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Related Concept Videos

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.
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...
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...
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 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...
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]...

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Related Experiment Video

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Automated Joint Space Detection Improves Bone Segmentation Accuracy
06:45

Automated Joint Space Detection Improves Bone Segmentation Accuracy

Published on: November 28, 2025

Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation.

C Herley1, Z Xiong, K Ramchandran

  • 1Hewlett-Packard Co., Palo Alto, CA.

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|January 1, 1997
PubMed
Summary
This summary is machine-generated.

This study introduces efficient pruning algorithms for image representation using filterbank trees. These algorithms find optimal tree structures for signal adaptive compression, achieving competitive compression rates without prior signal knowledge.

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Published on: August 30, 2013

Area of Science:

  • Digital Image Processing
  • Signal Processing
  • Computer Vision

Background:

  • Traditional wavelet trees offer good image representation but are often suboptimal.
  • Wavelet packet and double-tree libraries improve representation by allowing frequency-selective and spatially varying trees.
  • Existing methods have limitations in the size and adaptability of the basis library.

Purpose of the Study:

  • To develop efficient pruning algorithms for selecting space-varying filterbank tree representations.
  • To minimize additive cost functions, such as energy compaction and quantization distortion, for image representation.
  • To enhance signal adaptive compression schemes through improved basis selection.

Main Methods:

  • Development of efficient pruning algorithms for one- and two-dimensional filterbank trees.
  • Expansion of the library of bases beyond single-tree and double-tree approaches.
  • Application of algorithms to find the least-cost expansion using a rate-distortion cost function.

Main Results:

  • The new algorithms efficiently identify optimal tree structures from significantly larger libraries.
  • The proposed method overcomes the spatial variation constraints of double-tree bases.
  • The signal adaptive compression scheme demonstrates effectiveness across diverse image types.

Conclusions:

  • The developed pruning algorithms enable superior image representation through optimized filterbank trees.
  • The resulting compression scheme is universal, performing well without signal modeling or training data.
  • Experimental results show competitive compression rates compared to training-based methods.