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Image Processing & Machine Vision/Y. Gao 1
Image Enhancement – Introduction
1. Introduction
1.1 What is image enhancement?--- Modify the intensities of pixels in an image so that it
can be more suitable for a specific application.
Different enhancement process suits different application.
An enhancement method is good for an application but maybe bad for another application.
Image Processing & Machine Vision/Y. Gao 2
Image Enhancement – Introduction
1.2 Categories of methods for enhancement process.
Spatial domain: process pixels in the image plane directly.
Frequency domain: modify the Fourier transform of an image.
Image Processing & Machine Vision/Y. Gao 3
Image Enhancement -- Introduction
Mathematically, the enhancement process in the spatial domain can be described by the transform function
g(x,y) = T [f(x,y)]
f(x,y) --- input image,
g(x,y) --- output image,
T --- a transform
For a single pixel, we have s = T (r)
r --- grey-level value for an input pixel
s --- grey-level value for an output pixel
Image Processing & Machine Vision/Y. Gao 4
Image Enhancement – Introduction
Grey-level value transform graph
We use a graph to denote the transform function s = T (r) .
r
sT(r)
r
sT(r)
r
s
T(r)
No change image brighter image darker
Image Processing & Machine Vision/Y. Gao 5
Image Enhancement – Introduction
Two simple examples for contrast enhancement
x
Image Processing & Machine Vision/Y. Gao 6
Image Enhancement – Introduction
For the enhancement in the frequency domain, according to the convolution theorem we have
),( yxh ),( yxg),( yxf
),( vuH),( vuF ),( vuG
Image Processing & Machine Vision/Y. Gao 7
Image Enhancement – Introduction
)],(),([ ),(
),(),(),(),(),(),(
1- vuFvuHyxg
vuFvuHvuGyxfyxhyxg
=
=∗=
F
)],([ ),()],([ ),()],([ ),(
yxgvuGyxhvuHyxfvuF
=== F
FF
Image Processing & Machine Vision/Y. Gao 8
Image Enhancement -- outline
1. Introduction
2. Transforms
3. Histogram processing
4. Arithmetic/logic operations
5. Filtering
1) Spatial filters
i. Smoothing
ii. Sharpening
2) Frequency-domain filters
i. Smoothing
ii. Sharpening
Image Processing & Machine Vision/Y. Gao 9
Image Enhancement -- Transformations
2. Transformations
2.1. Image negatives
0
T(r)s
r
L-1
rLrTs −−== 1)(
L-1
Image Processing & Machine Vision/Y. Gao 10
Image Enhancement -- Transformations
Image Processing & Machine Vision/Y. Gao 11
Image Enhancement -- Transformations
2.2. Contrast stretching
For a low-contrast image, we feel it uncomfortable and sometimes can not see details clearly.
Image Processing & Machine Vision/Y. Gao 12
Image Enhancement -- Transformations
Mathematically, We have following transform for stretching contrast.
srrs
srrrss
rs
rLL
r
r
222
2
1112
12
1
1
)()1()1(
)()()(
+−⋅−−−−
+−⋅−−
⋅
{)( == rTs12
1
0
2
1
−<≤<≤<≤
Lrrrrrrr
Image Processing & Machine Vision/Y. Gao 13
Image Enhancement -- Transformations
Special cases:
When
ssrr
ssrrsrsr
L
2121
2121
2211
and
1 and 0,
and
≤≤
−===
== ----------------- No change
--- Thresholding, which produces a binary image
--- General, adjusting contrast
Image Processing & Machine Vision/Y. Gao 14
Image Enhancement -- Transformations
2.3. Compression of dynamic range
When the dynamic range of the grey-level in an image far exceedsthe capability of the display device, it needs to be compressed to display it better.
A good transform function (Log transform) is as follow,
)1log( rcs +⋅= c is a scaling constant
Image Processing & Machine Vision/Y. Gao 15
Image Enhancement -- Transformations
T(r)
s
r
Image Processing & Machine Vision/Y. Gao 16
Image Enhancement -- Transformations
For example, an image with dynamic range
So
[ ]105.2,0[],0 6×=R
4.6)1log( 105.2 6 ≈+ ×When the dynamic range of the display device is [0,255], we have
404.6
255 ≈=c
The transform result is shown as follows
Image Processing & Machine Vision/Y. Gao 17
Image Enhancement -- Transformations
Image Processing & Machine Vision/Y. Gao 18
Image Enhancement -- Transformations
2.4. Gray-level slicingSometimes need to highlight a specific range of gray levels in an image.
Two approaches:1. A high value for all gray levels in the range of
interest and a low value for all other gray levels.2. Brighten the desired range of gray levels but
preserve all other parts including background.
Image Processing & Machine Vision/Y. Gao 19
Image Enhancement -- Transformations
Image Processing & Machine Vision/Y. Gao 20
Image Enhancement -- Transformations
2.5. Bit-plane slicingSuppose we use one byte ( 8 bits ) to store the gray level valuefor each pixel, as follows
bbbbbbbb 01234567
For all pixels in an image we take all bits in the same position, which makes a bit-plane.
Image Processing & Machine Vision/Y. Gao 21
Image Enhancement -- Transformations
Usually higher order bit-plane contains much more visual data, Bit-plane is the most significant for the image but Bit-plane is the least significant
b7
b0
Image Processing & Machine Vision/Y. Gao 22
Image Enhancement -- Transformations
Image Processing & Machine Vision/Y. Gao 23
Image Enhancement -- Transformations
Image Processing & Machine Vision/Y. Gao 24
Image Enhancement -- Histogram
3. Histogram Processing
3.1. What is the histogram of an image?
The distribution of the pixel grey level is very important for an image display.
For a limited grey-level domain [0, L-1] we hope there is anuniform distribution of the pixel grey level instead of non-uniform distribution.
Based on the distribution of the pixel grey level we define the image histogram.
Image Processing & Machine Vision/Y. Gao 25
Image Enhancement -- Histogram
For a digital image with gray levels in the range [0, L-1], we define a discrete function
rnr kk
k np )( =
is the number of the pixels with the gray levelnk rkis the total number of pixels in the image.
We call the histogram of a digital image.
Four special histograms are as follows.
)(rkp
n
is the kth gray level.
Image Processing & Machine Vision/Y. Gao 26
Image Enhancement -- Histogram
)(r kp
r k
Dark image
)(r kp
r k
Bright image
)(r kp
r k
High-contrast image
)(r kp
r k
Low-contrast image
Image Processing & Machine Vision/Y. Gao 27
Image Enhancement -- Histogram
Image Processing & Machine Vision/Y. Gao 28
Image Enhancement -- Histogram
Compute Histogram H of gray-level image I
procedure histogram (I,H) {
“Initialize all bins to zero”
for i:=0 to MaxVal
H[i]:=0;
“Compute values”
for L:=0 to MaxRow
for P:=0 to MaxCol {
grayval:= I[L,P];
H[grayval]:= H[grayval] + 1;
}
}
Image Processing & Machine Vision/Y. Gao 29
Image Enhancement -- Histogram
3.2. Histogram equalization
Sometimes we need to convert a low-contrast image into a high-contrast image for displaying it better.
We can do it by extending its histogram ---- histogram equalization
For our discussion, we suppose the gray level r is a continuous quantity and normalized with the range [0,1].
And the enhancement transform function is s=T( r );
Image Processing & Machine Vision/Y. Gao 30
Image Enhancement -- Histogram
We also suppose that T( r ) satisfy the following two conditions,
(1) T( r ) is a single-valued and monotonically increasing for rbetween [0,1];
(2) for 1 )( 0 ≤≤ rT 1 0 ≤≤ r
Condition (1) preserves the order in the gray scale;Condition (2) guarantees a mapping with the range [0,1];
Image Processing & Machine Vision/Y. Gao 31
Image Enhancement -- Histogram
An example satisfying above conditions is as follow,
And the inverse transform can be denoted r 1 s 0 )(1 ≤≤= − sT
Image Processing & Machine Vision/Y. Gao 32
Image Enhancement -- Histogram
Assume the probability density functions of the gray level for theoriginal image and transformed image are respectively
Set
So,
)( and )( sr pp sr
1r0 )()(0
≤≤== ∫ dwwrTs r
rp
)(rdrds pr=
−=
=dsdrrpp r
T sr
ss )(
)(1
)( Q
10 1)( )(
1)()(1
≤≤==∴
−=
ssrp
rppr
rT sr
s
Image Processing & Machine Vision/Y. Gao 33
Image Enhancement -- Histogram
It means that the probability density of the gray level for thetransformed image is uniform
For the digital image, we have
So,
1r0 )()(0
≤≤== ∫ dwwrTs r
rp
1,...1,0 and 1 0 )( −=≤≤= Lkn rnrp kk
kr
1,.......,1,0 and 10
)()(00
−=≤≤
=== ∑∑==
Lkrn
T rpnrs j
k
jr
k
j
jkk
Image Processing & Machine Vision/Y. Gao 34
Image Enhancement -- Histogram
rk
17
67
57
47
37
27
10
7
6
5
4
3
2
1
0
=
=
=
=
=
=
=
=
rr
r
r
r
r
rr
nk
811222453296568501023790
02.003.006.008.016.021.025.019.0
nnrp k
kr=)( sk
00.198.095.089.081.065.044.019.0
1
1000
nk
500
rk
1000
nk
500
sk1
Image Processing & Machine Vision/Y. Gao 35
Image Processing & Machine Vision/Y. Gao 36
Image Enhancement -- Histogram
3.3. Histogram Matching (specification)
Sometimes we need to convert an image into another image with a specific gray level distribution.
Let and be the original and desired probabilitydensity functions respectively.
Firstly, we do the equalization transform for both of them.
)(rpr )(zpz
Image Processing & Machine Vision/Y. Gao 37
Image Enhancement -- Histogram
)()(0
dwwrTs r
rp∫==
We think s and v are the same. So,
substitute s with v,
)()(0
dwwzGv z
zp∫==
))(()(z )(
11
1
rTsvz
GGG
−−
−
==∴=Q
Image Processing & Machine Vision/Y. Gao 38
Image Enhancement -- Histogram
The procedure can be summarized as follows,(1) Equalize the original image using
(2) Equalize the desired image using
(3) Apply the inverse transformation functionwe can get the value z.
)()(0
dwwrTs r
rp∫==
)()(0
dwwzGv z
zp∫== )(1 sz G−=
Image Processing & Machine Vision/Y. Gao 39
Image Enhancement -- Histogram
The procedure above is also described as the following graph,
Image Processing & Machine Vision/Y. Gao 40
Image Enhancement -- Histogram
3.4. Local enhancement
When histogram equalization and histogram matching are usedto all sub-images in an image, we call it local enhancement.
Take the neighborhood of a point, for example 5x5, do theenhancement for all points.
An example is as follow.
Image Processing & Machine Vision/Y. Gao 41
Image Enhancement -- Histogram
Image Processing & Machine Vision/Y. Gao 42
Image Enhancement – Arithmetic/Logic
4.1. Image Subtraction
The result is that the different parts between two image are kept but the same parts between them are removed (become dark).
A typical application is the medical X-ray image test for a specificbody area of the patient.
We can do it by injecting some dye into the parts to be tested to let them differ from others.
),(),(),( yxhyxfyxg −=
Image Processing & Machine Vision/Y. Gao 43
Image Enhancement – Arithmetic/Logic
Image Processing & Machine Vision/Y. Gao 44
Image Enhancement – Arithmetic/Logic
4.2. Image Averaging
Assume a noisy image can be expressed as the following
is an original image
is the noise
And the noise is uncorrelated and has zero average value.
),),(),( yxyxfyxg (+= η
),(),(yxyxf
η
Image Processing & Machine Vision/Y. Gao 45
Image Enhancement – Arithmetic/Logic
When we acquire a set of noisy images , we compute)},({ yxgithe average value
When
),(1),(
y)](x,),([1),(1),(
1
11
yxM
yxf
yxfM
yxM
yxg
M
ii
M
ii
M
iig
∑
∑∑
=
==
+=
+==
η
η
),(),( 0),(1 ,1
yxfyxgyxM
MM
i→∴→∞→ ∑
=ηι
Image Processing & Machine Vision/Y. Gao 46
Image Processing & Machine Vision/Y. Gao 47
Image Enhancement – Filtering
5. Filtering
Removing or intensifying some components in an image by usingenhancement processing is called filtering.
Using spatial masks to process an image is called spatial filteringUsing the Fourier transform to process an image is called Frequency domain filtering.
Image Processing & Machine Vision/Y. Gao 48
Image Enhancement – Filtering
5.1. Spatial Filtering
We need to know that
1) A great change of gray level between the pixels usually stands for some obvious contours in an image and it corresponds to the high frequency components in an image.
2) Averaging the pixel grey levels in an image will let a digitalimage become blurring.
3) Intensifying the high frequency components in an image willlet a digital image have obvious contours.
Image Processing & Machine Vision/Y. Gao 49
Image Enhancement – Filtering
An example
High frequency components
Low frequency components
Image Processing & Machine Vision/Y. Gao 50
Image Enhancement – Filtering
5.1.1. Mask
Applying a mask to an image is a basic concept in image processing – convolution.
(1) What is a mask: A mask is a set of pixel positions with values called weights. A 3x3 mask is as follows.
1 w2 w3
w4 w5 w6
w7 w8 w9
21 1
242
1 2 1
1 w2 w3
w4 w5 w6
w7 w8 w9
11 1
111
1 1 1
w1 w2 w3
w4 w5 w6
w7 w8 w9
Image Processing & Machine Vision/Y. Gao 51
Image Enhancement – Filtering
(2) How to use a mask: For any point (x,y) in the image, align the center of the mask to it. We have
w1 w2 w3
w4 w5 w6
w7 w8 w9
z1 z 2 z3
z 4 z5 z6
z7 z8 z9
zwyxf ii
inew ∑==
9
1),(
(x,y)
Image Processing & Machine Vision/Y. Gao 52
Image Enhancement – Filtering
5.1.2. Smoothing filters
For low-pass filtering --- called smooth processing
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1 11 1
1 1 1
1 1 1
1 1 1
×251
×91
1 1 11 1
Image Processing & Machine Vision/Y. Gao 53
Effects of mask sizes
Image Enhancement – Filtering
Image Processing & Machine Vision/Y. Gao 54
Image Enhancement – Filtering
Image Processing & Machine Vision/Y. Gao 55
Image Enhancement – Filtering
Median filter – less blurring and effective in removing salt and pepper noise.
Image Processing & Machine Vision/Y. Gao 56
Image Enhancement – Filtering
5.1.3. Sharpening filters
For High-pass filtering
w
1− 1− 1− 1− 1−
1− 1− 1− 1−1− 1− 1− 1− 1−
1− 1− 1− 1−1−
1−
w1− 1−
1− 1− 1−
1− 1− ×251
×91
1− 1− 1− 1−1−
Image Processing & Machine Vision/Y. Gao 57
Image Enhancement – Filtering
Other forms of filters
Image Processing & Machine Vision/Y. Gao 58
Image Processing & Machine Vision/Y. Gao 59
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