Convolution

Convolution is a mathematical operation that combines two signals and outputs a third signal.

Assuming we have two functions, \(f(t)\) and \(g(t)\), convolution is an integral that expresses the amount of overlap of one function \(g\) as it is shifted over function \(f\)

Convolution is expressed as:

$$(f*g)(t) \approx^{def} \int_{-\infty}^{\infty}f(\tau)g(t-\tau)dr$$

Depending on the application of convolution, functions may be replaced with signals, images, or other types of data. Convolution and its applications can be implemented in several ways in MATLAB^®.

Convolution in Signal Processing

Convolution is used in digital signal processing to study and design linear time-invariant (LTI) systems such as digital filters.

The output signal, \(y[n]\), in LTI systems is the convolution of the input signal, \(x[n]\) and impulse response \(h[n]\) of the system

In practice, the convolution theorem is used to design the filter in the frequency domain. The convolution theorem states that the convolution in the time domain equals the multiplication in the frequency domain.

MATLAB functions such as conv and filter allow you to perform convolution and build filters from scratch. Signal Processing Toolbox™ and DSP System Toolbox™ have several functions and Simulink^® blocks for direct implementation of digital filters such as designfilt, lowpass, and highpass.

Convolution in Image Processing

In image processing, convolutional filtering can be used to implement algorithms such as edge detection, image sharpening, and image blurring.

This is done by selecting the appropriate kernel (convolution matrix).

Image Processing Toolbox™ has functions such as fspecial and imfilterv to design filters to emphasize certain features or remove other features in images.

Convolution Neural Networks (CNNs)

Convolution plays a key role in convolutional neural networks (CNNs). CNNs are a type of deep network commonly used to analyze images. CNNs eliminate the need for manual feature extraction, which is why they work very well for complex problems like image classification and medical image analysis. CNNs are effective for non-image data analysis such as audio, time-series, and signal data.

CNNs have several layers, the most common of which are convolution, ReLu, and pooling.

Convolution layers act as filters – each layer applies a filter and extracts specific features from the image. These filter values are learned by the network when the network is trained. The initial layers typically extract low-level features while the deeper layers extract high-level features from the data.

For more information on convolution and its applications in signal processing, image processing, deep learning, and other areas, see Signal Processing Toolbox, DSP System Toolbox, Image Processing Toolbox and Deep Learning Toolbox™ for use with MATLAB.