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Related Concept Videos

Downsampling01:20

Downsampling

When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
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...
Deconvolution01:20

Deconvolution

Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique is vital in scenarios where the system's characteristics are unknown, and they must be inferred from the observable signals.
Deconvolution involves several mathematical techniques to derive the impulse response. One common approach is polynomial division. In this method, the input and output sequences are treated as coefficients of...
Properties of Fourier series II01:21

Properties of Fourier series II

Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...
Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...

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Related Experiment Video

Updated: Jul 7, 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

Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage.

A Charnbolle1, R A DeVore, N Y Lee

  • 1CEREMADE, Univ. de Paris-Dauphine, France. antonin.chambolle@ceremade.dauphine.fr

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 16, 2008
PubMed
Summary
This summary is machine-generated.

This study links wavelet image processing to variational problems, showing wavelet shrinkage minimizes a specific energy functional. New methods offer improved noise removal and parameter selection for better image quality.

Related Experiment Videos

Last Updated: Jul 7, 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:

  • Image Processing
  • Variational Methods
  • Wavelet Theory

Background:

  • Wavelet-based algorithms are crucial in image processing.
  • Variational problems offer a framework for image analysis and restoration.
  • Existing methods like VisuShrink and SureShrink have limitations in noise removal.

Purpose of the Study:

  • To establish the connection between wavelet-based image processing and variational problems.
  • To derive and analyze wavelet shrinkage as a minimizer of a defined variational problem.
  • To develop improved error bounds and parameter selection for wavelet-based noise removal.

Main Methods:

  • Derivation of wavelet-based algorithms as minimizers of variational problems.
  • Application of nonlinear wavelet image compression theory in L(2)(I).
  • Analysis of images corrupted with i.i.d., mean zero, Gaussian noise.

Main Results:

  • Wavelet shrinkage is shown to be the exact minimizer of a specific variational problem.
  • Accurate error bounds for noise removal using wavelet shrinkage are derived.
  • A novel signal-to-noise ratio (SNR) is introduced for better visual noise perception.
  • Proposed shrinkage parameters, based on image properties (alpha and norm), outperform VisuShrink.

Conclusions:

  • The study provides a strong theoretical foundation linking wavelets and variational methods in image processing.
  • The proposed noise removal technique and parameter selection offer superior performance compared to VisuShrink.
  • While effective, the method may not always surpass SureShrink, which uses level-dependent parameters.