Browsing by Subject "HILBERT TRANSFORM"

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  • Hytönen, Tuomas; Vuorinen, Emil (2018)
    We consider boundedness of a certain positive dyadic operator T-sigma : L-p (sigma; l(2)) -> L-P (omega). that arose during our attempts to develop a two-weight theory for the Hilbert transform in L-P. Boundedness of T-sigma is characterized when p is an element of [2, infinity) in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.
  • Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2020)
    We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral T-n in R-n and a bounded singular integral T-m in R-m we prove parallel to T-n(1), [b, T-m(2)]]parallel to(Lp(mu)-> Lp(lambda)) less than or similar to([mu]Ap, [lambda]Ap) parallel to b parallel to(BMOprod)(nu), where p is an element of (1, infinity), mu, lambda is an element of A(p) and nu := mu(1/p) lambda(-1/p) is the Bloom. weight. Here T-n(1) is T-n acting on the first variable, T-m(2) is T-m acting on the second variable, A(p) stands for the bi-parameter weights of R-n x R-m and BMOprod(nu) is weighted product BMO space.
  • Ketola, Juuso H. J.; Heino, Helinä; Juntunen, Mikael A. K.; Nieminen, Miika T.; Siltanen, Samuli; Inkinen, Satu I. (2021)
    In interior computed tomography (CT), the x-ray beam is collimated to a limited field-of-view (FOV) (e.g. the volume of the heart) to decrease exposure to adjacent organs, but the resulting image has a severe truncation artifact when reconstructed with traditional filtered back-projection (FBP) type algorithms. In some examinations, such as cardiac or dentomaxillofacial imaging, interior CT could be used to achieve further dose reductions. In this work, we describe a deep learning (DL) method to obtain artifact-free images from interior CT angiography. Our method employs the Pix2Pix generative adversarial network (GAN) in a two-stage process: (1) An extended sinogram is computed from a truncated sinogram with one GAN model, and (2) the FBP reconstruction obtained from that extended sinogram is used as an input to another GAN model that improves the quality of the interior reconstruction. Our double GAN (DGAN) model was trained with 10 000 truncated sinograms simulated from real computed tomography angiography slice images. Truncated sinograms (input) were used with original slice images (target) in training to yield an improved reconstruction (output). DGAN performance was compared with the adaptive de-truncation method, total variation regularization, and two reference DL methods: FBPConvNet, and U-Net-based sinogram extension (ES-UNet). Our DGAN method and ES-UNet yielded the best root-mean-squared error (RMSE) (0.03 +/- 0.01), and structural similarity index (SSIM) (0.92 +/- 0.02) values, and reference DL methods also yielded good results. Furthermore, we performed an extended FOV analysis by increasing the reconstruction area by 10% and 20%. In both cases, the DGAN approach yielded best results at RMSE (0.03 +/- 0.01 and 0.04 +/- 0.01 for the 10% and 20% cases, respectively), peak signal-to-noise ratio (PSNR) (30.5 +/- 2.6 dB and 28.6 +/- 2.6 dB), and SSIM (0.90 +/- 0.02 and 0.87 +/- 0.02). In conclusion, our method was able to not only reconstruct the interior region with improved image quality, but also extend the reconstructed FOV by 20%.
  • Di Plinio, Francesco; Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2020)
    We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative L-P spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property alpha. (C) 2020 Elsevier Inc. All rights reserved.
  • Di Plinio, Francesco; Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2020)
    We prove L-p bounds for the extensions of standard multilinear Calderon-Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space-in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L-p spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.
  • Hytönen, Tuomas P. (2017)
    This exposition presents a self-contained proof of the A(2) theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L-2 (w). The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journe T(1) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided. (C) 2016 Elsevier GmbH. All rights reserved.
  • Hytonen, Tuomas; Petermichl, Stefanie; Volberg, Alexander (2019)
    We prove the matrix A(2) conjecture for the dyadic square function, that is, an estimate of the form vertical bar vertical bar S-w vertical bar vertical bar(L2cd(w)-> Lr2) less than or similar to [W](A2), where the focus is on the sharp linear dependence on the matrix A(2) constant. Moreover, we give a mixed estimate in terms of A(2) and A(infinity) constants. The key to the proof is a sparse domination of a process inspired by the integrated form of the matrix-weighted square function.
  • Vuorinen, Emil (2017)
    We consider two-weight L-p -> L-q-inequalities for dyadic shifts and the dyadic square function with general exponents 1 <p, q <infinity. It is shown that if a so-called quadratic A(p,q)-condition related to the measures holds, then a family of dyadic shifts satisfies the two-weight estimate in an R-bounded sense if and only if it satisfies the direct and the dual quadratic testing condition. In the case p = q = 2 this reduces to the result by T. Hytonen, C. Perez, S. Treil and A. Volberg (2014). The dyadic square function satis fi es the two-weight estimate if and only if it satis fi es the quadratic testing condition, and the quadratic A(p,q)-condition holds. Again in the case p = q = 2 we recover the result by F. Nazarov, S. Treil and A. Volberg (1999). An example shows that in general the quadratic A(p,q)-condition is stronger than the Muckenhoupt type A(p,q)-condition.