Lecture 5
Image Characterization
ch. 4 of Machine Vision by Wesley E. Snyder & Hairong Qi
Spring2016
18-791(CMUECE):42-735(CMUBME):BioE 2630(Pitt)
Dr.JohnGaleotti
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Digital Images
§Howaretheyformed?
§Howcantheyberepresented?
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1
Image Representation
§Hardware
§Storage
§Manipulation
§Human
§Conceptual
§Mathematical
3
Iconic Representation
§Whatyouthinkofasanimage,…
§Camera
§X-Ray
§CT
§MRI
§Ultrasound
§2D,3D,…
§etc
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2
Iconic Representation
§Andwhatyoumightnot
Range Image
Corresponding
Intensity Image
Images from CESAR lab at Oak Ridge National Laboratory,
Sourced from the USF Range Image Database:
http://marathon.csee.usf.edu/range/DataBase.html
Acknowledgement thereof requested with redistribution.
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Functional Representation
§AnEquation
§Typicallycontinuous
§Fittotheimagedata
§Sometimes theentireimage
§Usuallyjustasmallpieceofit
§Examples(QuadraticSurfaces):
§Explicit:
§Implicit:
z = ax 2 + by 2 + cxy + dx + ey + f
0 = ax 2 + by 2 + cz 2 + dxy + exz + fyz + gx + hy + iz + j
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3
Linear Representation
§Unwindtheimage
§ “Raster-scan”it
§Entireimageisnowavector
§ Nowwecandomatrixoperationson it!
§ Oftenusedinresearchpapers
Probabilistic & Relational
Representations
§ Probability & Graphs
§ Discussed later (if at all)
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Spatial Frequency
Representation
§Think“Fourier
Transform”
§MultipleDimensions!
§Variesgreatlyacross
differentimageregions
§HighFreq.=Sharpness
§StevenLehar ’sdetails:
http://sharp.bu.edu/~sle
har/fourier/fourier.html
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4
Image Formation
§Samplingananalogsignal
§Resolution
§ #Samplesperdimension,OR
§ Smallestclearlydiscernablephysicalobject
§DynamicRange
§ #bits/pixel(quantizationaccuracy),OR
§ Rangeofmeasurableintensities
§ Physicalmeaningofmin&maxpixelvalues
§ light, density,etc.
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Dynamic Range Example
(A slice from a Renal Angio CT: 8 bits, 4 bits, 3 bits, 2 bits)
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5
FYI: Python Code for the
Dynamic Range Slide
importSimpleITK assitk
# aprocessedslicefromhttp://pubimage.hcuge.ch:8080/DATA/CENOVIX.zip
img =sitk.ReadImage( “UpperChestSliceRenalAngioCT-Cenovix.tif")
out4=sitk.Image(512,512,sitk.sitkUInt8)
out3=sitk.Image(512,512,sitk.sitkUInt8)
out2=sitk.Image(512,512,sitk.sitkUInt8)
y=0
while y<512:
x=0
while x<512:
#print"img ",x,y,"=",img[x,y]
out4[x,y]=(img[x,y]>> 4)<<4
out3[x,y]=(img[x,y]>> 5)<<5
out2[x,y]=(img[x,y]>> 6)<<6
x= x+1
y= y+ 1
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An Aside: The
Correspondence Problem
§MyDefinition:
§ Giventwodifferentimagesofthesame(orsimilar)
objects,
foranypointinoneimage
determinetheexactcorresponding pointintheother
image
§Similar(identical?)toregistration
§Quitepossibly, itisTHEproblemincomputer
vision
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6
Image Formation: Corruption
Ideal
f(x,y)
Camera,CT,
MRI,…
Measured
g(x,y)
§Thereisanidealimage
§ Itiswhatwearephysicallymeasuring
§Nomeasuringdeviceisperfect
§ Measuringintroducesnoise
§ g(x,y) = D( f(x,y) ), where D is the distortion function
§Often,noise isadditiveandindependent ofthe
idealimage
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Image Formation: Corruption
§Noiseisusually nottheonlydistortion
§Iftheotherdistortions are:
§ linear&
§ space-invariant
thentheycanalways berepresentedwiththe
convolution integral!
§Totalcorruption:
g ( x, y) =
∫∫ f (α, β ) h ( x − α, y − β ) dα d β + n ( x, y)
−∞…∞
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7
The image as a surface
§Intensity→ height
§ In2Dcase,but conceptsextendtoND
§z = f ( x, y )
§Describesasurface inspace
§ Becauseonlyone z valueforeachx, y pair
§ Assumesurfaceiscontinuous (interpolatepixels)
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Isophote
§“Uniformbrightness”
§C = f ( x, y )
§Acurve (2D)orsurface(3D)inspace
§Alwaysperpendiculartoimagegradient
§Why?
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8
Isophotes & Gradient
§Isophotes arelike
contourlinesona
topography
(elevation)map.
§Atanypoint,the
gradientisalways
atarightangleto
theisophote!
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Ridges
§Onedefinition:
§Localmaximaoftherateofchangeofgradient
direction
§Sound confusing?
§Justthinkofridgelinesalongamountain
§Ifyouneedit,look itup
§ SnyderreferencesMaintz
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9
Medial Axis
§Skeletalrepresentation
§Definedforbinaryimages
§Thisincludes segmentedimages!
§“Ridgesinscale-space”
§Detailshavetowait(ch. 9)
Image courtesy of TranscenData Europe
http://www.fegs.co.uk/motech.html
http://sog1.me.qub.ac.uk/Research/medial/medial.php
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Neighborhoods
§Terminology
§ 4-connected vs.8-connected
§ Side/Face-connectedvs.vertex-connected
§ Maximally-connectedvs.minimallyconnected (ND)
§Connectivity paradox
§ Duetodiscretization
Is this
shape
closed?
§Candefineotherneighborhoods
§ Adjacencynot necessarilyrequired
?
Is this
pixel
connected
to the
outside?
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10
Curvature
§Computecurvatureateverypointina(range)
image
§ (Oron asegmented3Dsurface)
§Basedondifferentialgeometry
§Formulas areinyourbook
§2scalarmeasuresofcurvaturethatareinvariant
toviewpoint, derivedfromthe2principal
curvatures,(K1 , K2 ):
§ Meancurvature (arithmeticmean)
§ Gausscurvature(product)
§ =0ifeither K1 =0or K2 =0
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