Ingrid Daubechies

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Wavelet theory is an attempt to address the pervasive problem of describing the frequency content of a function locally in time. The wavelet approach is to analyze a function using an appropriate family of dilates and translates of one or more wavelets. Although this term is relatively new, wavelet-like techniques have been independently invented over the(More)
We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted ppenalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p <(More)
ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase(More)
A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed. This method involves two steps. First, a wavelet transform used in order to obtain a set of biorthogonal subclasses of images: the original image is decomposed at different scales using a pyramidal algorithm architecture. The(More)
Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly(More)
We are interested here in wavelet frames and their construction via multiresolution analysis (MRA); of particular interest to us are tight wavelet frames. The redundant representation offered by wavelet frames has already been put to good use for signal denoising, and is currently explored for image compression. Motivated by these and other applications, we(More)
Under certain conditions (known as the restricted isometry property, or RIP) on the m N matrix ˆ (where m < N ), vectors x 2 RN that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y WD ˆx even though ˆ 1.y/ is typically an .N m/–dimensional hyperplane; in addition, x is then equal to the element in ˆ 1.y/ of minimal(More)