Download Lecture Recap: Image Processing - Histogram Segmentation, Region Properties, Pyramid and more Study notes Computer Science in PDF only on Docsity!
Alper Yilmaz, Fall 2005 UCF
CAP 5415 Computer Vision
Fall 2005
Dr. Alper Yilmaz
Univ. of Central Florida
www.cs.ucf.edu/courses/cap5415/fall Office: CSB 250
Alper Yilmaz, Fall 2005 UCF
Histogram Segmentation Code
%smooth for i=1: hst=conv2(hst,gauss,'same'); end %compute derivative dr1=conv2(hst,dr,'same'); %find peaks and valleys pw =dr1(1:254).*dr1(2:255); peaks_valleys(find(pw<0))=1; control=find(peaks_valleys==1); for i=1:size(control,2) if (dr1(control(i)-2)>0) valleys(control(i))=1; end if (dr1(control(i)-2)<0) peaks(control(i))=1; end end peaklocs=find(peaks==1); valleylocs=find(valleys==1);
current_valley=1; for i=1:size(peaklocs,2) Va = hst(valleylocs(current_valley)); Vb = hst(valleylocs(current_valley+1)); P = hst(peaklocs(i)); Wvalleylocs(current_valley); = valleylocs(current_valley+1)- N = 0 ; forj=valleylocs(current_valley):valleylocs (current_valley+1) N = N + hst(j); end val1 = 1-((Va+Vb)/(2P)); val2 = 1-(N/(WP)); if (val1>0 && val1>0) peakiness(peaklocs(i)) = val1*val2; end; current_valley = current_valley +1; end
Alper Yilmaz, Fall 2005 UCF
Plots
Derivative
Histogram
Peaks
Valleys
Peakiness
-500 0 50 100 150 200 250 300
0
500
(^00 50 100 150 200 250 )
1000
2000
(^00 50 100 150 200 250 )
1
(^00 50 100 150 200 250 )
1
(^00 50 100 150 200 250 )
Alper Yilmaz, Fall 2005 UCF
Recap
Region Properties
z Area
z Centroid
z Moments
z Perimeter
z Compactness
z Orientation
∑∑ = =
=
m x
n y
A Bxy 0 0
(, )
A
yBx y y A
xBxy x
m x
n y
m x
n
= ∑∑=^0 y =^0 =^ ∑∑=^0 =^0
(,) (,)
μ pq = ∫ ∫( x − x ) p ( y − y ) qB ( x , y ) d ( x − x ) d ( y − y )
A
C^ Perimeter 4 π
2
Alper Yilmaz, Fall 2005 UCF
Recap
Region Orientation
(x,y)
y 0
r
s θ
x 1
y 1
x 0
θ
θ
sin
cos
0 1
0 1
y y s
x x s
( ) ( ρ θ) θ
ρ θ θ
cos sin
sin cos
0
0
y s
x s
Alper Yilmaz, Fall 2005 UCF
Recap
Region Orientation
(x,y)
y 0
r
x x 0
y
2 0
2 0 r^2 =( x − x ) +( y − y )
( ) ( ρ θ) θ
ρ θ θ
cos sin
sin cos
0
0
y s
x s
( ) ( ) 2 2 2 r = x +ρ sinθ− s cosθ + y −ρcosθ− s sin θ
r^2 = x^2 + y^2 +ρ 2 + s^2 − 2 s ( x cosθ+ y sinθ) + 2 ρ( x sinθ− y cosθ)
Substitute x 0 and y (^0)
Alper Yilmaz, Fall 2005 UCF
Recap
Region Orientation
r^2 2 s 2 ( x cosθ y sinθ) s
= − + ∂
∂
s = x cos θ + y sinθ
Substitute s back to r
r^2 = x^2 + y^2 +ρ 2 + s^2 − 2 s ( x cosθ+ y sinθ) + 2 ρ( x sinθ− y cosθ)
Alper Yilmaz, Fall 2005 UCF
Recap
Region Orientation
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (^2) ( )^2
2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2
sin cos
sin cos 2 sin cos
sin cos 2 sin cos 2 sin cos
1 cos 1 sin 2 sin cos 2 sin cos
cos 2 sin cos sin 2 sin cos
cos sin 2 sin cos
r x y
r x y x y
r x y xy x y
r x y xy x y
r x y x xy y x y
r x y x y x y
r^2 = x^2 + y^2 +ρ 2 + s^2 − 2 s ( x cosθ+ y sinθ) + 2 ρ( x sinθ− y cosθ)
Alper Yilmaz, Fall 2005 UCF
Homework from last lecture
z Show that following holds
where
E 1 (^) = a sin 2 θ − b sinθcosθ+ c cos^2 θ
θ sin 2 θ
( )cos 2
E 2 = a + c − a − c − b
E 1 = E 2
Alper Yilmaz, Fall 2005 UCF
Pyramid Representation
Gaussian
Laplacian
Alper Yilmaz, Fall 2005 UCF
Gaussian Pyramid
z Very useful for representing images
z Image Pyramid is built by using multiple
copies of image at different scales.
z Each level in the pyramid is ¼ of the size of
previous level
z The highest level is of the highest resolution
z The lowest level is of the lowest resolution
Alper Yilmaz, Fall 2005 UCF
Alper Yilmaz, Fall 2005 UCF
- Resample the checkerboard by
taking one sample at each circle. In
the case of the top left board, new
representation is reasonable.
- Top right also yields a reasonable
representation. Bottom left is all black
(dubious) and bottom right has checks
that are too big.
Alper Yilmaz, Fall 2005 UCF
Pyramid With Every Other Pixel
Constructing a pyramid by taking every
second pixel leads to layers that badly
misrepresent the top layer
Alper Yilmaz, Fall 2005 UCF
Sampling
z 1D
z 2D
gray level x
gray level x
gray level x
y
gray level x
y
Alper Yilmaz, Fall 2005 UCF
Smoothing
z High frequencies (edges) lead to trouble with
sampling.
z Solution: suppress edges before sampling
z Common solution: use a Gaussian
- Convolve image with Gaussian filter
Alper Yilmaz, Fall 2005 UCF
Applications of Scaled Images
z Search for correspondence
- look at coarse scales, then refine with finer scales
z Edge tracking
- “Good” edge at a fine scale has parents at a coarser scale
z Control of detail and computational cost in matching
- Finding stripes
- Very important in texture representation
Alper Yilmaz, Fall 2005 UCF
Gaussian Pyramid
z Let w be Gaussian filter
() ˆ( ) ( 2 )
2
2
g i wmg 1 i m m
l =^ ∑ l + =−
ˆ( 0 ) ( 4 ) ˆ( 1 ) ( 4 1 ) ˆ( 2 ) ( 4 2 )
( 2 ) ˆ( 2 ) ( 4 2 ) ˆ( 1 ) ˆ( 4 1 )
1 1 1
1 1
= − − + − − +
− − −
− −
l l l
l l l
w g w g w g
g w g w g w
ˆ( 0 ) ( 4 ) ˆ( 1 ) ( 5 ) ˆ( 2 ) ( 6 )
( 2 ) ˆ( 2 ) ( 2 ) ˆ( 1 ) ˆ( 3 )
1 1 1
1 1
− − −
− −
= − + − +
l l l
l l l
w g w g w g
g w g w g w
g 0 IMAGE
gl = REDUCE [ gl − 1 ]
Alper Yilmaz, Fall 2005 UCF
Convolution Kernel w
z Symmetric
z Sum must be 1
[ w ( (^) − 2 ) , (^) w (− 1 ), (^) w ( ) 0 , (^) w ( ) 1 , (^) w ( ) 2 ]
w ( ) i = w (− i ) ⇒[ c , b , a , b , c ]
a + 2 b + 2 c = 1
Alper Yilmaz, Fall 2005 UCF
Reduce 1D
Convolution Kernel w
z All nodes at a given level must contribute the same total weight to the nodes at the next higher level
a + 2 c = 2 b
b (^) b
c c
a
Alper Yilmaz, Fall 2005 UCF
Approximate Gaussian (5x1)
c b a b c
Gaussian
Alper Yilmaz, Fall 2005 UCF
Two Dimensions?
z Gaussian is separable
I ( x , y ) = I ( x , y ) * G ( x , y )
)
I ( x , y ) = I ( x , y ) * G ( ) x * G ( y )
)
G ( ) x G ( y ) transpose
T
=
Alper Yilmaz, Fall 2005 UCF
Gaussian Pyramid Algorithm
z Apply 1D mask to alternate pixels along each row of image.
z Apply 1D mask to alternate pixels along each column of resulting image from previous step.
Alper Yilmaz, Fall 2005 UCF
Alper Yilmaz, Fall 2005 UCF
Expand 1D
i p
g i w p g
m
ln ln
) 2
3 2 ) ˆ( 2 ) ( 2
3 1 ) ˆ( 1 ) ( 2
3 ˆ( 0 ) (
) 2
3 1 ) ˆ( 1 ) ( 2
3 2 ( 3 ) ˆ( 2 ) (
, 1 , 1 , 1
, , 1 , 1
−
− = −
− − −
− −
ln ln l n
ln ln ln
w g w g w g
g w g w g
gl , n ( 3 )= w ˆ(− 1 ) gl , n − 1 ( 1 )+ w ˆ( 1 ) gl , n − 1 ( 2 )
Alper Yilmaz, Fall 2005 UCF
The Laplacian Pyramid
z Similar to edge detected images
z Most pixels are zero
z Can be used in image compression
Alper Yilmaz, Fall 2005 UCF
Constructing Laplacian Pyramid
z Compute Gaussian pyramid
z Compute Laplacian pyramid as follows:
g (^) k , gk − 1 , gk − 2 ,K g 2 , g 1
( )
( )
( )
1 1
2 2 3
1 1 2
1
L g
L g EXPANDg
L g EXPANDg
L g EXPANDg
k k k
k k k
k k k
=
= −
= −
= −
− − −
− − −
−
M
Alper Yilmaz, Fall 2005 UCF
Laplacian Pyramid
