Proceedings, XXth congress: Proceedings, XXth congress

Altan, M. Orhan

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Multivolume work

Persistent identifier:: 1663674213

Title:: Proceedings, XXth congress

Sub title:: Istanbul, 12 - 23 July 2004

Year of publication:: 2004

Place of publication:: Istanbul

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 1663674213

Language:: English

Additional Notes:: Erscheinungsdatum des Originals ist aus dem Copyrightjahr ermittelt.; Auch bezeichnet als XXth International Congress for Photogrammetry and Remote Sensing

Editor:: Altan, M. Orhan

Corporations:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Adapter:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Founder of work:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Other corporate:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Document type:: Multivolume work

Volume

Persistent identifier:: 166368779X

Title:: Proceedings, XXth congress

Scope:: IV, 226 Seiten

Year of publication:: 2004

Place of publication:: Istanbul

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 166368779X

Illustration:: Illustrationen, Diagramme

Signature of the source:: ZS 312(35,B8)

Language:: English

Additional Notes:: Erscheinungsdatum des Originals ist aus dem Copyrightjahr ermittelt.

Usage licence:: Attribution 4.0 International (CC BY 4.0)

Editor:: Altan, M. Orhan

Corporations:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Adapter:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Founder of work:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Other corporate:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2019

Document type:: Volume

Collection:: Earth sciences

Chapter

Title:: THE CURVELET TRANSFORM FOR IMAGE FUSION Myungjin Choi, Rae Young Kim, Moon-Gyu Kim

Document type:: Multivolume work

Structure type:: Chapter

International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 2. CONTINUOUS RIDGELET TRANSFORM The two-dimensional continuous ridgelet transform in R° can be defined as follows(Candes, 1999; Starck et al, 2002). We pick a smooth univariate function y:R—R with sufficient decay and satisfying the admissibility condition car (|l ae « o (1) which holds if, say, v has a vanishing mean [y (r)dt=0 - We will suppose that vw is normalized so that n 0 each b e R and each 0 e[0,2z) , we 2.52.92 4 lv ca^ ^aést: For each a» 0, = Baris : 2 2 define the bivariate ridgelet V bo R' —R by {x)= aux, cos 0 * x, sinQ — b)/a) (2) Y a.b,0 A ridgelet is constant along lines x, cos 0 + x, sin 0 = const . Transverse to these ridges it is a wavelet. 3552 12924 =. Given f e L (R^), we define its ridgelet coefficients by W y (a,b. 8) = [vio 00rends (3) If fe LARRY, then we have the exact reconstruction formula T dt r= | | [rr@bowapom? d 10 1 0 — 0 and a Parseval relation 2x o oc do J fo” dx = ] | fe (a,b ey “a e (5) —o0 0 Hence, much like the MA or Fourier transforms, the identity (4) expresses the fact that one can represent any arbitrary function as a continuous superposition of ridgelets. Discrete analogs of (4) and (5) exists; see (Candes, 1999) or (Donoho, 2000) for a slightly different approach. 2.1 Radon Transform A basic tool for calculating ridgelet coefficients is to view ridgelet analysis as a form of wavelet analysis in the Radon domain. The Radon transform A: (RY) — D ([0, 22], L (R)) is defined by Rf (0,1) = [rman cos 0+x, sin O-t) dx, dx) (6) 60 where ó is the Dirac distribution. The ridgelet coefficients 9$ / transform via (a,b, 0) of an object / are given by analysis of the Radon 9t (a,b, 0) - [nrto. a ^v - 01 axi (7) Hence, the ridgelet transform is precisely the application of a 1- dimensional wavelet transform to the slices of the Radon transform where the angular variable © is constant and f is varying. 2.2 Ridgelet Pyramids Let Q denote a dyadic square Q - [4 j28 un - 0/2 )x [£, 12 Ah + 1)/2°) and let Q be the collection of all such dyadic squares. We write Q, for the collection of all dyadic squares of scale s . Associated to the squares QeQg we construct a partition of energy as follows. With & a nice smooth window obeying S dt e (x x5 -k2 )»1 , we dilate and transport & to all squares Q at scale s , producing a collection of windows (c, ) such that the ER QeQs , make up a partition of unity. We also let 7, donote the transport operator acting on functions g via S sS 5 (709)x,x,)= 2 g@ x, mA ne aka) With these notations, it is not hard to see that da dO fo, = IE f, 0, TU 7 TV nb. EN. (8) rai 3. e 4 and, therefore, summing the above equalities across squares at a given scale gives 2 m fe Y fay = 2 Io fo Q TV > T Vaso 3 QeQ, The identity (9) expresses the fact that one can represent any function as a superposition of elements of the form QUT Wb o 5 that is, of ridgelet elements localized near the squares Q For the function T M as bo is the ridgelet with parameters obeying V qo 59.00 p yıng s ag=2 ‘a, bo=b+k; 275 cos0+k32 * sin0, 09-0 and, thus, œ, TV ano is a windowed ridgelet, supported near a,b, the square Q, hence the name local ridgelet transform. The previous paragraph discussed the construction of local ridgelets of fixed length, roughly 275 ( s fixed). Letting the Inte th re th OY tr: se sl; sl; TI ha Tt po sc de an

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