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Related Concept Videos

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...
Properties of Fourier Transform II01:24

Properties of Fourier Transform II

The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
The Frequency Shifting property of Fourier Transforms highlights that a shift in the frequency domain corresponds to a phase shift in the time domain. Mathematically, if x(t) has...
Properties of Fourier Transform I01:21

Properties of Fourier Transform I

The application of Fourier Transform properties in radio broadcasting is multifaceted, enabling significant advancements in the way signals are transmitted and received. Key areas where these properties are utilized include simultaneous multi-channel transmission, audio clip speed adjustments, live broadcast delays for different time zones, audio frequency adjustments, and signal demodulation.
In radio broadcasting, multiple audio signals often need to be transmitted simultaneously. The Fourier...
Parseval's Theorem for Fourier transform01:15

Parseval's Theorem for Fourier transform

Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
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Cylinders in Three-Dimensional Space01:28

Cylinders in Three-Dimensional Space

A cylindrical surface is generated when a two-dimensional profile curve is translated along a straight line in three-dimensional space. The translated copies of the curve form a surface composed of parallel rulings, each oriented in the same fixed direction. This construction allows many three-dimensional forms to be described using relatively simple planar equations.In Cartesian coordinates, a cylindrical surface is often recognized by an equation that omits one of the three variables. For...
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...

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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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Published on: December 30, 2025

A New k-Space Model for Non-Cartesian Fourier Imaging.

Chin-Cheng Chan1, Justin P Haldar1

  • 1Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, USA.

IEEE Transactions on Computational Imaging
|July 9, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a novel Fourier-domain model for Fourier imaging reconstruction, outperforming traditional voxel-based methods. The new model enhances image quality and computational efficiency in non-Cartesian MRI.

Keywords:
Linear Basis ExpansionsModel-Based Fourier ImagingNon-Cartesian MRISplines

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Area of Science:

  • Medical Imaging
  • Signal Processing
  • Computational Science

Background:

  • Model-based approaches are popular for Fourier imaging reconstruction, incorporating physical constraints and machine learning priors.
  • The conventional voxel-based model, while widely used, suffers from high computational costs, slow convergence, and artifacts.
  • Existing methods face limitations like undesirable approximation, wrap-around, and nullspace issues.

Purpose of the Study:

  • To reexamine the limitations of the traditional voxel-based model for Fourier imaging reconstruction.
  • To propose a new Fourier-domain basis expansion model to overcome existing and newly identified issues.
  • To improve image quality and computational efficiency in Fourier imaging reconstruction.

Main Methods:

  • Developed a novel Fourier-domain basis expansion model for image reconstruction.
  • Compared the new model against the standard image-domain voxel-based approach.
  • Evaluated the model's performance in non-Cartesian Magnetic Resonance Imaging (MRI) reconstruction.

Main Results:

  • The proposed Fourier-domain model demonstrates improved resilience to limitations of the voxel-based approach.
  • Results show enhanced image quality with reduced artifacts.
  • The new model offers reduced computational complexity, leading to faster computations and improved convergence.

Conclusions:

  • The Fourier-domain basis expansion model represents a significant advancement over traditional voxel-based methods.
  • This new approach enhances both image quality and computational efficiency in Fourier imaging.
  • The findings are particularly relevant for applications like non-Cartesian MRI reconstruction.