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^{1, 2}using a quasi-continuous localization model with sparsity prior on image space. It was demonstrated in both 2D/3D live cell imaging. However, it has several disadvantages to be further improved. Here, we proposed a new localization algorithm using annihilating filter-based low rank Hankel structured matrix approach (ALOHA). According to ALOHA principle, sparsity in image domain implies the existence of rank-deficient Hankel structured matrix in Fourier space. Thanks to this fundamental duality, our new algorithm can perform data-adaptive PSF estimation and deconvolution of Fourier spectrum, followed by truly grid-free localization using spectral estimation technique. Furthermore, all these optimizations are conducted on Fourier space only. We validated the performance of the new method with numerical experiments and live cell imaging experiment. The results confirmed that it has the higher localization performances in both experiments in terms of accuracy and detection rate.

*constant-variance*interpolation is needed to remove this bias. In this paper, we present a theoretical investigation of the role of the interpolation basis in a registration context. Contrarily to published analyses, ours is deterministic; it nevertheless leads to the same conclusion, which is that constant-variance interpolation is beneficial to image registration. In addition, we propose a novel family of interpolation bases that can have any desired order of approximation while maintaining the constant-variance property. Our family includes every constant-variance basis we know of. It is described by an explicit formula that contains two free functional terms: an arbitrary 1-periodic binary function that takes values from {-1, 1}, and another arbitrary function that must satisfy the partition of unity. These degrees of freedom can be harnessed to build many family members for a given order of approximation and a fixed support. We provide the example of a symmetric basis with two orders of approximation that is supported over [-3/2, 3/2] this support is one unit shorter than a basis of identical order that had been previously

*SUREshrink*in each individual wavelet subband to denoise images corrupted by additive Gaussian white noise. We show that, for various images and a wide range of input noise levels, the orthogonal fractional (α, τ)-B-splines give the best peak signal-to-noise ratio (PSNR), as compared to standard wavelet bases (Daubechies wavelets, symlets and coiflets). Moreover, the selection of the best set (α, τ) can be performed on the MSE estimate (SURE) itself, not on the actual MSE (Oracle). Finally, the use of complex-valued fractional B-splines leads to even more significant improvements; they also outperform the complex Daubechies wavelets.

_{v}

^{→}that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of L

_{v}

^{→}, generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression

*a*=2

^{i}. The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm.

^{γ}

_{τ}, for a γ-th order derivative with shift τ. Both order of derivation and shift can be chosen fractional. Our design leads us naturally to select the fractional B-splines as scaling functions. By putting the differential wavelet in the perspective of a derivative of a smoothing function, we find that signal singularities are compactly characterized by at most two local extrema of the wavelet coefficients in each subband. This property could be beneficial for signal analysis using wavelet bases. We show that this wavelet transform can be efficiently implemented using FFTs.

*non-stationary*multiresolution analysis, which involves a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. We consider a dual pair of such multiresolutions, where the scaling functions at a given scale are mutually biorthogonal with respect to translation. Also, they must have the shortest-possible support while reproducing a given set of exponential polynomials. This constitutes a generalization of the standard polynomial reproduction property. The corresponding refinement filters are derived from the ones that were studied by Dyn et al. in the framework of non-stationary subdivision schemes. By using different factorizations of these filters, we obtain a general family of compactly supported dual wavelet bases of

*L*

_{2}. In particular, if the exponential parameters are all zero, one retrieves the standard CDF B-spline wavelets and the 9/7 wavelets. Our generalized description yields equivalent constructions for E-spline wavelets. A fast filterbank implementation of the corresponding wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. This new scheme offers high flexibility and is tunable to the spectral characteristics of a wide class of signals. In particular, it is possible to obtain symmetric basis functions that are well-suited for image processing.

_{p}-norm of its rescaled wavelet coefficients. Indeed, the key condition for a wavelet basis to be an unconditional basis of the Besov space

*B*(

_{q}^{s}*L*(

_{p}*R*)) is that the

^{d}*s*-order derivative of the wavelet be in

*L*.

_{p}*q*different multiwavelet-like basis functions. The basis functions can be dissymmetric but should preserve the "partition of unity" property for high-quality signal interpolation. The scheme of decomposition and reconstruction of signals by the proposed basis functions can be implemented in a filterbank form using separable IIR implementation. An important property of the proposed scheme is that the prefilters for decomposition can be implemented by FIR filters. Recall that the shifted-linear interpolation requires IIR prefiltering, but we find a new configuration which reaches almost the same quality with the shifted-linear interpolation, while requiring FIR prefiltering only. Moreover, the present basis functions can be explicitly formulated in time-domain, although most of (multi-)wavelets don’t have a time-domain formula. We specify an optimum configuration of interpolation parameters for image interpolation, and validate the proposed method by computing PSNR of the difference between multi-rotated images and their original version.

^{+}. The corresponding filters allow a FFT-based implementation and thus provide a fast algorithm for the wavelet transform.

_{2}. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel transform that are needed to construct our new wavelet. We derive a Heisenberg-like uncertainty relation that links the localization of the Fresnelets with that of the original wavelet basis. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. We conclude that the Fresnel B-splines are particularly well suited for processing holograms because they tend to be well localized in both domains.

_{p}sense where p can take non-integer values. The underlying image model is specified using arbitrary shift- invariant basis functions such as splines. The solution is determined by an iterative optimization algorithm, based on digital filtering. Its convergence is accelerated by the use of first and second derivatives. For p equals 1, our modified pyramid is robust to outliers; edges are preserved better than in the standard case where p equals 2. For 1 < p < 2, the pyramid decomposition combines the qualities of l

_{1}and l

_{2}approximations. The method is applied to edge detection and its improved performance over the standard formulation is determined.

^{2}-error made by approximating the signal by its projection. This result provides a norm for evaluating the accuracy of a complete decimation/interpolation branch for arbitrary analysis and synthesis filters; such a norm could be useful for the joint design of an analysis and synthesis filter, especially in the non-orthonormal case. As an example, we use our framework to compare the efficiency of different wavelet filters, such as Daubechies' or splines. In particular, we prove that the error made by using a Daubechies' filter downsampled by 2 is of the same order as the error using an orthonormal spline filter downsampled by 6. This proof is valid asymptotically as the number of regularity factors tends to infinity, and for a signal that is essentially low- pass. This implies that splines bring an additional compression gain of at least 3 over Daubechies' filters, asymptotically.

^{(alpha}) (one-sided) or x

^{(alpha})

_{*}(symmetric); in each case, they are (alpha) -Hoelder continuous for (alpha) > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Biesz basis condition and the two-scale relation, which makes them suitable for the construction of new families of wavelet bases. What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order of approximately ((alpha) + 1), while they reproduce the polynomials of degree [(alpha) ]. We show how they yield continuous-order generalization of the orthogonal Battle- Lemarie wavelets and of the semi-orthogonal B-spline wavelets. As (alpha) increases, these latter wavelets tend to be optimally localized in time and frequency in the sense specified by the uncertainty principle. The corresponding analysis wavelets also behave like fractional differentiators; they may therefore be used to whiten fractional Brownian motion processes.

_{1}to m

_{p}where m equals m

_{1}...m

_{p}is a prime number factorization of m.

_{2}-error of wavelet-like expansions as a function of the scale a. This yields an extension as well as a simplification of the asymptotic error formulas that have been published previously. We use our bound determinations to compare the approximation power of various families of wavelet transforms. We present explicit formulas for the leading asymptotic constant for both splines and Daubechies wavelets. For a specified approximation error, this allows us to predict the sampling rate reduction that can obtained by using splines instead Daubechies wavelets. In particular, we prove that the gain in sampling density (splines vs. Daubechies) converges to (pi) as the order goes in infinity.

_{2}norm of the approximation error. We then define the centered pyramid and give explicit filter coefficients for odd and even spline approximation functions. Finally, we compare the centered pyramid to the ordinary one and highlight some applications.

_{2}. The approximation power of these representations is essentially as good as that of the corresponding Battle- Lemarie orthogonal wavelet transform, with the difference that the present wavelet synthesis filters have a much faster decay. This last property, together with the fact that these transformation s are almost orthogonal, may be useful for image coding and data compression.

_{i}with (i equals 1,2) (continuous/discrete representation). The corresponding input and output functions spaces are V(p

_{i}) and V(p

_{2}), respectively. The principle of the method is as follows: we start by interpolating the discrete input signal with a function s

_{1}(epsilon) V(p

_{1}). We then apply a linear operator T to this function and compute the minimum error approximation of the result in the output space V(p

_{2}). The corresponding algorithm can be expressed in terms of digital filters and a matrix multiplication. In this context, we emphasize the advantages of B-splines; and show how a judicious use of these basis functions can result in fast implementations of various types of operators. We present design examples of differential operators involving very short FIR filters. We also describe an efficient procedure for the geometric affine transformation of signals. The present formulation is general enough to include most earlier continuous/discrete signal processing techniques (e.g., standard bandlimited approach, spline or wavelet-based) as special cases.

_{2}and we give error bounds on the approximation. We define the concept of discrete multiresolutions and wavelet spaces and show that the oblique projections on certain subclasses of discrete multiresolutions and their associated wavelet spaces can be obtained using perfect reconstruction filter banks. Therefore we obtain a discrete analog of the Cohen-Daubechies- Feauveau results on biorthogonal wavelets.

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**Proceedings Volume Editor**(11)

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