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SUMMARY:Francesca Bartolucci (ETH - Zurich)
DTSTART:20230208T170000Z
DTEND:20230208T180000Z
DTSTAMP:20260423T021144Z
UID:HAeS/41
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HAeS/41/">Wh
 at's new in wavelet phase retrieval?</a>\nby Francesca Bartolucci (ETH - Z
 urich) as part of Harmonic analysis e-seminars\n\n\nAbstract\nWavelet phas
 e retrieval consists of the inverse problem of reconstructing a square-int
 egrable function $f$ from its scalogram\, that is from the absolute value 
 of its wavelet transform\n\\[\n    \\mathcal{W}_{\\phi}f(b\,a) = a^{-\\fra
 c{1}{2}} \\int_{\\R} f(x) \\overline{\\phi\\left(\\frac{x-b}{a}\\right)} \
 \\,\\mathrm{d} x\, \\qquad b \\in \\R\,~a \\in \\R_+. \n\\]\nThe wavelet t
 ransform emerged from the research activities aimed to develop new analysi
 s and processing tools to enhance signal theory\, and has proved to be ext
 remely efficient in various applications such as denoising and compression
 . However\, there is still limited knowledge of the problem of reconstruct
 ing a function from the absolute value of its wavelet transform. More prec
 isely\, wavelet phase retrieval aims to determine for which analyzing wave
 lets $\\phi$ and which choices of $\\Lambda \\subseteq \\R \\times \\R_+$ 
 as well as $\\mathcal{M} \\subseteq L^2(\\R)$ the forward operator \n\\[\n
 F_\\phi : \\mathcal{M} /\\!\\sim \\\, \\to\\\, [0\,+\\infty)^\\Lambda\,\\q
 quad F_\\phi f(b\,a) = \\lvert \\mathcal{W}_\\phi f (b\,a) \\rvert\, \\qua
 d (b\,a) \\in \\Lambda\,\n\\]\nis injective\, where $f\\sim g$ if and only
  if $f=\\text{e}^{i\\alpha}g$ for some $\\alpha\\in\\R$. In this talk\, we
  present old and new results on this question and conclude by discussing s
 ome open problems.\n
LOCATION:https://researchseminars.org/talk/HAeS/41/
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