january 19, 2006computer vision © 2006 davi geigerlecture 1.1 image measurements and detection davi...
Post on 19-Dec-2015
217 views
TRANSCRIPT
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.1
Image Measurements and Detection
Davi Geiger
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.2
Images are Intensity Surfaces andEdges are Intensity Discontinuities
I1(x, y)
I2(x, y)
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.3
s
dIs
syxI0
sin,cos1
,,,~
1
0
sin,cos1
,,,~ s
i
yiyxixIs
syxI
//4 /3
/6
/3 /
/
s=3 pixels
s=7 pixels
Discrete approximation
y
x
syxI ,,,~
on,AccumulatiIntensity :tsMeasuremen
97536
11,
4
7,
3
5,
2
3,
3
4,
4
5,
6
7,,
6
5,
4
3,
3
2,
2,
3,
4,
6,0
scales.different 4four at and directions 16sixteen alongintensity theofon accumulati theevaluate we,, pixeleach For
,,,s
yx
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.4
1
0
)sin,cos(1
),,,(~ s
i
yiyxixIs
syxI
//4 /3
/6
/3 /
/
, where
1
1
sin,cos1
1,
1,,,
~
1,,,
~ s
ic yiyxixI
syxI
sxyxI
s
ssyxI
Removing the center pixel: ),,,(~
syxIc
.35,34,32,3for3
2and1
613,67,65,6for1and3
247,45,43,4for2and2
23,,2,0for1and1
yx
yx
yx
yx
syxI ,,,~
on,AccumulatiIntensity :tsMeasuremen
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.5
/4
s=3
s=5
/3
/6
s=5
s=3s=5
s=3
s=5
s=7s=7
s=7
s=7
s=3
syxIsyxIsyxIsyxI ,,,1~
,,1,~
,,1,~
,,,~
4
1
syxIsyxIsyxIsyxI ,,1,~
,,,1~
,,,1~
,,,~
4
1
syxIsyxIsyxIsyxI ,,1,1~
,,1,~
,,,1~
,,,~
4
1
),,,(ˆ syxI for
for
for
In order to obtain more robust measures, for an angle , we can average the value of along to obtain),,,(
~syxI ),,,(ˆ syxI
syxI ,,,ˆ II,on AccumulatiIntensity :tsMeasuremen
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.6
.,,,ˆ,,,ˆ
sin,,,ˆ
cos,,,ˆ
sin,cos,,,ˆˆ,,,ˆ,,,ˆ
syxIsyxI
y
syxI
x
syxI
syxIsyxIsyxID
cc
Taking the derivative along the angle i.e., in a direction ,sin,cosˆ
Analogously, we obtain . We now have a bank of sixteen (16) distinct oriented filters at different scales, namely { ; and s .
syxID ,,,~
will have maximum response (highest value) among the possible orientations, while will have the minimum response.
),6,,(ˆ syxID
),32,,(ˆ syxID
/ 3 syxID ,,,ˆ
x
y
syxID ,,,ˆ s,Derivative :tsMeasuremen
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.7
3,0,,ˆ yxID
3,6
,,ˆ yxID
3,4
,,ˆ yxID
3,3
,,ˆ yxID
128
64
16
0
-16
-64
-128
syxID ,,,ˆ :sExperiment
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.8
3,2
,,ˆ yxID
3,3
2,,ˆ
yxID
3,4
3,,ˆ
yxID
3,6
5,,ˆ
yxID
255
128
32
0
-32
-128
-255
syxID ,,,ˆ :(cont.) sExperiment
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.9
/3
/6
syxID
y
syxI
x
syxI
syxIsyxIsyxID
,2
,,ˆ
cos,,,ˆ
sin,,,ˆ
cos,sin,,,ˆˆ,,,ˆ,,,ˆ
Corners and Junctions will respond to high values of at distant locationsfrom the center.
syxID ,,,ˆ
syxIDsyxID ,23
,,ˆ,3
,,ˆ
syxIDsyxID ,26
,,ˆ,6
,,ˆ
The maximum responses here among the different are
and
x
y
syxID ,,,ˆ s,Derivative :tsMeasuremen
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.10
yI
xyI
yxI
xI
yxH
2
22
2
2
2
ˆˆ
ˆˆ
,The Hessian allows for computing the second derivative.
Tvu uyxHvsyxID ˆ,ˆ,,,,ˆ
In order to compute the (second) derivative along the direction of a (first) derivative along a direction we compute the projectionsu
v
For example, for we obtain
and for we obtain
,ˆ0,1ˆ;ˆ0,1ˆ xvxu ,ˆ,,,,ˆ2
2
xIsyxIDvu
,ˆ1,0ˆ;ˆ0,1ˆ yvxu xyIsyxIDvu
ˆ,,,,ˆ 2
Measurements: 2nd Derivatives
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.11
ssyyxxIDsyxID
yxHsyxID T
,,sin,cosˆ,,,ˆ
sin,cos,sin,cos,,,,ˆ
In order to compute the (second) derivative along a direction of a (first) derivative along a direction we obtain a continuous and a lattice approximation as
x
y
sin
cossin,cosˆ,sin,cosˆ TTuv
x
y
Measurements: 2nd Derivatives (cont.)
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.12
Features such as edges, corners, junction, and eyes are obtained by making some decision from the image measurements.
Decisions are the result of some comparison followed by a choice. Examples: (i) if a measurement is above a threshold, we accept; otherwise, we reject; (ii) if a measurement is the largest compared to others, we select it.
Image Features – Decisions!
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.13
),,,(ˆ),
2,,(ˆmaxarg,max syxIDsyxIDyxs
TsyxIDsyxIDyx
),,,(ˆ),
2,,(ˆsuch that ,,
Edge threshold: Decision!
Edge orientation: Decision!
A step edge at The value is (equally) large for both andas shown (in red) for the scale s = 3 pixels. It is also large for values not shownHowever, the quantity is significantly larger for
),2/,,(~
syxID
),,,(~
),2/,,(~
syxIDsyxID
syxID ,,,ˆ
syxID ,
2,,ˆ
3,
4,
6,0
Decisions: Edgels (Edge-pixels and Orientation)
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.14
yx,3max
The gray level indicates the angle: the darkest one is 0 degrees. The larger the angle, the lighter its color, up to
),,,(ˆ),2
,,(ˆ,,, syxIDsyxIDsyx Strength of the Edgel
Edgel(x, y) if
else Nil
Tsyx ,,,
syxyxs ,,,maxarg,max
Decisions: Edgels (cont.)
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.15
Decisions: Connecting Edgels (Pseudocode)
end
;,,,xNeighbors-Link;,,yx,Link
NIL,xEdgel if;sin;cos
,,Neighbors-Link
end;,,Neighbors-Link;,,Neighbors-Link
NIL,Edgel if,Follower-Contour
., location seed a withStart edgels. link to Algorithm
maxn
n
max
max
nnn
nn
n
n
n
cc
cc
cc
cc
cc
yxyyx
yxxyxxx
yx
yxyx
yxyx
yx
max
(xc , yc)
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.16
Decisions: Connecting Edgels (Pseudocode)
endFalse;return
elseTrue;return
,2
3,I
~,
2
3,I
~
and ,2
,I~
,2
,I~
if
,,,Coherence
path. same on thereturn not do contours and changenot doescontrast that theso used is Coherence
end;,,,Neighbors-Link
;,,,Link,,,Coherence&,Edgel if
;sin;cos
,y,xNeighbors-Link
1maxmax
1maxmax
max
max
max
maxcc
Ts,yxs,yx
Ts,yxs,yx
yxyx
yxyxyxyx
yxyxyxyyyxxx
cccnnn
cccnnn
nncc
nnnn
nncc
nnccnn
ccn
ccn
c
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.17
We have considered at least four parameters: 1,,, TTTT H
How to estimate them? One technique is Histogram partitioning:plot the histogram and find the parameter that “best partitions it”.
syxID ,2/,,ˆ
ountingHistogram c:
T
Threshold Parameters: Estimation
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.18
yxyyxxyx sss ,sin,cos, maxmaxmax Angle change
y
x
yxs ,max
2
sin,cosmax yyxxs
2
,max
yxs
A contour segment
yxs ,max
22 sincos yx is the contour curvature multiplied by the arc length , where
yxs ,max .,max yxs where
Decisions: Local Angle Change
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.19
2
2
222
2
2
32
2
2
222
2
2
2
2
22
2
2
2
ˆˆˆˆˆ2ˆˆ
ˆ
1
ˆ
Thus, .ˆby given is which arc, theoflength by the divided change angle thebe willcurvature the
ˆˆˆˆˆ2ˆˆ
ˆ
1
ˆ
ˆ
ˆˆ
ˆˆˆˆ
ˆ
ˆ),(
ˆ
ˆ
i.e.,
Isocontour thealong movingwhen ˆ vector d)(normalize with theoccurs
change (angle)much howask simply can onecontour -iso theof curvature theestimate To
y)(x, from direction in the movingby contour -Iso thefollowcan one Thus,
IsoContour theo tangent tisˆ
I of change maximal ofdirection theisit as ,Isocontour the tonormal isˆ),(ˆ
xI
yI
xyI
xI
yI
yI
xI
II
I
xI
yI
xyI
xI
yI
yI
xI
I
xI
yI
yI
xyI
yxI
xI
xI
yI
I
IyxH
I
I
I
I
I
ICyxI
Curvature of Isocontour
January 19, 2006 Computer Vision © 2006 Davi Geiger Lecture 1.20
IIyxH
yxHyxH
IIyxHII
IyxHII
yxHII
I
ICyxI
ˆandˆ thengrepresenti),( of rseigenvecto two
for thesearch theas maximal isother theandlowest is onesuch that sderivative
orthogonal twoof smeasurmentby drepresente ,detector" edge"our interpret nowcan We
negative.-non are seigenvalue thei.e., matrix,
symmetric a is),( that Notezero. be to),( of eigenvaluesmallest expect the we
)0( curvature zero i.e., segments),(straight linesstraight are that contours-isofor Thus,
ˆˆ),()ˆ(ˆ
1
thereforeis eigenvalueminimun theand
,ˆ),()ˆ(ˆ
1
thereforeis eigenvalue maximum Thes.eigenvalueminimun and maximum
thetoingcorrespond),( of rseigenvecto theare ˆ andˆ vectors, two theseThus,
IsoContour theo tangent tisˆ
I of change maximal ofdirection theisit as ,Isocontour the tonormal isˆ),(ˆ
2min
2max
Curvature of Isocontour (cont.)