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Convolution: Math, Graphics, and Discrete Signals

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...

Deconvolution

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...

Convolution Properties I

Convolution Properties I

Convolution computations can be simplified by utilizing their inherent properties.
The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:

Convolution Properties II

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...

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Cortical Bone Assessment Using Ultrasonic Guided Waves: A Reproducibility Study in a Healthy Population

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Published on: January 31, 2025

Multi-Modal Convolutional Parameterisation Network for Guided Image Inverse Problems.

Mikolaj Czerkawski¹, Priti Upadhyay¹, Christopher Davison¹

¹Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1XW, UK.

Journal of Imaging

|March 27, 2024

Summary

This summary is machine-generated.

Deep internal learning solves image inverse tasks using a single sample. A new Multi-Modal Convolutional Parameterisation Network (MCPN) improves performance on multi-modal data for tasks like satellite image inpainting and super-resolution.

Keywords:

image inpainting image super-resolution image synthesis internal learning multi-modal learning

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Deep Neural Networks for Image-Based Dietary Assessment

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

Area of Science:

Computer Vision
Deep Learning
Image Processing

Background:

Deep internal learning utilizes deep neural networks trained on the sample itself, not external datasets, for image inverse tasks like inpainting and super-resolution.
Existing methods, such as Deep Image Prior, are limited when dealing with multi-modal image data.
Multi-modal representations are crucial in domains like satellite image processing.

Purpose of the Study:

To propose a novel Multi-Modal Convolutional Parameterisation Network (MCPN) architecture.
To enhance deep internal learning for multi-modal image inverse problems.
To demonstrate the effectiveness of MCPN over single-modal approaches.

Main Methods:

Developed MCPN, a convolutional neural network integrating a shared core network with modality-specific heads.
MCPN approximates shared information across multiple data modes.
Applied MCPN to guided image inverse problems, including inpainting and super-resolution.

Main Results:

MCPN significantly outperforms single-mode convolutional parameterisation networks.
The proposed method shows superior performance on multi-modal guided image inverse tasks.
Demonstrated improved results for inpainting and super-resolution using multi-modal data.

Conclusions:

MCPN effectively handles multi-modal data for image inverse tasks.
The proposed network architecture advances deep internal learning for complex image processing.
MCPN offers a significant improvement for satellite image processing and other multi-modal applications.