Convolutions

Craig Rykal John Rivera Justin Malaise Jeff Swanson Ben Rougier

Presented By { From{ University of Wisconsin

STOUT

AbstractThe convolution of two functions is an important concept in a number of areas of pure and applied mathematics such as Fourier Analysis, Differential Equations, Approximation Theory, and Image Processing. Nevertheless convolutions often seem unintuitive and difficult to grasp for beginners. This project explores the origins of the convolution concept as well as some computer graphic and physical interpretations of convolution which illustrate various ways the technique of smoothing can be used to solve some real world problems.

Under the direction of Dr. Steve Deckelman

Origins and Interpretations of the Concept of

Convolutions

Convolutions

R

dxxtkxftF )()()(

Convolutions can be thought of as a method of averaging unruly functions.

Unruly Functions include:

• Discontinuous functions• Functions with sharp or jagged edges

Weighted Averages

n

jjj y

1

Origin of ConvolutionsOrigin of Convolutions

Substitute into the function to receive)(xfy

n

jjj xf

1

)(

Now we need

n

jjj xf

1

)(

Therefore we create a new function

to return a function

Which returns a function

n

jjj xxf

1

)(

One more substitution of )(xg jj Gives us the function:

n

jjj xxfxg

1

)()(

This function will give us a new discrete function, but we need this function in continuous form

To accomplish this, change the function to an integration

This gives us:

R

dttxftg )()(

Which is a Convolution!

ExamplesExamples Take the function

xh

hxh

-hx

xf

for 0

for 1

for 0

)(

Convolve the function with itself to receive the new function

th

htth

thht

ht

tF

2for 0

20for 2

02for 2

2 for 0

)(

One more convolution gives use the continuous function

th

hthht

hthht

hthht

ht

tG

3 0

3 2/)3(

3

3 2/)3(

3 0

)(2

22

2

Convolutions are also useful in smoothing functions of more than one dimension

R R

dxdyytxskyxftsF ),(),(),(

Antialiasing Aliasing

Blurred Deconvolution

Radius of the Earth

Earth

Sun

EratosthenesEratosthenes

2.7787)( ALength KM

c Circumference

360

2.7787

c

2

cr

2.7

360*767c

A

230 B.C 230 B.C Alexandria, EgyptAlexandria, Egypt

59.6103r

Parallax - any alteration in the relative apparent positions of objects

produced by a shift in the position of the observer. 

R = radius of Earth.

= angle between observation points with relation to the center of the

Earth.

P & Q are observation points on the Earth.

X = distance between the center of the sun and the center of the Earth.

How To find the Distance: 

If we know |PQ| => |PQ| = R= |PQ| / R

angle of center of sun to P & Q

R/X = Cos () => X = R/Cos () = R(Sec ()

 The only problem is that there must be a very accurate measurement of in order to get an accurate distance. Hipparchus (130 BC) and Ptolemy (150 AD) used the value of the diameter of the earth given by Eratosthenes (195 BC) and estimated the distance to be 10 million miles. We know that the distance is 93 million miles away. Thus, even close but not precise values give huge errors.

Figuring out the Distance of The Earth to the Sun Using Parallax

P

SUN

Earth

Q

R

X

WE SEE:

TRUE IMAGE(shape):

x0

otherwise. 0

disc) inside(x )(

xf

The Diameter of a StarThe Diameter of a Star

Where D(x) is the characteristic function of the unit circle.

10)(

,11)(

xforxD

xforxD

Fundamental Problem:

Calculate from f

)()( 0

xx

Dxf

Note: x are all unknown quantities.

The Photographic PlateThe Photographic Plate

O

Let: “Brightness” at point source O:' “Brightness” at point source Y

:)(XK t “Brightness” at X

|||| Y

:)(' YXK t “Brightness” at X arising from point source Y.

:)(XK tApparent Brightness

True Brightness

(At time t)

X

Y

YX

n ,..., ,2 1

YYY n,...,,

21

|||||||||||| ,...,,21

YYY n

“Brightness” of Yj given point sources

Point sources of light

Given are very small

)( X"at Brightness" A(x)n

1jj YX jtK

We Get:

)(1

YX jt

n

jj K

)()()( YdAYXYf K t

to

Let’s pass from discrete to the continuous model:

TRUE IMAGE(brightness) = f(x)

ACTUAL IMAGE = f*Kt

:)(Yf :)( YXK t :)(YdA

Brightness at point Y. Blurring effect(kernel) of the atmosphere at time t.

Integrate with respect to Y.

sun

1 a.u.d

earth

earth

star

d = sec()

1 a.u. = distance from the sun to the earth

is found experimentally

The Fundamental Problem:The Fundamental Problem:

Extract f from f*Kt where Kt is an unknown random function.

Labeyrie’s IdeaLabeyrie’s Idea::

Use Averaging and Fourier Transforms

Knowing d and will allow us to find the diameter of the star.

1. Averaging Kt:

Average the image received at various times, )(),...,2(),1( ntttTo get a fixed Kt or the “Average Blur”:

KKjt

n

j n )(1

1~

: AVERAGE BLUR

n

jjtKf

n 1)(

*1

Average the Convolutions, K jt

n

j nf

)(1

1*

~

* Kf

2. Using the Convolution Theorem for Fourier Transforms:

K ttf * : FOURIER TRANSFORM of our image(convolution)

K ttf : Under the Convolution Theorem for Fourier Transforms

is RANDOM, due to Kt being RANDOM.

)( jt

Take the sequence:

)()3()2()1(

,...,,,ntttt

fAre the zeros of !will have in common with all other ‘s

)( jt

The only zeros(roots) the ‘s

)( jt

Superimposing

)( jt

forms

K jt

n

j

n

j jtf

)(11 )(

)ζ()ζ()ζ( Clearly the zeros of )ζ(

f

Will stand out as the zeros of )ζ(

)(ε

)ζ(2

xiζπ20)( xdAxx

Df e

Substitute y = x – x0

)(ε

)ζ(2

)xy(π2 0)( ydAy

Df e i

Move out the constants

)()ε

()ζ(2

yiζπ2xζiπ2 0

ydAy

Df ee

)()()ζ(2

wεiζπ2xζiπ22 0

ε wdAwDf ee

Substitute w = y/ y = w

By the definition of Fourier Transforms

)εζ()()(2

wεiζπ2

DwdAwD e

Which should be visually evident from our picture.

)εζ()ζ(0xζiπ22

ε

Df e

We see that the zeros of )ζ(f Are the same as the zeros of )εζ(

D

TO COMPUTE , we need to know the location of the rings of zeros of )εζ(

D

The zeros occur on the circle of radius = r0,

i.e.0)εζ(

D When = r0

ζ

)ζ(π2)ζ( 1JD

Where J1 is the first order Bessel function

The zeros Occur on the circle of radius = r1

i.e. 0)ζ(

D When = r1 NOTE: r1 can be found analytically.

!5!42!4!32!3!22!2!122

)(9

9

7

7

5

5

3

3

1

xxxxxxJ

So If0)rε( 0

D and 0)r( 1

D

then

rr 10ε

rr

0

1ε And thus

Accuracy:• Labeyrie’s method gives results consistent to Michelson’s interferometer results.• The method has been applied to over 30 stars already.

ReferencesReferences

• “Fourier Analysis” by T.W. Korner, Cambridge University Press, 1988

• “Convolutions and Computer Graphics” by Anne M. Burns, College Mathematics Journal, 1992

• Dr. Steve Deckelman