Browsing by Author "Bartan, B."

Now showing 1 - 3 of 3

Open Access
Digital computation of fractional Fourier and linear canonical transforms and sparse image representation
(IEEE, 2018) Koc, A.; Özaktaş, Haldun M.; Bartan, B.; Gundogdu, E.; Çukur, Tolga
Fast and accurate digital computation of the fractional Fourier transform (FRT) and linear canonical transforms (LCT) are of utmost importance in order to deploy them in real world applications and systems. The algorithms in O(NlogN) to obtain the samples of the transform from the samples of the input function are presented for several different types of FRTs and LCTs, both in 1D and 2D forms. To apply them in image processing we consider the problem of obtaining sparse transform domains for images. Sparse recovery tries to reconstruct images that are sparse in a linear transform domain, from an underdeter- mined measurement set. The success of sparse recovery relies on the knowledge of domains in which compressible representations of the image can be obtained. In this work, we consider two- and three-dimensional images, and investigate the effects of the fractional Fourier (FRT) and linear canonical transforms (LCT) in obtaining sparser transform domains. For 2D images, we investigate direct transforming versus several patching strategies. For the 3D case, we consider biomedical images, and compare several different strategies such as taking 2D slices and optimizing for each slice and direct 3D transforming.
Open Access
Discrete scaling based on operator theory
(Elsevier, 2020-11-04) Koç, Aykut; Bartan, B.; Özaktaş, Haldun Memduh
Signal scaling is a fundamental operation of practical importance in which a signal is made wider or narrower along the coordinate direction(s). Scaling, also referred to as magnification or zooming, is complicated for signals of a discrete variable since it cannot be accomplished simply by moving the signal values to new coordinate points. Most practical approaches to discrete scaling involve interpolation. We propose a new approach based on hyperdifferential operator theory that does not involve conventional interpolation. This approach provides a self-consistent and pure definition of discrete scaling that is fully consistent with discrete Fourier transform theory. It can potentially be applied to other problems in signal theory and analysis such as transform designs. Apart from its theoretical elegance, it also provides a basis for numerical implementation.
Open Access
Sparse representation of two-and three-dimensional images with fractional fourier, hartley, linear canonical, and haar wavelet transforms
(Elsevier Ltd, 2017) Koç A.; Bartan, B.; Gundogdu, E.; Çukur, T.; Özaktaş, Haldun M.
Sparse recovery aims to reconstruct signals that are sparse in a linear transform domain from a heavily underdetermined set of measurements. The success of sparse recovery relies critically on the knowledge of transform domains that give compressible representations of the signal of interest. Here we consider two- and three-dimensional images, and investigate various multi-dimensional transforms in terms of the compressibility of the resultant coefficients. Specifically, we compare the fractional Fourier (FRT) and linear canonical transforms (LCT), which are generalized versions of the Fourier transform (FT), as well as Hartley and simplified fractional Hartley transforms, which differ from corresponding Fourier transforms in that they produce real outputs for real inputs. We also examine a cascade approach to improve transform-domain sparsity, where the Haar wavelet transform is applied following an initial Hartley transform. To compare the various methods, images are recovered from a subset of coefficients in the respective transform domains. The number of coefficients that are retained in the subset are varied systematically to examine the level of signal sparsity in each transform domain. Recovery performance is assessed via the structural similarity index (SSIM) and mean squared error (MSE) in reference to original images. Our analyses show that FRT and LCT transform yield the most sparse representations among the tested transforms as dictated by the improved quality of the recovered images. Furthermore, the cascade approach improves transform-domain sparsity among techniques applied on small image patches.