image processing 1 (ip1) bildverarbeitung 1seppke... · 2. gonzales & woods "digital image...
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MIN-FakultätFachbereich InformatikArbeitsbereich SAV/BV (KOGS)
ImageProcessing1(IP1)Bildverarbeitung1
Lecture 6 – ImagePropertiesand Filters
WinterSemester2014/15
Slides:Prof.BerndNeumannSlightly revised by:Dr.BenjaminSeppke&Prof.SiegfriedStiehl
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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
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Empirical Mean and Variance
Empirical mean =averageof allpixels of animage
Empirical variance =averageof squared deviation of allpixels frommean
IP1– Lecture 6:ImagePropertiesand Filters
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:
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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.
IP1– Lecture 6:ImagePropertiesand Filters
hf(z)
qN-1
z
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Example of Greyvalue Histogram
IP1– Lecture 6:ImagePropertiesand Filters
Image
Ahistogramcan be "sharpened"bydiscountingpixels at edges(more about edges later):
2550
255(darkest)0
Histogram
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HistogramModification (1)
Greyvaluesmaybe remapped into new greyvalues to• facilitate image analysis• improve subjective imagequality
Example:Histogramequalization
IP1– Lecture 6:ImagePropertiesand Filters
1.CuthistogramintoNstripesofequalarea(N=newnumberofgreyvalues)2.Assignnewgreyvaluestoconsecutivestripes
Examples show improved resolution ofimage parts with most frequent greyvalues(road surface)
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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.
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IP1– Lecture 6:ImagePropertiesand Filters
T (i) = round 255 h( j)Nj=0
i∑
"
#$
%
&'
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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
IP1– Lecture 6:ImagePropertiesand Filters
m
n
pm = gmnn∑
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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
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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.
IP1– Lecture 6:ImagePropertiesand Filters
= +
p(r) = 1σ 2π
e−r2
2σ 2
E[x] is"expectedvalue"of x
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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?
IP1– Lecture 6:ImagePropertiesand Filters
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
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Example of Averaging
IP1– Lecture 6:ImagePropertiesand Filters
Intensity averagingwith 5x5mask
11 1111 1111 1111 1111 11
11111
125
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SimpleSmoothing Operations
1. Averaging
2. Removal of outliers
3. Weighted average
IP1– Lecture 6:ImagePropertiesand Filters
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
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Bimodal AveragingTo avoid averagingacross edges,assumebimodal greyvalue distributionandselect average value of modalitywith largest population.Determine:
IP1– Lecture 6:ImagePropertiesand Filters
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
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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
IP1– Lecture 6:ImagePropertiesand Filters
σ ij2 =
1D
(gmngmn∈ D∑ − gij )
2gij =1D
gmngmn∈ D∑
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MedianFilterMedianof adistributionP(x):xm suchthatP(x < xm) = 1/2
MedianFilter:1.Sort pixels inD according to greyvalue2.Choose greyvalue inmiddle position
Example:
IP1– Lecture 6:ImagePropertiesand Filters
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
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Local Neighbourhood OperationsManyuseful image transformationsmaybe defined as aninstance of alocalneighbourhood operation:
IP1– Lecture 6:ImagePropertiesand Filters
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
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Example of Sharpening
IP1– Lecture 6:ImagePropertiesand Filters
intensity sharpeningwith 3x3mask
"unsharpmasking"=subtraction of blurred image
gij = gij −1
|D|gmn
gmn∈ D∑
-1 -1 -1
-1 -1
-1 -1 -1
9
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