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Related Concept Videos

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 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...
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...
State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
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...

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Related Experiment Video

Updated: Jun 18, 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

Steerable wavelet frames based on the Riesz transform.

Stefan Held1, Martin Storath, Peter Massopust

  • 1Zentrum Mathematik, Technische Universität München, Germany. held@ma.tum.de

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|November 26, 2009
PubMed
Summary
This summary is machine-generated.

We introduce the monogenic wavelet, an n-dimensional extension of the analytical wavelet compatible with rotations. This new tool decomposes wavelet coefficients into amplitude, phase, and phase direction for advanced image analysis.

Related Experiment Videos

Last Updated: Jun 18, 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

Area of Science:

  • Multidimensional signal processing
  • Image analysis and reconstruction
  • Hypercomplex analysis

Background:

  • The 1-D analytical wavelet is a fundamental tool in signal processing.
  • Extending wavelet analysis to higher dimensions while maintaining rotational compatibility is a significant challenge.
  • Existing methods often lack directional steerability.

Purpose of the Study:

  • To extend the concept of the analytical wavelet to n-dimensions.
  • To develop a rotation-compatible wavelet transform for image analysis.
  • To introduce a method for decomposing wavelet coefficients into amplitude, phase, and phase direction.

Main Methods:

  • Definition of the monogenic wavelet based on the hypercomplex monogenic signal.
  • Utilizing Riesz transforms and isotropic wavelet frames.
  • Application of Clifford frames to demonstrate wavelet frame generation.
  • Development of directionally steerable wavelet frames.

Main Results:

  • The monogenic wavelet is constructed as a rotation-compatible extension of the 1-D analytical wavelet.
  • Wavelet coefficients are decomposed into amplitude, phase, and phase direction.
  • The monogenic wavelet generates a wavelet frame using Clifford frames.
  • The resulting wavelet frames are steerable with respect to direction.

Conclusions:

  • The monogenic wavelet provides a versatile framework for n-dimensional signal and image analysis.
  • This approach enables directional steerability in wavelet transforms.
  • Applications in descreening and contrast enhancement demonstrate practical utility.