Convolution in the One Dimensional Domain

Convolution is a widely used mathematical operator that processes an image by computing—for each pixel—a weighted sum of the values of that pixel and its neighbors. Depending on the choice of weights, a wide variety of image processing operations can be implemented. Convolution and correlation are the two fundamental mathematical operations involved in linear neighborhood-oriented image processing algorithms. The two operations differ in a very subtle way.

Convolution in the One-Dimensional Domain

The convolution between two discrete one-dimensional (1D) arrays A(x) and B(x), denoted by A ∗ B, is mathematically described by the equation :
Convolution (One-Dimensional Domain)

Example

Step 1 :


In the following example A is the sample matrix on which operation is to be performed and B is the convolution matrix which is used to perform operation on matrix A.
Convolution
We have to calculate A*B. Now in order to cover all matrix elements we need to shift our convolution matrix (B) one to the left side. And the calculation will be as :
Convolution (Step 1)
Step 2 (after shifting convolution to the one step right)


Convolution (Step 2)
Step 3 (continue shifting convolution to the one step right)



Step 4 (continue shifting convolution to the one step right)


Convolution (Step 4)
Step 5 (continue shifting convolution to the one step right)


Convolution (Step 5)
Step 6 (continue shifting convolution to the one step right)


Convolution (Step 6)
Step 7 (continue shifting convolution to the one step right and one step out of the range of sample matrix)


Convolution (Step 7)

The final result of the convolution operation is the array [4 7 11 15 13 7 2]

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