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Related Concept Videos

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...
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
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...
Transformations of Functions III01:20

Transformations of Functions III

Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
Basic signals of Fourier Transform01:07

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The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
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Related Experiment Videos

The finite ridgelet transform for image representation.

Minh N Do1, Martin Vetterli

  • 1Audiovisual Communications Laboratory, Department of Communication Systems, Swiss Federal Institute of Technology, Lausanne, Switzerland. minhdo@uiuc.edu

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 2, 2008
PubMed
Summary
This summary is machine-generated.

We developed an orthonormal finite ridgelet transform (FRIT) for digital images. This new transform is efficient and superior to wavelets for denoising and approximating images with straight edges.

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Area of Science:

  • Image processing
  • Harmonic analysis
  • Applied mathematics

Background:

  • The ridgelet transform offers sparse representations for functions with line discontinuities.
  • Existing ridgelet transforms are not directly applicable to discrete, finite-size images.

Purpose of the Study:

  • To introduce an orthonormal version of the ridgelet transform for discrete images.
  • To develop a computationally efficient and invertible transform for image analysis.

Main Methods:

  • Construction of the finite ridgelet transform (FRIT) using the finite Radon transform (FRAT).
  • Novel ordering of FRAT coefficients to mitigate periodization effects.
  • Analysis of FRAT as a frame operator to derive frame bounds.

Main Results:

  • The proposed FRIT is invertible, nonredundant, and computed via fast algorithms.
  • FRIT provides a family of directional and orthonormal bases for images.
  • Numerical results demonstrate FRIT's effectiveness in image approximation and denoising.

Conclusions:

  • The finite ridgelet transform is a powerful tool for image processing, particularly for images with linear features.
  • FRIT outperforms the wavelet transform for approximating and denoising images containing straight edges.