image processing 1 (ip1) bildverarbeitung 1seppke... · 2. gonzales & woods "digital image...

MIN-FakultätFachbereich InformatikArbeitsbereich SAV/BV (KOGS)

ImageProcessing1(IP1)Bildverarbeitung1

Lecture 6 – ImagePropertiesand Filters

WinterSemester2014/15

Slides:Prof.BerndNeumannSlightly revised by:Dr.BenjaminSeppke&Prof.SiegfriedStiehl

GlobalImageProperties

Globalimage properties refer to animage as awhole ratherthan components.Computation of globalimage propertiesis often required for image enhancement,preceding imageanalysis.We treat

– empiricalmean and variance– histograms– projections– cross-sections– frequency spectrum

IP1– Lecture 6:ImagePropertiesand Filters

02.11.15 University of Hamburg, Dept. Informatics 2

Empirical Mean and Variance

Empirical mean =averageof allpixels of animage

Empirical variance =averageof squared deviation of allpixels frommean


g = 1MN

gmnn=0

N−1∑m=0

M−1∑ with image size: M ×N

Simplified notation: g = 1K

gkk=0K−1

∑

Incremental computation: g0 = 0 gk =gk−1(k −1)+ gk

k with k = 2…K

σ 2 =1K

(gkk=1

K∑ − g)2 = 1

Kgk2

k=1

K∑ − g 2

σ 02 = 0 σ k

2 =σ k-1

2 + gk−12( )(k −1)+ gk

2

k−gk−1(k −1)+ gk

k"

#$

%

&'

22

with k = 2…K

Incremental computation:


GreyvalueHistogramsAgreyvalue histogram hf (z) of animage f provides the frequency ofgreyvalues z inthe image.

Thehistogramof animage with N quantization levels is represented by a1DarraymitN elements.

Agreyvalue histogramdescribes discrete values,agreyvalue distributiondescribes continuous values.


hf(z)

qN-1

z


Example of Greyvalue Histogram


Image

Ahistogramcan be "sharpened"bydiscountingpixels at edges(more about edges later):

2550

255(darkest)0

Histogram


HistogramModification (1)

Greyvaluesmaybe remapped into new greyvalues to• facilitate image analysis• improve subjective imagequality

Example:Histogramequalization


1.CuthistogramintoNstripesofequalarea(N=newnumberofgreyvalues)2.Assignnewgreyvaluestoconsecutivestripes

Examples show improved resolution ofimage parts with most frequent greyvalues(road surface)


HistogramModification (2)Two algorithmic solutions:

N pixels perimage,greyvalues i=0...255,histogram h(i)

1."Cuttingup ahistogram into stripesof equal area"

stripe area S = N/256

Cutting up anarbitrary histogram into equal stripes may requireassignment of differentnew greyvalues to pixels of the sameoldgreyvalue.

2.Gonzales&Woods"DigitalImageProcessing"(2ndedition)

old greyvalue i,new greyvalue k = T(i)

Transformationfunction

Histogram is only coarsely equalized.

7


T (i) = round 255 h( j)Nj=0

i∑

"

#$

%

&'

02.11.15 University of Hamburg, Dept. Informatics

Projections

Aprojectionof greyvalues inanimage is the sumof allgreyvalues orthogonalto abase line:

Often used:• "row profile"=row vector of all(normalized)column sums• "column profile"=column vector of all(normalized)rowsums


m

n

pm = gmnn∑


Cross-sections

Across-section of agreyvalue image is avector of allpixels alongastraight line through the image.• fasttest for localizingobjects

• commonly taken alongarow or column or diagonal

9


02.11.15 University of Hamburg, Dept. Informatics

NoiseDeviations fromanidealimage can often be modelled as additivenoise:

Typical properties:

• mean 0,variance σ2 > 0• spatially uncorrelated:E[ rij rmn] = 0 for ij ≠ mn• temporally uncorrelated:E[ rij,t1 rij,t2] = 0 for t1 ≠ t2

• Gaussian probability density:

Noisearises from analogsignal generation (e.g.amplification)andtransmission.

There are several other noisemodels other than additivenoise.


= +

p(r) = 1σ 2π

e−r2

2σ 2

E[x] is"expectedvalue"of x


NoiseRemoval by Averaging

There are basically 2ways to "average out"noise:• temporalaveraging if several samples gij,t of the samepixel

butat differenttimes t = 1 ... T are available• spatial averaging if gmn ≈ gij for allpixels gmn inaregion around gij

How effective is averagingof K greyvalues?


Principle: samplemean approaches densitymeanrK =1K

rkk=1

K

∑ ⇒ 0

rK =1K

rkk=1

K

∑ is random variablewith mean and variance depending onK

mean

E (rK −E rK[ ])2"# $%= E rK2"# $%= E

1K 2 ( rk )

2

k=1

K

∑"

#'

$

%(=

1K 2 E rk

2"# $%=σ 2

Kk=1

K

∑ variance

Example:Inorder cut the standard deviation inhalf,4values have to averaged

E rK[ ] = 1K

E rk[ ]k=1

K

∑ = 0


Example of Averaging


Intensity averagingwith 5x5mask

11 1111 1111 1111 1111 11

11111

125


SimpleSmoothing Operations

1. Averaging

2. Removal of outliers

3. Weighted average


gij =1D

gmngmn∈ D∑ with D the set of all greyvalues around gij

Example of3-by-3region D

gij =1wk∑

wkgkgk∈ D∑ with wk = weights in D

Notethat these operations are heuristics and notwell founded!

Example of weightsin3-by-3region

gij =1D

gmn gmn∈ D∑ if gij −

1D

gmngmn∈ D∑ ≥ S with threshold S

gij else

%

&''

(''

ij

1 2 1

2 2

1 2 1

3


Bimodal AveragingTo avoid averagingacross edges,assumebimodal greyvalue distributionandselect average value of modalitywith largest population.Determine:


A

B

g D =1D gmn

gmn∈ D∑

A = gk with gk ≥ gD{ } B= gk with gk < gD{ }

!gD =

1A gkgk∈ A∑ if A ≥ B

1B gkgk∈ B∑ else

%

&

''

(

''

Example:gD =16.7 A, B g´D= 1311 14 15

13 12 25

15 19 26


Averaging with Rotating MaskReplace center pixel by averageover pixels from the most homogeneous subsettaken from the neighbourhood of center pixel.

Measure for (lackof)homogeneity is dispersion σ2 (=empirical variance)of thegreyvalues of aregion D:

Possible rotated masks in5x5neighbourhood of center pixel:

Algorithm:1.Consider each pixel gij2.Calculate dispersion inmask for allrotated positions of mask3.Choose mask with minimum dispersion4.Assign average greyvalue of chosen mask to gij


σ ij2 =

1D

(gmngmn∈ D∑ − gij )

2gij =1D

gmngmn∈ D∑


MedianFilterMedianof adistributionP(x):xm suchthatP(x < xm) = 1/2

MedianFilter:1.Sort pixels inD according to greyvalue2.Choose greyvalue inmiddle position

Example:


gij = max a( ) with gk ∈ D and {gk < a} <D2

11 14 15

13 12 25

15 19 26

111213141515192526

greyvalue of center pixel ofregion is set to 15

MedianFilter reduces influence of outliers ineither direction!02.11.15 University of Hamburg, Dept. Informatics 16

Local Neighbourhood OperationsManyuseful image transformationsmaybe defined as aninstance of alocalneighbourhood operation:


Generate anew imagewith pixels by applying operator f to allpixels gij of animage

gmn = f (g1,g2,...,gK ) g1,g2,...,gK ∈ Dij

gmn

ij

Dijmn

Pixelindices i, j may be incremented by steps largerthan 1 to obtain reducednew image.

example ofneighbour-

hood


Example of Sharpening


intensity sharpeningwith 3x3mask

"unsharpmasking"=subtraction of blurred image

gij = gij −1

|D|gmn

gmn∈ D∑

-1 -1 -1

-1 -1

-1 -1 -1

9


image processing 1 (ip1) bildverarbeitung 1seppke... · 2. gonzales & woods "digital image...

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