conv2

R2026b

2-D convolution

Syntax

C = conv2(A,B)

C = conv2(u,v,A)

C = conv2(___,shape)

Description

C = conv2(A,B) returns the two-dimensional convolution of matrices A and B.

• If A is a matrix and B is a row vector (or A is a row vector and B is a matrix), then C is the convolution of each row of the matrix with the vector.

• If A is a matrix and B is a column vector (or A is a column vector and B is a matrix), then C is the convolution of each column of the matrix with the vector.

C = conv2(u,v,A) returns the 2-D convolution of A with the outer product of the vectors u and v. The vectors u and v can be row or column vectors. This operation is equivalent to convolving A with the convolution of the column-vector form of u and the row-vector form of v, that is, conv2(A,conv2(u(:),v(:)')).

C = conv2(___,shape) returns a subsection of the convolution according to shape. For example, C = conv2(A,B,"same") returns the central part of the convolution, which is the same size as A.

Examples

2-D Convolution

In applications such as image processing, it can be useful to compare the input of a convolution directly to the output. The conv2 function allows you to control the size of the output.

Create a 3-by-3 random matrix A and a 4-by-4 random matrix B. Compute the full convolution of A and B, which is a 6-by-6 matrix.

A = rand(3);
B = rand(4);
Cfull = conv2(A,B)

Cfull = 6×6

    0.7861    1.2768    1.4581    1.0007    0.2876    0.0099
    1.0024    1.8458    3.0844    2.5151    1.5196    0.2560
    1.0561    1.9824    3.5790    3.9432    2.9708    0.7587
    1.6790    2.0772    3.0052    3.7511    2.7593    1.5129
    0.9902    1.1000    2.4492    1.6082    1.7976    1.2655
    0.1215    0.1469    1.0409    0.5540    0.6941    0.6499

Compute the central part of the convolution Csame, which is a submatrix of Cfull with the same size as A. Csame is equal to Cfull(3:5,3:5).

Csame = conv2(A,B,"same")

Csame = 3×3

    3.5790    3.9432    2.9708
    3.0052    3.7511    2.7593
    2.4492    1.6082    1.7976

Extract 2-D Pedestal Edges

The Sobel edge-finding operation uses a 2-D convolution to detect edges in images and other 2-D data.

Create and plot a 2-D pedestal with interior height equal to one.

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

Convolve the rows of A with the vector u, and then convolve the rows of the result with the vector v. The convolution extracts the horizontal edges of the pedestal.

u = [1 0 -1]';
v = [1 2 1];
Ch = conv2(u,v,A);
mesh(Ch)

To extract the vertical edges of the pedestal, reverse the order of convolution with u and v.

Cv = conv2(v,u,A);
mesh(Cv)

Compute and plot the combined edges of the pedestal.

figure
mesh(sqrt(Ch.^2 + Cv.^2))

Input Arguments

A — Input array

vector | matrix

Input array, specified as a vector or matrix.

The input array A can be a sparse vector or matrix of type double, single, or logical. (since R2026b)

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

B — Second input array

vector | matrix

Second input array, specified as a vector or a matrix to convolve with A. The array B does not have to be the same size as A.

The input array B can be a sparse vector or matrix of type double, single, or logical. (since R2026b)

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

u — Input vector

row or column vector

Input vector, specified as a row or column vector.

The input vector u can be a sparse vector of type double, single, or logical. (since R2026b)

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

v — Second input vector

row or column vector

Second input vector, specified as a row or column vector.

The input vector v can be a sparse vector of type double, single, or logical. (since R2026b)

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

shape — Subsection of convolution

"full" (default) | "same" | "valid"

Subsection of the convolution, specified as one of these values:

• "full" — Return the full 2-D convolution.

• "same" — Return the central part of the convolution, which is the same size as A.

• "valid" — Return only parts of the convolution that are computed without zero-padded edges.

Output Arguments

C — 2-D convolution

vector | matrix

2-D convolution, returned as a vector or matrix. When A and B are matrices, then the convolution C = conv2(A,B) has size size(A)+size(B)-1. When [m,n] = size(A), p = length(u), and q = length(v), then the convolution C = conv2(u,v,A) has m+p-1 rows and n+q-1 columns.

When one or more input arguments to conv2 are of type single, then the output is of type single. Otherwise, conv2 converts inputs to type double and returns type double.

Data Types: double | single

Limitations

• For a column vector u and a row vector v, the syntax conv2(u,v,A) performs the convolution operation $$A\ast (u\otimes v)$$, where $$\ast$$ is the convolution operator and $$\otimes$$ is the outer product operator. Ideally, in the absence of round-off errors, this operation satisfies $$A\ast (u\otimes v)=A\ast (u\ast v)=(A\ast u)\ast v=(A\ast v)\ast u$$ due to the commutativity and associativity of convolution. However, because of round-off errors, the commutativity and associativity properties do not always hold. For this reason, operations such as conv2(u,v,A), conv2(A,u*v), and conv2(conv2(A,u),v) might produce different results.

More About

2-D Convolution

For discrete, two-dimensional matrices A and B, the following equation defines the convolution of A and B:

$$C(j,k)={\displaystyle \sum _p{\displaystyle \sum _{q}A(p,q)B(j-p+1,k-q+1)}}$$

p and q run over all values that lead to legal subscripts of A(p,q) and B(j-p+1,k-q+1).

Using this definition, conv2 calculates the direct convolution of two matrices, rather than the FFT-based convolution.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The conv2 function supports tall arrays with the following usage notes and limitations:

• If shape is "full" (default), then the inputs A and B must not be empty and only one them can be a tall array.

• If shape is "same" or "valid", then B cannot be a tall array.

• u and v cannot be tall arrays.

For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The conv2 function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The conv2 function supports GPU array input with these usage notes and limitations:

• Sparse gpuArray input is not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

The conv2 function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2026b: Perform convolutions using sparse inputs

The conv2 function now supports sparse inputs. You can perform convolutions using sparse inputs to significantly reduce memory usage and improve processing performance when the inputs contain many elements that are zero.