Convolutions for Image Processing

Convolutions for Image Processing
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Introduction

Defined:
- In its most general usage, a convolution is a product of two functions
- As we are using it here, a convolution is a way of calculating the color of a pixel in the destination image from that same pixel and its neighbors in the source image (the product of the source and a kernel)
Kernels:
- A kernel contains weights that are applied to the source pixels to obtain the destination pixels

The Kernel as a Grid/Matrix

images/source-destination-images.gif

images/kernel.gif

images/spatial_convolution.gif

A Formal Definition

Letting \(s_{i,j}\) denote the value in source pixel \((i,j)\), \(k_{r,c}\) denote the value in element \((r, c)\) of the kernel, and \(d_{i,j}\) denote the value in destination pixel \((i, j)\), a 3x3 spatial convolution can be defined as follows:

\( d_{i,j} = \sum_{r=-1}^{1} \sum_{c=-1}^{1} s_{i+r, j+c} k_{r,c} \)

An Example

The Source
images/spatial_convolution_source.gif

The Kernel
images/spatial_convolution_kernel.gif

The Convolution
images/spatial_convolution_destination1.gif

The Identify Kernel

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A Blurring Kernel

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A Blurring Kernel (cont.)

An Edge-Detection Kernel

\( 0-1 0 -1 4-1 0-1 0 \)

What if the value of the source pixel and its neighbors is given by \(x\)?
What if the source pixel is brighter than its vertical and horizontal neighbors?

A Sharpening Kernel

Find the edges (i.e., start with an edge detection kernel)
Add the edges to the original image (i.e., add an identity kernel)

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There's Always More to Learn