Generalized frames in the space of strong limit power functions

By using the existence of a larger orthonormal basis, the space of strong limit power functions is extended. We use the windowed Fourier transform and wavelet transform to analyze strong limit power signals and we construct generalized frame decompositions using the discretized versions of these transforms.Publisher's VersionAuthor Post Prin

Fractional Hankel and Bessel wavelet transforms of almost periodic signals

The main objective of this paper is to study the Hankel, fractional Hankel, and Bessel wavelet transforms using the Parseval relation. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit power signals, and these transform methods.Publisher's Versio

Inverse solution of thermoacoustic wave equation for cylindrical layered media

Thermoacoustic imaging is a crossbred approach taking advantages of electromagnetic and ultrasound disciplines, together. A significant number of current medical imaging strategies are based on reconstruction of source distribution from information collected by sensors over a surface covering the region to be imaged. Reconstruction in thermoacoustic imaging depends on the inverse solution of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image quality. In our paper, inverse solution of the thermoacoustic wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section is investigated for cross-sectional thermoacustic imaging of breast and brain. By using Green’s functions and surface integral methods we derive an exact analytic inverse solution of thermoacoustic wave equation in frequency domain. Our inverse solution is an extension of conventional solution to layered cylindrical structures. By carrying out simulations, using numerical test phantoms consisting of thermoacoustic sources distributed in the layered model, our layered medium assumption solution was tested and benchmarked with conventional solutions based on homogeneous medium assumption in frequency domain. In thermoacoustic image reconstruction, where the medium is assumed as homogeneous medium, the solution of nonhomogeneous thermoacoustic wave equation results in geometrical distortions, artifacts and reduced image resolution due to inconvenient medium assumptions

On the fractional Fourier and continuous fractional wave packet transforms of almost periodic functions

Abstract We state the fractional Fourier transform and the continuous fractional wave packet transform as ways for analyzing persistent signals such as almost periodic functions and strong limit power signals. We construct frame decompositions for almost periodic functions using these two transforms. Also a norm equality of this signal is given using the continuous fractional wave packet transform

On the fractional sums of some special functions

We obtain new relations involving the Lerch transcendent and establish some closed-form expressions using special functions like the Riemann and Hurwitz zeta functions and fractional sums. We also get some formulae for the specific values of the derivative of Lerch transcendent.Publisher's Versio

An inverse source problem connected with thermoacoustic imaging in multi-layer planar medium

We derived analytical forward and inverse solution of thermoacoustic wave equation for nonhomogeneous medium. We modelled the nonhomogeneous medium as a multi-layer planar medium and defined initial conditions, continuity conditions on the layer boundaries and radiation conditions at infinity assuming the source distribution existing in all layers. These solutions of thermoacoustic wave equation are based on the method of Green’s functions for layered planar media. For qualitative testing and comparison of the point-spread functions associated with the homogeneous and layered solutions, we performed numerical simulations. Our simulation results showed that the conventional inverse solution based on homogeneous medium assumption, as expected, produced incorrect locations of point sources, whereas our inverse solution involving the multi-layer planar medium produced point sources at the correct source locations. Also, we examined whether the performance of our layered inverse solution is sensitive to medium parameters used as priority information in the measured data. Our inverse solutions based on multi-layer planar media are applicable for cross-sectional two-dimensional imaging of abdominal structure and the organs such as breast and skin.This work was supported by TUBITAK of Turkey through ARDEB-1003 Program under Grant 213E038. We are immensely grateful to Prof. Dr. Mustafa Karaman who provided insight and expertise that assisted the research. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliationsPublisher's Versio

Cross-sectional thermoacoustic imaging using multi-layer cylindrical media

For cross-sectional two-dimensional thermoacustic imaging of breast and brain, we explored solution of the wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section. To obtain the forward and inverse solutions of the thermoacoustic wave equation, we derived the Green's function involving Bessel and Hankel functions by employing the geometrical and acoustic parameters (densities and velocities) of layered media together with temporal initial condition, radiation conditions and continuity conditions on the layers' boundaries. The image reconstruction based on this approach involves the layer parameters as the apriori information which can be estimated from the acquired thermoacoustic data. To test and compare our layered solution with conventional solution based on homogeneous medium assumption, we performed simulations using numerical test phantoms consisting of sources distributed in the layered structure.This work was supported by TUBITAK of Turkey through ARDEB-1003 Program under Grant 213E038Publisher's Versio

Inverse solution of thermoacustic wave equation for cylindirical multi-layer mediums

Termoakustik görüntüleme, elektromanyetik enerji uyarımı ile ultrason dalgaları oluşumunu sağlayan yeni bir yöntemdir. Bu sistemin görüntüleme işlemi termoakustik dalga denkleminin ters çözümüne dayanmaktadır. Ters çözümde görüntülenecek dokunun homojen yapıda olduğu varsayımı, görüntü kalitesinin azalmasına neden olur. Bu çalışmada, meme ve beyinin görüntülemesinde uygulanabilecek üç boyutlu eş merkezli silindirik çok katmanlı yapılar için termoakustik dalga denkleminin analitik ters çözümü elde edilmiştir. Çözümü sayısal olarak test etmek için, noktasal kaynaklar içeren üç katmanlı fantom kullanılarak numerik simulasyonlar elde edilmiştir.Thermoacoustic imaging is a new modality of generating ultrasonic waves with excitation of electromagnetic pulsed energy. Thermoacoustic imaging is based on the inverse soluion of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image qualtiy. In this study, we obtain an analytical inverse solution of thermoacoustic wave equation for 3-dimensional concentric cylindrical multi layer structures. This solution can be applied for imaging of breast and brain. To test our solution, we use a 3-layer test phantom consisting of point sources in numerical simulations.Publisher's Versio

Inverse solution of thermoacoustic wave equation for multi-layer planar medium

Bu çalışmada, farklı akustik parametrelere sahip çok katmanlı düzlemsel ortam için tüm katmanlarda kaynak dağılımı olduğu varsayımı altında termoakustik dalga denkleminin analitik olarak düz ve ters çözümü elde edilmiştir. Çok katmanlı düzlemsel ortam için elde edilen analitik çözüm katmanlı düzlemsel green fonksiyonlarına dayanmaktadır. Çok katmanlı düzlemsel modelleme meme, deri ve karın bölgesi görüntülemelerine uygun bir modellemedir. Elde edilen analitik çözüm ile literatürde var olan homojen ortam varsayımına dayanan çözüm her katmanda noktasal kaynak alınarak sayısal olarak karşılaştırılmıştır.In this study, we derive analytical forward and inverse solution of thermoacoustic wave equation for inhomogeneous multi-layer planar medium, where the source distribution can exist in all layers with different acoustic parameters. Our solutions are based on methods of the Greens functions for layered planar media. This model is applicable for imaging of tissue structures such as skin, breast and abdominal imaging. We have tested and compared our layered solution and conventional solution for point sources located in a two-layer by using numerical simulations.Publisher's Versio

Thermoacoustic image reconstruction based on layered tissue model

We derived analytical forward and inverse solution of thermoacoustic wave equation for inhomogeneous multi layered planar and cylindrical mediums with the source distribution existing in all layers. These solutions are applicable for imaging of organs such as breast and brain, whose structures are suitable for multi-layer modelling. For qualitative testing and comparison of the point-spread-functions associated with the homogeneous and layered solutions, we performed numerical simulations. Our simulation results show that the conventional inverse solution based on homogeneous medium assumption, as expected, produces incorrect locations of point sources and significantly increased side lobes, whereas our inverse solution involving the multi-layered medium produces point sources at the correct locations with lower side lobes.This work was supported by TUBITAK of Turkey through ARDEB-1003 Program under Grant 213E038Publisher's Versio