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Related Concept Videos

Properties of the z-Transform I01:17

Properties of the z-Transform I

The z-transform is a fundamental tool in digital signal processing, enabling the analysis of discrete-time systems through its various properties. It is an invaluable tool for analyzing discrete-time systems, offering a range of properties that simplify complex signal manipulations. One fundamental property is linearity. For any two discrete-time signals, the z-transform of their linear combination equals the same linear combination of their individual z-transforms. This property is essential...
Properties of the z-Transform II01:16

Properties of the z-Transform II

The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency...
Definition of z-Transform01:26

Definition of z-Transform

The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is an essential analytical tool, analogous to the Laplace transform used in continuous-time systems. It plays a crucial role in the analysis of signals and systems, complementing the discrete-time Fourier transform. Both the z-transform and the Laplace transform convert differential or difference equations into algebraic equations, simplifying the process of solving complex problems.
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...
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 series I01:20

Properties of Fourier series I

The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...

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Related Experiment Video

Updated: Jun 17, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Wavelet steerability and the higher-order Riesz transform.

Michael Unser1, Dimitri Van De Ville

  • 1Biomedical Imaging Group (BIG), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland. michael.unser@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|December 25, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces a novel framework for designing steerable, reversible signal transformations in multiple dimensions. The generalized Riesz transform enables steerable wavelet frames with perfect reconstruction, advancing signal processing techniques.

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Last Updated: Jun 17, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Published on: July 19, 2016

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Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing

Published on: February 13, 2018

Area of Science:

  • Signal Processing
  • Mathematical Physics
  • Image Analysis

Background:

  • Designing steerable and reversible signal transformations (frames) in multiple dimensions is crucial for advanced signal processing.
  • Existing methods, like Simoncelli's steerable pyramid, primarily rely on filterbank design.
  • A continuous-domain framework for such transformations is needed.

Purpose of the Study:

  • To establish a general continuous-domain framework for designing steerable, reversible signal transformations in multiple dimensions (d ≥ 2).
  • To introduce and analyze a novel Nth-order extension of the Riesz transform with desirable properties.
  • To demonstrate the application of this framework in creating steerable wavelet frames.

Main Methods:

  • Introduction of a self-reversible, Nth-order extension of the Riesz transform.
  • Mathematical proof of the transform's properties: shift-invariance, scale-invariance, inner-product preservation, and steerability.
  • Development of a perfect reconstruction filterbank algorithm based on the proposed transform.

Main Results:

  • The generalized Riesz transform maps wavelet frames/bases into steerable wavelet frames while preserving frame bounds.
  • The framework allows for wavelets with arbitrary steerability order in any dimension.
  • A novel family of multidimensional Riesz-Laplace wavelets was designed, mimicking Nth-order Gaussian derivatives.

Conclusions:

  • The proposed framework provides a functional counterpart to filterbank-based steerable pyramids.
  • This approach enables the design of highly flexible and steerable multidimensional wavelet transformations.
  • The Riesz-Laplace wavelets offer a new tool for analyzing signals with directional and scale-invariant properties.