conv2 - 2-D convolution

Syntax

Description

C = conv2(A,B) computes the two-dimensional convolution of matrices A and B. If one of these matrices describes a two-dimensional finite impulse response (FIR) filter, the other matrix is filtered in two dimensions. The size of C is determined as follows: if [ma,na] = size(A), [mb,nb] = size(B), and [mc,nc] = size(C), then mc = max([ma+mb-1,ma,mb]) and nc = max([na+nb-1,na,nb]).

C = conv2(hcol,hrow,A) first convolves A with the vector hcol along the rows and then with the vector hcol along the columns. The size of C is determined as follows: if n1 = length(hcol) and n2 = length(hrow), then mc = max([ma+n1-1,ma,n1]) and nc = max([na+n2-1,na,n2]).

C = conv2(...,'shape') returns a subsection of the two-dimensional convolution, as specified by the shape parameter:

full	Returns the full two-dimensional convolution (default).
same	Returns the central part of the convolution of the same size as A.
valid	Returns only those parts of the convolution that are computed without the zero-padded edges. Using this option, size(C) = max([ma-max(0,mb-1),na-max(0,nb-1)],0).

Algorithms

conv2 uses a straightforward formal implementation of the two-dimensional convolution equation in spatial form. If a and b are functions of two discrete variables, n₁ and n₂, then the formula for the two-dimensional convolution of a and b is

In practice however, conv2 computes the convolution for finite intervals.

Note that matrix indices in MATLAB software always start at 1 rather than 0. Therefore, matrix elements A(1,1), B(1,1), and C(1,1) correspond to mathematical quantities a(0,0), b(0,0), and c(0,0).

Examples

Example 1

For the 'same' case, conv2 returns the central part of the convolution. If there are an odd number of rows or columns, the "center" leaves one more at the beginning than the end.

This example first computes the convolution of A using the default ('full') shape, then computes the convolution using the 'same' shape. Note that the array returned using 'same' corresponds to the red highlighted elements of the array returned using the default shape.

A = rand(3); 
B = rand(4); 
C = conv2(A,B)   % C is 6-by-6

C =
   0.1838  0.2374  0.9727  1.2644  0.7890  0.3750
   0.6929  1.2019  1.5499  2.1733  1.3325  0.3096
   0.5627  1.5150  2.3576  3.1553  2.5373  1.0602
   0.9986  2.3811  3.4302  3.5128  2.4489  0.8462
   0.3089  1.1419  1.8229  2.1561  1.6364  0.6841
   0.3287  0.9347  1.6464  1.7928  1.2422  0.5423

Cs = conv2(A,B,'same')   % Cs is the same size as A: 3-by-3
Cs =
   2.3576  3.1553  2.5373
   3.4302  3.5128  2.4489
   1.8229  2.1561  1.6364

Example 2

In image processing, the Sobel edge finding operation is a two-dimensional convolution of an input array with the special matrix

s = [1 2 1; 0 0 0; -1 -2 -1];

These commands extract the horizontal edges from a raised pedestal.

A = zeros(10);
A(3:7,3:7) = ones(5);
H = conv2(A,s);
mesh(H)

Transposing the filter s extracts the vertical edges of A.

V = conv2(A,s');
figure, mesh(V)

This figure combines both horizontal and vertical edges.

figure
mesh(sqrt(H.^2 + V.^2))

See Also

conv | convn | filter2 | xcorr2

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