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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 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...
Symmetry Elements in a Crystal01:27

Symmetry Elements in a Crystal

Crystal symmetry operations are isometric transformations that map objects onto indistinguishable copies while preserving distances, angles, and volumes. The simplest symmetry operation is translation, which shifts the entire infinite crystal lattice parallelly by a translation vector.Crystallographic rotations involve rotations by an angle of 2π/n around an axis without changing the positions of points on the axis. It is called the rotational axis of the symmetry, denoted by n. The combination...
Transformations of Functions II01:29

Transformations of Functions II

Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c, where c is a constant.

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Related Experiment Video

Updated: Jul 7, 2026

Quantifying Intermembrane Distances with Serial Image Dilations
07:45

Quantifying Intermembrane Distances with Serial Image Dilations

Published on: September 28, 2018

Image coding using translation invariant wavelet transforms with symmetric extensions.

J Liang, T W Parks

    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 introduces novel wavelet transform algorithms that overcome translation sensitivity in 2-D signals. These algorithms offer translation invariance, reduced edge effects, and size-limitedness for improved image coding.

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    Area of Science:

    • Signal Processing
    • Image Analysis
    • Applied Mathematics

    Background:

    • Conventional wavelet transforms exhibit translation sensitivity, impacting 2-D signal analysis.
    • Edge effects and size limitations are common challenges in signal processing applications.

    Discussion:

    • This work proposes novel wavelet transform algorithms designed for translation invariance.
    • The algorithms simultaneously address reduced edge effects and size-limitedness.
    • Symmetric extensions are incorporated into the translation invariant biorthogonal wavelet transform.

    Key Insights:

    • Developed wavelet transform algorithms achieve translation invariance for 2-D signals.
    • The proposed methods effectively reduce edge effects and ensure size-limitedness.
    • Demonstrated successful application in image coding with significant improvements.

    Outlook:

    • Further research can explore advanced applications in image compression and feature extraction.
    • Optimization of the transform for real-time processing is a potential future direction.
    • Investigating the theoretical properties for diverse signal types could expand applicability.