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Related Concept Videos

Discrete-Time Fourier Series01:20

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
Trigonometric Fourier series01:17

Trigonometric Fourier series

Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
Fast Fourier Transform01:10

Fast Fourier Transform

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
Discrete-time Fourier transform01:26

Discrete-time Fourier transform

The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
One of the notable...
Discrete Fourier Transform01:15

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...

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Maximum entropy Fourier synthesis with application to diffraction tomography.

A Mohammad-Djafari, G Demoment

    Applied Optics
    |May 11, 2010
    PubMed
    Summary
    This summary is machine-generated.

    A new maximum entropy method enhances diffraction tomography object reconstruction. This approach improves accuracy by incorporating noise considerations and offers a unique solution compared to classical methods.

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    Area of Science:

    • Physics
    • Applied Mathematics
    • Imaging Science

    Background:

    • The generalized Radon theorem in diffraction tomography links the diffracted field's Fourier transform (FT) to the object's 2D FT.
    • Linear approximations (Born, Rytov) define algebraic contours (semicircles) but yield insufficient data for unique object reconstruction.
    • Classical methods struggle with noise and uniqueness in tomographic reconstruction.

    Purpose of the Study:

    • To propose a novel maximum entropy method for reconstructing objects in diffraction tomography.
    • To address the limitations of existing methods regarding data sufficiency and noise sensitivity.
    • To develop a robust reconstruction algorithm from Fourier domain data or direct field measurements.

    Main Methods:

    • Introduced a new definition of entropy for an object as a function of R(2) to C.
    • Incorporated a chi-squared statistic into the entropy measure to account for noise.
    • Utilized variational techniques and a conjugate-gradient iterative method to minimize the objective function.

    Main Results:

    • The proposed maximum entropy method allows for unique object reconstruction.
    • The method effectively handles noise in the diffracted field measurements.
    • Simulated results demonstrate the efficacy and advantages over classical reconstruction techniques.

    Conclusions:

    • The maximum entropy method provides a more robust and accurate approach to diffraction tomography.
    • This technique offers a viable solution for unique object reconstruction in the presence of noise.
    • Further investigation into computational cost and practical implementation is warranted.