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![An example of a fractal known as the Julia set [47] for a given constant/offset c. Available in colour in the online version of this article.](profile/Shekhar-Chandra-2/publication/326965061/figure/fig1/AS:658393085784064@1533984552077/An-example-of-a-fractal-known-as-the-Julia-set-47-for-a-given-constant-offset-c.png)
An example of a fractal known as the Julia set [47] for a given constant/offset c. Available in colour in the online version of this article.
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![Figure 3. An example of a fractal known as the Julia set [47] for a...](profile/Shekhar-Chandra-2/publication/326965061/figure/fig1/AS:658393085784064@1533984552077/An-example-of-a-fractal-known-as-the-Julia-set-47-for-a-given-constant-offset-c_Q320.jpg)
We propose a sparse imaging methodology called Chaotic Sensing (ChaoS) that enables the use of limited yet deterministic linear measurements through fractal sampling. A novel fractal in the discrete Fourier transform is introduced that always results in the artefacts being turbulent in nature. These chaotic artefacts have characteristics that are i...
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... examples include the Cantor set, the Terdragon set, the Mandelbrot set and the Julia sets. An example of a Julia set [47] is given in figure 3. In this work, we will create a new fractal for the DFT shown in figure 1 and described in the next section. ...
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... crucially, it remains to be seen whether other fractals can be utilised for making limited measurements, particularly in DFT space. For example, the Julia set shown in figure 3 naturally exists in the complex plane. Given recent work on studying the actual pattern of DFT coefficients utilised in medical image reconstruction [41] showed that there is a certain distribution of coefficients mostly near the origin, fractal patterns that sample near the origin, such as the Dragon or Julia sets, could prove very useful in medical image reconstruction. ...
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... Fractals: Fractals are constructed from a set of simple deterministic rules, yet exhibit complex behaviour at multiple scales. For example, the Sierpinski carpet is formed by simply dividing a rectangle into 9 equal parts and removing the centre. The process is repeated with the remaining rectangles ad infinitum to produce a pattern that has an area of zero. This particular fractal sees repeated use in RF design [45] and more recently in MR imaging hardware [46]. Other examples include the Cantor set, the Terdragon set, the Mandelbrot set and the Julia sets. An example of a Julia set [47] is given in figure 3. In this work, we will create a new fractal for the DFT shown in figure 1 and described in the next section. figure 1, the base pattern of this fractal in discrete Fourier space repeats itself at multiple scales with finer and finer resolutions at higher Fourier frequencies. This produces a multi-band response in image space, in the same way as fractal antennas were designed [45]. These antennas are capable of receiving and transmitting at multiple bands because they have the same shape, i.e. it is self-similar, at the required (different) scales for those frequency ...
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... crucially, it remains to be seen whether other fractals can be utilised for making limited measurements, particularly in DFT space. For example, the Julia set shown in figure 3 naturally exists in the complex plane. Given recent work on studying the actual pattern of DFT coefficients utilised in medical image reconstruction [41] showed that there is a certain distribution of coefficients mostly near the origin, fractal patterns that sample near the origin, such as the Dragon or Julia sets, could prove very useful in medical image ...
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... Fractals: Fractals are constructed from a set of simple deterministic rules, yet exhibit complex behaviour at multiple scales. For example, the Sierpinski carpet is formed by simply dividing a rectangle into 9 equal parts and removing the centre. The process is repeated with the remaining rectangles ad infinitum to produce a pattern that has an area of zero. This particular fractal sees repeated use in RF design [45] and more recently in MR imaging hardware [46]. Other examples include the Cantor set, the Terdragon set, the Mandelbrot set and the Julia sets. An example of a Julia set [47] is given in figure 3. In this work, we will create a new fractal for the DFT shown in figure 1 and described in the next section. figure 1, the base pattern of this fractal in discrete Fourier space repeats itself at multiple scales with finer and finer resolutions at higher Fourier frequencies. This produces a multi-band response in image space, in the same way as fractal antennas were designed [45]. These antennas are capable of receiving and transmitting at multiple bands because they have the same shape, i.e. it is self-similar, at the required (different) scales for those frequency ...
Context 6
... crucially, it remains to be seen whether other fractals can be utilised for making limited measurements, particularly in DFT space. For example, the Julia set shown in figure 3 naturally exists in the complex plane. Given recent work on studying the actual pattern of DFT coefficients utilised in medical image reconstruction [41] showed that there is a certain distribution of coefficients mostly near the origin, fractal patterns that sample near the origin, such as the Dragon or Julia sets, could prove very useful in medical image ...
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Magnetic Resonance Imaging at field strengths up to 3 T, has become a default diagnostic modality for a variety of disorders and injuries, due to multiple reasons ranging from its non-invasive nature to the possibility of obtaining high resolution images of internal organs and soft tissues. Despite tremendous advances, MR imaging of certain anatomi...
Citations
In this letter, we demonstrate how characteristics of a permutation of the Hadamard transform (HT), known as the binary discrete Hartley transform (BDHT), can be leveraged to develop a Hadamard slice theorem (HST). In doing so, we establish an orthogonal binary two dimensional (2D) transform consisting of square wave basis functions. Typically, the 2D-HT is separated into a series of one dimensional (1D) transforms of the columns and rows of an image. This process yields binary, checkerboard pattern basis functions similar in appearance to the 2D separable discrete Hartley transform (SDHT). Instead, basis functions of our proposed non-separable 2D Hartley-Hadamard transform (HHT) are analogous to sinusoids of the non-separable 2D-discrete Hartley transform (DHT). This new transform closely mimics the 2D-DHT, while preserving the benefits and characteristics of the HT (such as being multiplication free). To our knowledge, this is the first non-separable 2D-HT, with the accompanying HST being the first to relate projections of the discrete Radon transform (DRT) to the 2D-HT of an image.
Sampling strategies are important for sparse imaging methodologies, especially those employing the discrete Fourier transform (DFT). Chaotic sensing is one such methodology that employs deterministic, fractal sampling in conjunction with finite, iterative reconstruction schemes to form an image from limited samples. Using a sampling pattern constructed entirely from periodic lines in DFT space, chaotic sensing was found to outperform traditional compressed sensing for magnetic resonance imaging; however, only one such sampling pattern was presented and the reason for its fractal nature was not proven. Through the introduction of a novel image transform known as the kaleidoscope transform, which formalises and extends upon the concept of downsampling and concatenating an image with itself, this paper: (1) demonstrates a fundamental relationship between multiplication in modular arithmetic and downsampling; (2) provides a rigorous mathematical explanation for the fractal nature of the sampling pattern in the DFT; and (3) leverages this understanding to develop a collection of novel fractal sampling patterns for the 2D DFT with customisable properties. The ability to design tailor-made fractal sampling patterns expands the utility of the DFT in chaotic imaging and may form the basis for a bespoke chaotic sensing methodology, in which the fractal sampling matches the imaging task for improved reconstruction.
Magnetic resonance (MR) imaging visualises soft tissue contrast in exquisite detail without harmful ionising radiation. In this work, we provide a state-of-the-art review on the use of deep learning in MR image reconstruction from different image acquisition types involving compressed sensing techniques, parallel image acquisition and multi-contrast imaging. Publications with deep learning-based image reconstruction for MR imaging were identified from the literature (PubMed and Google Scholar), and a comprehensive description of each of the works was provided. A detailed comparison that highlights the differences, the data used and the performance of each of these works were also made. A discussion of the potential use cases for each of these methods is provided. The sparse image reconstruction methods were found to be most popular in using deep learning for improved performance, accelerating acquisitions by around 4–8 times. Multi-contrast image reconstruction methods rely on at least one pre-acquired image, but can achieve 16-fold, and even up to 32- to 50-fold acceleration depending on the set-up. Parallel imaging provides frameworks to be integrated in many of these methods for additional speed-up potential. The successful use of compressed sensing techniques and multi-contrast imaging with deep learning and parallel acquisition methods could yield significant MR acquisition speed-ups within clinical routines in the near future.
A laser ablation process assisted by the feedback of a sensor with chaotic electronic
modulation is reported. A synchronous bistable logic circuit was analyzed for switching optical signals in a laser-processing technique. The output of a T-type flip-flop configuration was employed in the photodamage of NiO films. Multiphotonic effects involved in the ablation threshold were evaluated by a vectorial two-wave mixing method. A photoinduced thermal phenomenon was identified as the main physical mechanism responsible for the nonlinearity of index under nanosecond irradiation at 532 nm wavelength. Comparative experiments for destroying highly transparent human cells were carried out. Potential applications for developing hierarchical functions yielding laser-induced controlled explosions with immediate applications for biomedical photothermal processes can be contemplated.
The evolution of the optical absorptive effects exhibited by plasmonic nanoparticles was systematically analyzed by electronic signals modulated by a Rössler attractor system. A sol-gel approach was employed for the preparation of the studied Au nanoparticles embedded in a TiO2 thin solid film. The inclusion of the nanoparticles in an inhomogeneous biological sample integrated by human cells deposited in an ITO glass substrate was evaluated with a high level of sensitivity using an opto-electronic chaotic circuit. The optical response of the nanoparticles was determined using nanosecond laser pulses in order to guarantee the sensing performance of the system. It was shown that high-intensity irradiances at a wavelength of 532 nm could promote a change in the absorption band of the localized surface plasmon resonance associated with an increase in the nanoparticle density of the film. Moreover, it was revealed that interferometrically-controlled energy transfer mechanisms can be useful for thermo-plasmonic functions and sharp selective optical damage induced by the vectorial nature of light. Immediate applications of two-wave mixing techniques, together with chaotic effects, can be contemplated in the development of nanostructured sensors and laser-induced controlled explosions, with potential applications for biomedical photo-thermal processes.
As an emerging field for sampling paradigms, compressive sensing (CS) can sample and represent signals at a sub-Shannon–Nyquist rate. To realize CS from theory to practice, the random sensing matrices must be substituted by faster measurement operators that correspond to feasible hardware architectures. In recent years, binary matrices have attracted much research interest because of their multiplier-less and faster data acquisition. In this work, we aim to pinpoint the potential of chaotic binary sequences for constructing high-efficiency sensing implementations. In particular, the proposed chaotic binary sensing matrices are verified to meet near-optimal theoretical guarantees in terms of both the restricted isometry condition and coherence analysis. Simulation results illustrate that the proposed chaotic constructions have considerable sampling efficiency comparable to that of the random counterparts. Our framework encompasses many families of binary sensing architectures, including those formed from Logistic, Chebyshev, and Bernoulli binary chaotic sequences. With many chaotic binary sensing architectures, we can then more easily apply CS paradigm to various fields.
