Lecture 2-Filtering.pdf

CAP 5415 Computer Vision
Fall 2011
Dr. Mubarak Shah
Univ. of Central Florida
www.cs.ucf.edu/~vision/courses/cap5415/fall2012
Office 247-F HEC

General
 Binary
 Gray Scale
 Color

Binary Images
0: Black
1: White p
q
X
Y
Row 1
Row q
1 1 1
0 0 0 0
0
1

Color Image
Red, Green, Blue Channels

Image Noise
 Light Variations
 Camera Electronics
 Surface Reflectance
 Lens

Image Noise
 I(x,y) : the true pixel values
 n(x,y) : the noise at pixel (x,y)
     
y
x
n
y
x
I
y
x
I ,
,
,
ˆ 



Gaussian Noise
  2
2
2
, 
n
e
y
x
n



Definitions
 Derivative: Rate of change
– Speed is a rate of change of a distance
– Acceleration is a rate of change of speed
 Average (Mean)
– Dividing the sum of N values by N

Derivative
x
x f
x
f
x
x
x
f
x
f
dx
df







 
 )
(
)
(
)
(
lim 0
on
accelerati
speed
dt
dv
a
dt
ds
v 


Examples
3
4
2
4
2 x
x
dx
dy
x
x
y




x
x
e
x
dx
dy
e
x
y







)
1
(
cos
sin

Discrete Derivative
)
(
)
(
)
(
lim 0 x
f
x
x
x
f
x
f
dx
df
x






 

)
(
1
)
1
(
)
(
x
f
x
f
x
f
dx
df





)
(
)
1
(
)
( x
f
x
f
x
f
dx
df






Discrete Derivative
Finite Difference
)
(
)
1
(
)
1
( x
f
x
f
x
f
dx
df






)
(
)
1
(
)
( x
f
x
f
x
f
dx
df





)
(
)
1
(
)
( x
f
x
f
x
f
dx
df




 Backward difference
Forward difference
Central difference

Example
0
5
20
15
5
10
5
0
)
(
0
0
5
15
0
5
5
0
)
(
20
20
20
25
10
10
15
10
)
(









x
f
x
f
x
f
Derivative Masks
Backward difference
Forward difference
Central difference
[-1 1]
[1 -1]
[-1 0 1]

Derivatives in 2 Dimensions
)
,
( y
x
f
Given function

























y
x
f
f
y
y
x
f
x
y
x
f
y
x
f )
,
(
)
,
(
)
,
(
Gradient vector
2
2
)
,
( y
x f
f
y
x
f 


Gradient magnitude
y
x
f
f
1
tan


Gradient direction

Derivatives of Images














1
0
1
1
0
1
1
0
1
3
1
x
f
Derivative masks














1
1
1
0
0
0
1
1
1
3
1
y
f

















20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
I

















0
0
0
0
0
0
0
10
10
0
0
0
10
10
0
0
0
10
10
0
0
0
0
0
0
x
I

Derivatives of Images

















0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y
I

















20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
20
20
20
10
10
I

Correlation
   
 



k l
l
j
k
i
h
l
k
f
h
f ,
,
Kernel
Image


f
f
h1 h2 h3
h4 h5 h6
h7 h8 h9
h
f1 f2 f3
f4 f5 f6
f7 f8 f9

9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
*
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f









f

Convolution
   
 


k l
l
j
k
i
h
l
k
f
h
f ,
,
*
Kernel
Image


h
f h7 h8 h9
h4 h5 h6
h1 h2 h3
h9 h8 h7
h6 h5 h4
h3 h2 h1
h1 h2 h3
h4 h5 h6
h7 h8 h9
h
flip
X 
flip
Y 
f1 f2 f3
f4 f5 f6
f7 f8 f9

1
9
2
8
3
7
4
6
5
5
6
4
7
3
8
2
9
1
*
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f
h
f









f

Convolution
x
x
f
h
)
1
,
1
(
)
1
,
1
(
)
1
,
0
(
)
1
,
(
)
1
,
1
(
)
1
,
1
(
)
0
,
1
(
)
,
1
(
)
0
,
0
(
)
,
(
)
0
,
1
(
)
,
1
(
)
1
,
1
(
)
1
,
1
(
)
1
,
0
(
)
1
,
(
)
1
,
1
(
)
1
,
1
(
)
,
(
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f
h
y
x
f





























 





1
1
1
1
)
,
(
)
,
(
i j
j
i
h
i
y
i
x
f
h
f
-1,0 0,1 1,1
-1,0 0,0 1,0
-1,-1 0,-1 1,-1
Coordinates

Averages
 Mean
n
I
n
I
I
I
I
n
i
i
n





 1
2
1 
 Weighted mean
n
I
w
n
I
w
I
w
I
w
I
n
i
i
i
n
n






 1
2
2
1
1 

Gaussian Filter
2
2
2
)
( 
x
e
x
g


 
011
.
13
.
6
.
1
6
.
13
.
011
.
)
( 
x
g
2
2
2
2
(
)
,
( 
y
x
e
y
x
g



1



Properties of Gaussian
 Most common natural model
 Smooth function, it has infinite number of
derivatives
 Fourier Transform of Gaussian is Gaussian.
 Convolution of a Gaussian with itself is a
Gaussian.
 There are cells in eye that perform Gaussian
filtering.

Alper Yilmaz, Mubarak Shah, UCF
Filtering
 Modify pixels based on some function of the
neighborhood
10 30 10
20 11 20
11 9 1
5.7
 
p
f

Linear Filtering
 The output is the linear combination of the
neighborhood pixels
1 3 0
2 10 2
4 1 1
Image
1 0 -1
1 0.1 -1
1 0 -1
Kernel
= 5
Filter Output


Filtering Examples
0 0 0
0 1 0
0 0 0
* 

Filtering Examples
0 0 0
0 0 1
0 0 0
* 

Filtering Examples
1 1 1
1 1 1
1 1 1
9
1
* 

Filtering Examples
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
25
1
* 

Blurring Examples
original
original
pixel offset
pixel offset
filtered
filtered
0
0
8
8
4
0.3
0.3
2.4
8
4
6
4.8

Filtering Gaussian
* 

Gaussian vs. Smoothing
Gaussian Smoothing Smoothing by Averaging

Noise Filtering
Gaussian Noise
After Gaussian Smoothing
After Averaging

MATLAB Functions
 conv: 1-D Convolution.
– C = conv(A, B) convolves vectors A and B.
 conv2: Two dimensional convolution.
– C = conv2(A, B) performs the 2-D convolution
of matrices A and B.

MATLAB Functions
 filter2: Two-dimensional digital filter.
– Y = filter2(B,X) filters the data in X with the 2-D
FIR filter in the matrix B.
– The result, Y, is computed using 2-D correlation and is
the same size as X.
– filter2 uses CONV2 to do most of the work. 2-
D correlation is related to 2-D convolution by a
180 degree rotation of the filter matrix.

MATLAB Functions
 gradient: Approximate gradient.
– [FX,FY] = gradient(F) returns the numerical
gradient of the matrix F. FX corresponds to dF/dx,
FY corresponds to dF/dy.
 mean: Average or mean value.
– For vectors, mean(X) is the mean value
(average) of the elements in X.

MATLAB Functions
 special: Create predefined 2-D filters
– H = fspecial(TYPE) creates a two-dimensional filter H of the
specified type. Possible values for TYPE are:
– 'average' averaging filter;
– 'gaussian' Gaussian lowpass filter
– 'laplacian' filter approximating the 2-D Laplacian operator
– 'log' Laplacian of Gaussian filter
– 'prewitt' Prewitt horizontal edge-emphasizing filter
– 'sobel' Sobel horizontal edge-emphasizing filter
– Example: H=fspecial('gaussian',7,1) creates a 7x7 Gaussian
filter with variance 1.

Lecture 2-Filtering.pdf

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