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Related Concept Videos

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...
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.
Vector Transformation in Rotating Coordinate Systems01:16

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Consider a vector rotating about an axis with an angular velocity, such that its tip sweeps a circular path.
Convolution Properties II01:17

Convolution Properties II

The important convolution properties include width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2 seconds and 1 second results in a function with a width of 3 seconds.
The area property asserts that the area under the...
Properties of DTFT I01:24

Properties of DTFT I

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...
Basic Operations on Signals01:22

Basic Operations on Signals

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Related Experiment Video

Updated: Jul 7, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Coordinate-transformed filter for shift-invariant and scale-invariant pattern recognition.

D Cojoc, M T Molina, J García

    Applied Optics
    |July 10, 1997
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a novel phase-only filter (POF) for pattern recognition. The enhanced POF achieves shift and scale invariance, improving target detection across different sizes.

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    Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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    Published on: August 30, 2013

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    Published on: February 23, 2017

    Area of Science:

    • Optics and Photonics
    • Computer Vision
    • Signal Processing

    Background:

    • Traditional pattern recognition methods struggle with variations in object scale.
    • Phase-only filters (POFs) offer advantages in optical pattern recognition but often lack scale invariance.

    Purpose of the Study:

    • To develop a shift- and scale-invariant pattern recognition technique using a modified phase-only filter.
    • To enable robust detection of targets regardless of their size in optical systems.

    Main Methods:

    • A variable radial coordinate transformation was applied to the phase-only filter (POF).
    • The transformation was dependent on the target spectrum's angular energy distribution.
    • The filter was radially stretched to ensure consistent energy contribution to the correlation peak for various target scales.

    Main Results:

    • The modified POF demonstrated shift and scale invariance in pattern recognition.
    • Computer simulations and optical experiments validated the filter's performance.
    • A maximum recognition ratio of 1:1.5 for scaled objects was achieved.

    Conclusions:

    • The proposed variable radial coordinate transformation effectively enhances POFs for scale-invariant pattern recognition.
    • This method provides a robust solution for detecting targets of varying sizes in optical and computational systems.