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Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations. [1] The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures).
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods.
the solution of the initial-value problem = is the convolution (). Through the superposition principle , given a linear ordinary differential equation (ODE), L y = f {\displaystyle Ly=f} , one can first solve L G = δ s {\displaystyle LG=\delta _{s}} , for each s , and realizing that, since the source is a sum of delta functions , the solution ...
The matrix operation being performed—convolution—is not traditional matrix multiplication, despite being similarly denoted by *. For example, if we have two three-by-three matrices, the first a kernel, and the second an image piece, convolution is the process of flipping both the rows and columns of the kernel and multiplying locally ...
In artificial neural networks, a convolutional layer is a type of network layer that applies a convolution operation to the input. Convolutional layers are some of the primary building blocks of convolutional neural networks (CNNs), a class of neural network most commonly applied to images, video, audio, and other data that have the property of uniform translational symmetry.
In general, the objective of deconvolution is to find the solution f of a convolution equation of the form: f ∗ g − 1 = h {\displaystyle f*g^{-1}=h\,} Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with a filter or distortion function g , before we recorded it.
In mathematics and physics, the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.