2-D Convolution

Compute 2-D discrete convolution of two input matrices

Libraries:
Computer Vision Toolbox / Filtering

Description

The 2-D Convolution block computes the two-dimensional convolution of two input matrices. Assume that matrix A has dimensions (Ma, Na) and matrix B has dimensions (Mb, Nb). When the block calculates the full output size, the equation for the 2-D discrete convolution is:

$C (i, j) = \sum_{m = 0}^{(M a - 1)} \sum_{n = 0}^{(N a - 1)} A (m, n) * B (i - m, j - n)$

where $0 \leq i < M a + M b - 1$ and $0 \leq j < N a + N b - 1$ .

Examples

Blur Image Using Gaussian Kernel

Perform 2-D convolution to blur an image using the Gaussian kernel.

Open Script

Ports

Input

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I1 — Input matrix
matrix

Input matrix, specified as either a matrix of intensity values or a matrix that represents one plane of an RGB video stream.

I2 — Input matrix
matrix

Input matrix, specified as either a matrix of intensity values or a matrix that represents one plane of an RGB video stream.

Output

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Output — Convolution
matrix

Convolution of the input matrices, returned as a matrix.

Dependencies

The dimensions of the output are dictated by the Output size parameter.
If the data type of the input is floating point, the output of the block is also floating point.
If all(size(I1)<size(I2)), the block returns an error.

Parameters

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Main Tab

Output size — Output size
matrix

Dimensions of the output.

The table describes the block output, given the following input dimensions:

I1 — (Ma, Na)
I2 — (Mb, Nb)

Output size	Output	Output Dimension
`Full`	Full two-dimensional convolution	(Ma+Mb-1, Na+Nb-1).
`Same as input port I1`	Central part of the convolution with the same dimensions as the input at port I1
`Valid`	Only the parts of the convolution that are computed without the zero-padded edges of any input.	(Ma-Mb+1, Na-Nb+1)

Normalized output — Normalized output
matrix

Normalize output by dividing the output by sqrt(sum(dot(I1p,I1p))*sum(dot(I2,I2))), where I1p is the portion of the I1 matrix that aligns with the I2 matrix.

Note

When you select the Normalized output check box, the block input cannot be fixed point.

Data Types Tab

For details on the fixed-point block parameters, see Specify Fixed-Point Attributes for Blocks (DSP System Toolbox).

Block Characteristics

Data Types	`double` \| `fixed point` \| `integer` \| `single`
Multidimensional Signals	`no`
Variable-Size Signals	`yes`

Algorithms

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2-D Convolution

In convolution, the value of an output element is computed as a weighted sum of neighboring elements.

For example, suppose the first input matrix represents an image and is defined as:

I1 = [17  24   1   8  15
     23   5   7  14  16
      4   6  13  20  22
     10  12  19  21   3
     11  18  25   2   9]

The second input matrix also represents an image and is defined as:

I2 = [8   1   6
     3   5   7
     4   9   2]

The following figure shows how to compute the (1,1) output element by using these steps:

Rotate the second input matrix, I2, 180 degrees around its center element.
Slide the center element of I2 so that it lies on top of the (0,0) element of I1.
Multiply each element of the rotated I2 matrix by the element of I1 underneath.
Sum the individual products from step 3.

The (1,1) output element is $0 \cdot 2 + 0 \cdot 9 + 0 \cdot 4 + 0 \cdot 7 + 17 \cdot 5 + 24 \cdot 3 + 0 \cdot 6 + 23 \cdot 1 + 5 \cdot 8 = 220$ .

Computing the (1,1) Output of Convolution

The normalized convolution of the (1,1) output element is 220/sqrt(sum(dot(I1p,I1p))*sum(dot(I2,I2))) = 0.3459, where I1p = [0 0 0; 0 17 24; 0 23 5].

Output Sizes

The equations below describe how the block computes the convolution depending on how you set the Output size parameter. For the cases below, the inputs are set as:

I1 — dimensions (4,3)
I2 — dimensions (2,2)
When Output size is set to Full, the block uses the following equations:
The resulting matrix is:
When Output size is set to Same as input port I1, the output is the central part of $C f u l l$ with the same dimensions as the input at port I1, (4,3). However, since a 4-by-3 matrix cannot be extracted from the exact center of $C f u l l$ , the block leaves more rows and columns on the top and left side of the $C f u l l$ matrix and outputs:
When the Output size is set to Valid, the block uses the following equations to determine the number of rows and columns of the output matrix:
In this case, it is always possible to extract the exact center of $C f u l l$ . Therefore, the block outputs: