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# conv2

2-D convolution

## Syntax

C = conv2(A,B)
C = conv2(h1,h2,A)
C = conv2(...,shape)

## 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(h1,h2,A) first convolves each column of A with the vector h1 and then convolves each row of the result with the vector h2. The size of C is determined as follows: if n1 = length(h1) and n2 = length(h2), 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).

 Note:   All numeric inputs to conv2 must be of type double or single.

## Examples

### Shape for Subsection of 2-D Convolution

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```

### Extract Edges from Raised Pedestal

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);
figure, 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))
```

expand all

### 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

$c\left({n}_{1},{n}_{2}\right)=\sum _{{k}_{1}=-\infty }^{\infty }\sum _{{k}_{2}=-\infty }^{\infty }a\left({k}_{1},{k}_{2}\right)b\left({n}_{1}-{k}_{1},{n}_{2}-{k}_{2}\right)$

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).