials,thestudentsandresearcherswillbecomefamiliarwithhowtoconcretelyuse/implementwaveletsandframeletstosolve some practical problems in applications. Present some most recent developments on wavelets andframelets. Thisallows the studentsand researchers tobecomeawarewhatare thecurrent frontiersofwavelettheoryandwhatarethepossiblefurtherdevelopmentsandapplicationsofwaveletsandframelets.Thestudents'knowledgeabout the course contentwill be raised to the level such that theywill beable tousewavelets andframeletsfortheirownapplicationsandresearch.

Modules A: Fast Framelets and Wavelet Transforms

TopicsofLecture:• Ashortnaïveintroductiontowaveletsforeveryone:Whatisawavelet?• Perfect reconstruction, sparsityandvanishingmomentsof frameletand

wavelettransforms• Multilevelstructureofframelet/wavelettransformsandtheirvariants• How to implement fast framelet and wavelet transforms for practical

data• Howtoprocesswaveletorframeletcoefficients:waveletshrinkage

TopicsofTutorial:• Introductiontoframeletandwavelettransforms• Demonstration and examples on wavelet transforms using matlab

toolbox• Exploreframelettransformsanddiscusstheirdifferencestowavelets• Waveletapplicationstothesignaldenoisingproblem

Modules B: Mathematical Theory of Wavelets and Framelets

TopicsofLecture:• Framesandbases inHilbert spaces, in finitedimensional spaces, and in

shift-invariantspaces• Samplingtheoremsinshift-invariantspacesforsignalprocessing• Multiresolutionanalysisandorthogonalwavelets• CompactlysupportedDaubechiesorthogonalwavelets• Refinablefunctionsandtheoryoftightframelets• Basicdesirablepropertiesofwaveletsandframelets

TopicsofTutorial:• Explore sampling theorems in shift-invariant spaces generated by B-

splines• Designandimplementorthogonalwavelets• Howtodesignandimplementtightframeletfilterbanks• Use subdivision schemes and cascade algorithms to plot wavelets and

refinablefunctions

Modules C: Applications of Wavelets and Framelets

TopicsofLecture:• Tensorproductionandhigh-dimensionalwaveletsandframelets• Introductionofsubdivisionschemesandcascadealgorithms• Waveletsubdivisionforcomputergraphicsandgeometricmodeling• Dual-tree complex wavelet transform and directional complex tight

frameletsTopicsofTutorial:

• Exploretensorproductwavelet/frameletalgorithmsforhigh-dimensionaldata

• Imagemodelsandwavelet-basedalgorithmsforimageprocessing• Usesubdivisionschemestogeneratesubdivisioncurvesforgeometric

modeling• Applicationsofdirectionalframeletsandwaveletstoimageprocessing• Problemsolvingsessionwithexamplesinimageprocessing

You Should Attend If…

• You are students at all levels (BTech/MSc/MTech/PhD/Post Doc) orFacultyfromreputedacademicinstitutionsandtechnicalinstitutions.

• You are executives, engineers and researchers from manufacturing,serviceandgovernmentorganizationsincludingR&Dlaboratories.

Max. No. of Participants

50

Fees

Participantsfromabroad:US$200MSc/M.Phil/B.Tech/M.Tech.Students:Rs.1,000/-Ph.D.Student/PostDoctoralParticipants:Rs.2,000/-FacultyParticipants:Rs.2,500/-GovernmentResearchOrganizationParticipants:Rs.3,000/-IndustryParticipants:Rs.5,000/-Theabovefeeincludeslunch,instructionalmaterials,24hoursinternetfacility.

Accommodation

The participants may be provided with hostel accommodation, depending onavailability,onpaymentbasis.

Foranyquery,pleasesendanemailtonirajshukla@iiti.ac.in.

TheFaculty

Prof. BinHan is a full professor ofmathematicsat theDepartmentofMathematical and StatisticalSciences in the University ofAlberta,Canada.Theresearchareaof Bin Han includes applied

harmonic analysis, wavelet analysis and theirapplications in computer graphics, image/signalprocessing,andnumericalalgorithms.BinHanservesasan editor for four academic SCI journals includingApplied and Computational Harmonic Analysis andJournalofApproximationTheory.BinHan is anauthorof more than 85 papers in top academic SCI journalsandistheauthorofthebook:Frameletsandwavelets:algorithms,analysis,andapplications.According toSCIcitationforrecent10-yearwork,heisrankedwithintop200inmathematicswithseveralofhispaperscited299,132 and 118 times. He has been invited to presentmany invited talks including fiveplenary talks inmajorinternational conferences in the area of wavelets,approximation theory, and applied and computationalharmonicanalysis.Forexample,hepresenteda1-hourplenary talk in the 13th International Conference onApproximation Theory in March 2010 at San Antonio,USA. He served as the director of the AppliedMathematicsInstituteattheUniversityofAlbertafrom2012to2015.Hehassupervised5PhDstudents,4MScstudents,and5postdoctoralfellows.Formoredetails,pleasevisithttps://sites.ualberta.ca/~bhan/.

Dr.NirajK.Shukla isworkingasanAssistant Professor in Discipline ofMathematics, IIT Indore. Hisresearch interest isWavelet,FrameandHarmonicAnalysis.

Formoredetails,pleasevisithttp://iiti.ac.in/people/~nirajshukla/.

Duration:

December28,2017-January04,2018

CourseCo-ordinatorDr. Niraj Kumar Shukla Assistant Professor Discipline of Mathematics IIT Indore, Khandwa Road, Simrol Indore- 453 552, Madhya Pradesh Tel: +91 732 4306 506 +91 731 2438 947 Email: nirajshukla@iiti.ac.in http://iiti.ac.in/people/~nirajshukla/ ............................................................. Course Web-page: http://iiti.ac.in/people/~nirajshukla/Gian Registration Link: http://gian.iiti.ac.in/register.php