Convolution

Computational Photography WS 07/08Image Processing Basics

Björn Bollensdorff – bbolle@cs.tu-berlin.de

Overview

• Introduction (multiplication)• Discrete convolution• Continuous convolution• Properties• Filter• Excursion: frequency domain• 2D Convolution

Multiplication

• using a simpleroperation to generate ahigher order operation

• multiple summations• general formula

Introduction

Doing the same with functions

• Applying theoperations on eachvalue separated

• The result is afunction

Introduction

Multiplication and addition oftwo functions

f1+f2 f1• f2

Introduction

Convolution

• Operation that uses addition and multiplication• result is a function• It is a way to combine to functions• It is like weighting one function with the other• Flipping one function and then summing up the

products for each positions for a given offset n

Discrete Convolution

Flipping the function

f2(-k)

Discrete Convolution

Multiply and add

Discrete Convolution

Multiply and add

1•1 =1

Discrete Convolution

Multiply and add

1•3 + 2•1 =5

Discrete Convolution

Multiply and add

1•1 + 2•3 + 1•1 = 8

Discrete Convolution

Multiply and add

1•2 + 2•1 + 1•3 + 2•1 = 9

Discrete Convolution

Multiply and add

2•2 + 1•1 + 2•3 = 11

Discrete Convolution

Multiply and add

1•2 + 2•1 = 4

Discrete Convolution

Multiply and add

2•2 = 4

Discrete Convolution

Multiply and add

Discrete Convolution

Result

final length = length(f1(n)) + length(f2(n)) - 1

Discrete Convolution

ComparisonCrosscorrelationConvolution

Discrete Convolution

Continuous Convolution

• Summation becomes an integral

Continuous Convolution

Properties

• Commutative

• Associative

• Distributive

Properties

Filter

• signal y(n) is aconvolution of u(n)with the Transferfunction h(n)

• Filtering can be doneby the convolution oftwo signals

Filter

Excursion Frequency Domain

• Another representation of the same signal• Shows of which frequencies the signal consists• Frequency in images represents the intensity

changes

Excursion

Discrete Fourier Transformation

Example Rectangle Function

• the rectanglefunction in thespatial domainbecomes a sinc in thefrequency domain

• two overlaid sincs

Excursion

Example Lena

picture 1 picture 2

Excursion

Convolution Theorem

Convolution in spatial domain becomesmultiplication in frequency domain

Fast convolution using FFT

Properties

Derivation of the theorem

Properties

2D-Convolution

• We can do it in 2D too• usually one function is a small one and called

(convolution) kernel• sometimes only the cross correlation is used

2D-Convolution

What should we do with theborder?

• different approaches• ignore them ⇒ image gets smaller• suppose they are black• mirror the border• suppose the image continuous with the

last pixels

2D-Convolution

Example finding edges

10-120-210-1

Differentiating to find the steep parts of the picture

121000-1-2-1

2D-Convolution

Example Simple NoiseReduction

• cutting of the high frequency noise by low passfiltering with a sinc like kernel

• for not changing the aspect of the image thesum of the kernel must be 1

111151111

1/13

2D-Convolution

Sources

• Noll, P. Script (1999) Signale und Systeme. TU Berlin• Noll, P. Script (WS 2006/7) Nachrichten Übertragung I TU

Berlin• Albiol, A., Naranjo, V., Prades J., Tratamiento digital de la

señal, teoría y aplicaciones. Universidad Politécnica deValencia

• Picture 1, 2:http://www.vis.ne.jp/mt/archives/000680.html

• Applet: http://www.jhu.edu/~signals/convolve/index.html• Proof:

http://mathworld.wolfram.com/ConvolutionTheorem.html

Sources