1. technology and computing

LECTURE NOTES
ON
DIGITAL IMAGE PROCESSING
MR. G.Sasi M.E
ASST. PROFESSOR
DEPT OF ELECTRONICS & COMMUNICATION ENGINEERING
NPRCET
Syllabus
DIGITAL IMAGE PROCESSING
UNIT I DIGITAL IMAGE FUNDAMENTALS AND TRANSFORMS
Elements of visual perception – Image sampling and quantization basic relationship between pixels – Basic geometric transformations –
Introduction to fourier transform and DFT – Properties of 2D fourier transform – FFT – Separable image transforms – Walsh- Hadamard
– Discrete cosine transform, Haar, Slant-Karhunen – Love transforms.
UNIT II IMAGE ENHANCEMENT TECHNIQUES
Spatial domain methods: Basic grey level transformation – Histogram equalization – Image subtraction – Image averaging – Spatial
filtering – Smoothing, sharpening filters – Laplacian filters – Frequency domain filters – Smoothing – Sharpening filters –
Homomorphic filtering.
UNIT III IMAGE RESTORATION
Model of image degradation/restoration process – Noise models – Inverse filtering – Least mean square filtering – Constrained least
mean square filtering – Blind imager– Pseudo inverse – Singular value decomposition.
UNIT IV IMAGE COMPRESSION
Lossless compression: variable length coding – LZW coding – Bit plane coding– Predictive coding– DPCM. Lossy Compression:
Transform coding – Wavelet coding – Basics of image compression standards – JPEG, MPEG, basics of vector quantization.
UNIT V IMAGE SEGMENTATION AND REPRESENTATION
Edge detection – Thresholding – Region based segmentation – Boundary representation – Chair codes– Polygonal approximation –
Boundary segments – Boundary descriptors – Simple descriptors– Fourier descriptors – Regional descriptors – Simple descriptors–
Texture.
TEXT BOOK
1. Rafael C. Gonzalez, Richard E. Woods, “Digital Image Processing”, 2nd Edition,
Pearson Education, 2003.
REFERENCES
1. William K. Pratt, “Digital Image Processing” , John Willey ,2001
2. Millman Sonka, Vaclav Hlavac, Roger Boyle, Broos/Colic, Thompson Learniy,
Vision, “Image Processing Analysis and Machine”, 1999.
UNIT I DIGITAL IMAGE FUNDAMENTALS AND TRANSFORMS
Elements of visual perception – Image sampling and quantization basic relationship between pixels – Basic geometric transformations –
Introduction to fourier transform and DFT – Properties of 2D fourier transform – FFT – Separable image transforms – Walsh- Hadamard
– Discrete cosine transform, Haar, Slant-Karhunen – Love transforms.
Projections
• There
are two types of projections (P) of interest to us:
1. Perspective Projection
– Objects closer to the capture device appear bigger. Most
image formation situations can be considered to be under
this category, including images taken by camera and the
human eye.
2. Ortographic Projection
– This is “unnatural”. Objects appear the same size regardless of their distance to the “capture device”.
•
Both types of projections can be represented via mathematical formulas. Ortographic projection is easier and is sometimes used as a
mathematical convenience.
°
Inside the Camera - Sensitivity
• Once we
have cp(x0, y 0, λ) the characteristics of the capture device take
over.
• V (λ)
is the sensitivity function of a capture device. Each capture device
has such a function which determines how sensitive it is in capturing
the range of wavelengths (λ) present in cp(x0, y 0, λ).
The result is an “image function” which determines the amount of
reflected light that is captured at the camera coordinates (x0, y 0).
•
Z
f (x0, y 0 ) =
°
cp(x0, y 0, λ)V (λ)dλ
(1)
Let us determine the image functions for the above sensitivity functions imaging the same scene:
1. This is the most realistic of the three. Sensitivity is concentrated in a band around λ0 .
0
Z
0
cp (x0 , y 0 , λ)V1 (λ)dλ
f1 (x , y ) =
2. This is an unrealistic capture device which has sensitivity only to a single wavelength λ0 as
determined by the delta function. However there are devices that get close to such “selective”
behavior.
f2 (x0 , y 0 ) =
Z
cp (x0 , y 0 , λ)V2 (λ)dλ =
= cp (x0 , y 0 , λ0 )
Z
cp (x0 , y 0 , λ)δ(λ − λ0 )dλ
3. This is what happens if you take a picture without taking the cap off the lens of your camera.
0
0
Z
f3 (x , y ) =
= 0
°
0
0
cp (x , y , λ)V3 (λ)dλ =
Z
cp (x0 , y 0 , λ) 0 dλ
Sensitivity and Color
• For a camera that captures color images,
imagine that it has three
sensors at each (x0, y 0 ) with sensitivity functions tuned to the colors or
wavelengths red, green and blue, outputting three image functions:
0
0
fR (x , y ) =
fG (x0, y 0 ) =
fB (x0, y 0 ) =
• These
Z
Z
Z
cp(x0, y 0, λ)VR (λ)dλ
cp(x0, y 0, λ)VG (λ)dλ
cp(x0, y 0, λ)VB (λ)dλ
three image functions can be used by display devices (such
as
your monitor or your eye) to show a “color” image.
°
Digital Image Formation
• The image function fC (x0, y 0 ) is still a function
0 , y0
[ymin
max ] which vary in a continuum given by
of x0 ∈ [x0 min , x0max] and y0 ∈
the respective intervals.
• The
values taken by the image function are real numbers which again
vary in a continuum or interval fC (x0, y 0) ∈ [fmin, fmax].
• Digital computers
cannot process parameters/functions that vary in a
continuum.
• We
°
have to discretize:
Quantization
• fC (i, j) (i = 0, . . . , N − 1, j = 0, . . . , M − 1).
We have the second step of
discretization left.
• fC (i, j) ∈ [fmin , fmax], ∀(i, j).
• Discretize the
values fC (i, j) to P levels as follows:
Let ∆Q = fmax− Pfmin .
fˆC(i, j) = Q(fC (i, j))
where
Q(fC (i, j)) = (k + 1/2)∆Q + fmin
if and only if fC (i, j) ∈ [fmin + k∆Q, fmin + (k + 1)∆Q )
if and only if fmin + k∆Q ≤ fC (i, j) < fmin + (k + 1)∆Q
for k = 0, . . . , P − 1
°
(4)
Quantization to P levels
• Typically P = 28 = 256 and we have log2 (P ) = log2(28) = 8 bit
quantization.
• We
have thus achieved the second step of discretization.
• From
now on omit references to fmin, fmax and unless otherwise stated
assume that the original digital images are quantized to 8 bits or 256
levels.
• To denote this refer to fˆC (i, j)
255, i.e., let us say that
as taking integer values k where 0 ≤ k ≤
fˆC(i, j) ∈ {0, . . . , 255}
°
(5)
(R,G,B) Parameterization of Full Color Images
fˆR (i, j), fˆG(i, j), fˆB(i, j) →
,
•
fˆ
R
(i, j), fˆG(i, j)
,
full color image
, ⇒
and fˆB(i, j) are called the (R, G, B) parameterization of
the “color space” of the full color image.
• There
are other parameterizations, each with its own advantages and
disadvantages.
°
Grayscale Images
“Grayscale” image fˆgray (i, j)
• A grayscale
or luminance image can be considered to be one of the
components of a different parameterization.
• Advantage:
It captures most of the “image information”.
• Our emphasis
in this class will be on general processing.
Hence we
will mainly work with grayscale images in order to avoid the various
nuances involved with different parameterizations.
°
Images as Matrices
• Recalling the image formation operations we have discussed, note that
the image fˆgray (i, j) is an N × M matrix with integer entries in the range
0, . . . , 255.
• From
now on suppress (ˆ)gray and denote an image as a matrix “A” (or
B, . . ., etc.) with elements A(i, j) ∈ {0, . . . ,
255} for i = 0, . . . , N − 1 , j =
0, . . . , M − 1.
• So we
• Some
will be processing matrices!
processing
will take
we will do
an image A
with A(i, j) ∈ {0, . . . , 255} into a new matrix B
which may not have integer entries!
In these cases we must suitably scale and
round the elements of B in order to display it
as an image.
Computer Vision &
Digital Image Processing
Fourier Transform Properties, the
Laplacian, Convolution and
Correlation
Periodicity of the Fourier transform
• The discrete Fourier transform (and its inverse) are
periodic with period N.
F(u,v) = F(u+N,v) = F(u,v+N) = F(u+N,v+N)
• Although F(u,v) repeats itself infinitely for many
values of u and v, only N values of each variable
are required to obtain f(x,y) from F(u,v)
– i.e. Only one period of the transform is necessary to
specify F(u,v) in the frequency domain.
– Similar comments may be made for f(x,y) in the spatial
domain
Conjugate symmetry of the Fourier
transform
• If f(x,y) is real (true for all of our cases), the Fourier
transform exhibits conjugate symmetry
F(u,v)=F*(-u,-v)
or, the more interesting
|F(u,v)| = |F(-u,-v)|
where F*(u,v) is the complex conjugate of F(u,v)
Implications of periodicity & symmetry
• Consider a 1-D case:
– F(u) = F(u+N) indicates F(u) has a period of length N
– |F(u)| = |F(-u)| shows the magnitude is centered about the
origin
• Because the Fourier transform is formulated for
values in the range from [0,N-1], the result is two
back-to-back half periods in this range
• To display one full period in the range, move (shift)
the origin of the transform to the point u=N/2
9-4
Periodicity properties
Fourier spectrum
with back-to-back
half periods in the
range
[0,n-1]
Shifted spectrum
with a
full period
in the
same range
Periodicity properties: 2-D Example
Distributivity & Scaling
• The Fourier transform (and its inverse) are
distributive over addition but not over multiplication
• So,
ℑ{ f1 ( x, y) + f 2 ( x, y)} = ℑ{ f1 ( x, y)} + ℑ{ f 2 ( x, y)}
ℑ{ f1 ( x, y) × f 2 ( x, y)} ≠ ℑ{ f1 ( x, y)}× ℑ{ f 2 ( x, y)}
• For two scalars a and b,
af ( x, y) ⇔ aF (u,
v)
1
f (ax, by) ⇔
F (u / a, v / b)
ab
Average Value
• A widely used expression for the average value of a
2-D discrete function is:
f ( x, y) =
N −1 N −1
1
N2
∑∑ f ( x, y)
x =0 y =0
• From the definition of F(u,v), for u=v=0,
F (0,0) =
• Therefore,
1
N
N −1 N −1
∑∑ f ( x, y)
f ( x, y) =
x =0 y =0
1
F (0,0)
N
The Laplacian
• The Laplacian of a two variable function f(x,y) is
given as:
∇ 2 f ( x, y) =
∂2 f ∂2 f
2 +
2
∂x
∂y
• From the definition of the 2-D Fourier transform,
ℑ{∇ 2 f ( x, y)} ⇔ −(2π ) 2 (u 2 + v 2 )F (u, v)
• The Laplacian operator is useful for outlining edges
in an image
The Laplacian: Matlab example
% Given F(u,v), use the Laplacian
% to construct an edge outlined
% representation of the f(x,y)
[f,fmap]=bmpread('lena128.bmp');
F=fft2(f);
Fedge=zeros(128);
for u=1:128
for v=1:128
Fedge(u,v)=(2*pi).^2*(u.^2+v.^2)*F(u,v);
end end
fedge=ifft2(Fedge);
image(real(fedge));colormap(gray(256);
0
Convolution & Correlation
• The convolution of two functions f(x) and g(x) is
denoted f(x)*g(x) and is given by:
+∞
f ( x) * g ( x) = ∫ f (α ) g ( x − α )dα
−∞
• Where α is a dummy variable of integration.
• Example: Consider the following functions f(α ) and
g(α )
f(α)
g(α)
1
1/2
1
α
1
α
1-D convolution example
• Compute g(-α) by folding g(α) about the origin
g(α)
g(−α)
1/2
1/2
α
1
α
-1
• Compute g(x-α) by displacing g(-α) by the value x
g(−α)
g(x−α)
1/2
-1
1/2
α
-1
x
α
1-D convolution example (continued)
• Then, for any value x, we multiply g(x-α) and f(α)
and integrate from -∞ to +∞
• For 0≤x ≤ 1 we have
For 1 ≤ x ≤ 2 we have
f(α)g(x- α)
f(α)g(x- α)
1
1
1
α
1
α
1-D convolution example (continued)
• Thus we have
0 ≤ x ≤1
⎧x / 2
⎪
f ( x) * g ( x) = ⎨1 − x / 2 1 ≤ x ≤ 2
⎪0
elsewhere.
⎩
• Graphically,
f(x)*g(x)
1/2
1
2
x
Convolution and impulse functions
• Of particular interest will be the convolution of a
function f(x) with an impulse function δ(x-x0)
+∞
∫ f ( x)δ ( x − x
0
)dx = f ( x0 )
−∞
• The function δ(x-x0) may be viewed as having an
area of unity in an infinitesimal neighborhood
around x0 and 0 elsewhere. That is
+∞
∫δ (x − x
0
−∞
x0+
)dx = ∫ δ ( x − x0 )dx = 1
x0−
Convolution and impulse functions
(continued)
• We usually say that δ(x-x0) is located at x=x0 and the
strength of the impulse is given by the value of f(x) at x=x0
• If f(x)=A then, Aδ(x-x0) is impulse of strength A at x=x0.
• Graphically this is:
A
x0
Aδ(x-x0)
x
Convolution with an impulse function
• Given f(x) is
f(α)
A
• and g(x)=δ(x+T)+ δ(x)+ δ(x-T)
α
α
x
T
-T
g(α)
Convolution with an impulse function
(continued)
• f(x)*g(x) is
A
−Τ
α
Τ
x
Convolution and the Fourier transform
• f(x)*g(x) and F(u)G(u) form a Fourier transform pair
• If f(x) has transform F(u) and g(x) has transform
G(u) then f(x)*g(x) has transform F(u)G(u)
f ( x) * g ( x) ⇔ F (u)G(u)
f ( x) g ( x) ⇔ F (u) * G(u)
• These two results are commonly referred to as the
convolution theorem
Frequency domain filtering
• Enhancement in the frequency domain is straightforward
– Compute the Fourier transform
– Multiply the result by a filter transform function
– Take the inverse transform to produce the enhanced image
• In practice, small spatial masks are used considerably more
than the Fourier transform because of their simplicity of
implementation and speed of operation
• However, some problems are not easily addressable by
spatial techniques
– Such as homomorphic filtering and some image restoration
techniques
Lowpass frequency domain filtering
• Given the following relationship
G(u, v) = H (u, v)F (u, v)
• where F(u,v) is the Fourier transform of an image to
be smoothed
• The problem is to select an H(u,v) that yields an
appropriate G(u,v)
• We will consider zero-phase-shift filters that do not
alter the phase of the transform (i.e. they affect the
real and imaginary parts of F(u,v) in exactly the
same manner)
Ideal lowpass filter (ILPF)
• A transfer function for a 2-D ideal lowpass filter (ILPF) is
given as
⎧1 if D(u, v) ≤ D 0
H (u, v) = ⎨
⎩0 if D(u, v) > D 0
• where D0 is a stated nonnegative quantity (the cutoff
frequency) and D(u,v) is the distance from the point (u,v) to
the center of the frequency plane
D(u, v) = u 2 + v 2
H(u,v)
v
u
Ideal lowpass filter (ILPF) (continued)
• The point D0 traces a circle from the frequency origin giving a locus of
cutoff frequencies (all are at distance D0 from the origin)
• One way to establish a set of “standard” loci is to compute circles that
encompass various amounts of the total signal power PT
N −1 N −1
• PT is given by
PT = ∑∑ P(u, v)
u =0 v =0
•
where P(u,v) is given as
P(u, v) = F (u, v) = R 2 (u, v) + I 2 (u, v)
2
• For the centered transform, a circle of radius r encompasses β percent of
the power, where
⎡
⎤
β = 100 ⎢∑∑ P(u, v) / PT ⎥ (the summation
⎣ u
v
⎦
is over all points (u, v) encompassed by the circle)
UNIT 1
2 marks
1. Define Image?
2. What is Dynamic Range?
3. Define Brightness?
4. Define Tapered Quantization?
5. What do you meant by Gray level?
6. What do you meant by Color model?.
7. List the hardware oriented color models?
8. What is Hue of saturation?
9. Explain separability property in 2D fourier transform
10. What are the properties of Haar and slant transform.
11. Define Resolutions?
12. What is meant by pixel?
13. Define Digital image?
14. What are the steps involved in DIP?
16. Specify the elements of DIP system?
18. What are the types of light receptors?
19. Differentiate photopic and scotopic vision?
26. Define sampling and quantization
27. Find the number of bits required to store a 256 X 256 image with 32 gray levels?
28. Write the expression to find the number of bits to store a digital image?
30. What do you meant by Zooming and shrinking of digital images?
32. Write short notes on neighbors of a pixel.
33. Explain the types of connectivity.
34. What is meant by path?
36. What is geometric transformation?
40. What is Image Transform?
16 MARKS
UNIT I
1. Explain the steps involved in digital image processing.
(or)
Explain various functional block of digital image processing
# Image acquisition
# Preprocessing
# Segmentation
# Representation and Description
# Recognition and Interpretation
2. Describe the elements of visual perception.
# Cornea and Sclera
# Choroid – Iris diaphragm and Ciliary body
# Retina- Cones and Rods
3. Describe image formation in the eye with brightness adaptation and
discrimination
# Brightness adaptation
# Subjective brightness
# Weber ratio
#Mach band effect
#simultaneous contrast
4. Write short notes on sampling and quantization.
# Sampling
# Quantization
# Representing Digital Images
5. Describe the functions of elements of digital image processing system
with a diagram.
# Acquisition
# Storage
# Processing
# Communication
# Display
6. Explain the basic relationships between pixels?
# Neighbors of a pixel
# Connectivity, Adjacency, Path
# Distance Measure
# Arithmetic and Logic Operations
7. Explain the properties of 2D Fourier Transform.
# Separability
# Translation
# Periodicity and Conjugate Symmetry
# Rotation
# Distribution and Scaling
# Average Value
# Laplacian
# Convolution and correlation
# Sampling
8. ( i )Explain convolution property in 2D fourier transform.
* lD Continuous
* lD Discrete
* lD convolution theorem
* 2D continuous
* 2D Discrete
* 2D convolution theorem
(ii) Find F (u) and |F (u)|
9. Explain Fast Fourier Transform (FFT) in detail.
# FFT Algorithm
# FFT Implementation
10. Explain in detail the different separable transforms
# Forward lD DFT & 2D DFT
# Inverse lD DFT & 2D DFT
# Properties
11. Explain Hadamard transformation in detail.
# lD DHT
# lD Inverse DHT
# 2D DHT
# 2D Inverse DHT
12. Discuss the properties and applications of
1)Hadamard transform
2)Hotelling transform
# Properties of hadamard:
Real and orthogonal
fast transform
faster than sine transform
Good energy compaction for image
# Appl:
Image data compression,
filtering and design of course
# Properties of hotelling:
Real and orthogonal
Not a fast transform
Best energy compaction for image
# Appl:
Useful in performance evaluation & for finding performance
bounds
13. Explain Haar transform in detail.
# Def P= 2P+q-l
# Find h k (z)
14. Explain K-L transform in detail.
Consider a set of n or multi-dimensional discrete signal represented as column
vector xl,x2,…xn each having M elements,
X=
Xl
X2
.
.
Xn
The mean vector is defined as Mx=E{x}
Where E{x} is the expected value of x.
M
For M vector samples mean vector is Mx=l/M ∑ Xk
K=l
T
The co-variant matrix is, Cx=E{(X-Mx)(X-Mx)}
M
T
For M samples, Cx=l/M ∑ (xk-Mx)(xk-Mx).
K=l
K-L Transform Y= A (X- MX)
UNIT II IMAGE ENHANCEMENT TECHNIQUES
Spatial domain methods: Basic grey level transformation – Histogram equalization – Image subtraction – Image averaging – Spatial
filtering – Smoothing, sharpening filters – Laplacian filters – Frequency domain filters – Smoothing – Sharpening filters –
Homomorphic filtering.
Dynamic Range, Visibility and Contrast Enhancement
• Contrast
enhancing point functions we have discussed earlier expand
the dynamic range occupied by certain “interesting” pixel values in the
input image.
• These pixel values in the input image may be difficult to distinguish
and the goal of contrast enhancement is to make them “more visible”
in the output image.
• Don’t forget we have a limited dynamic range (0 − 255) at our disposal.
Point Functions and Histograms
• In
general a point operation/function B(i, j) = g(A(i, j)) results in a new
histogram hB (l) for the output image that is different from hA(l).
• The relationship
between hB (l) and hA(l) may not be straightforward as
we have already discussed in Lecture 2.
• You must
g(l):
–
learn how to calculate hB (l) given hA(l) and the point function
Usually via writing a matlab script that computes hB (l)
from hA(l) and g(l).
– By sketching hB (l) given the sketches for hA(l)
and g(l).
°
“Unexpected” Effect of some Point Functions
− B has ∼ 10 times as
few distinct pixel values.
− Note also the vertical axis
scaling in hB (l).
Stretched/Compressed Pixel Value Ranges
• B(i, j) = g(A(i, j))
Suppose g(l) represents
an overall point function which includes contrast stretching/compression, emphasis/de-emphasis, rounding, normalizing etc.
• Given an
image matrix A, B(i, j) = g(A(i, j)) is also an image matrix.
may not be “continuous” or connected and it also may not be
composed of connected line segments.
• g(l)
Image Segmentation
• If one
views an image as depicting a scene composed of different
objects, regions, etc. then segmentation is the decomposition of an
image into these objects and regions by associating or “labelling” each
pixel with the object that it corresponds to.
• Most
humans can easily segment an image.
•
Computer automated segmentation is a difficult problem, requiring
sophisticated algorithms that work in tandem.
• “High level”
segmentation, such as segmenting humans, cars etc.,
from an image is a very difficult problem. It is still considered unsolved
and is actively researched.
• Based
on point processing, histogram based image segmentation is a
very simple algorithm that is sometimes utilized as an initial guess at
the “true” segmentation of an image.
°
Histogram Based Image Segmentation
• For
a given image, decompose the range of pixel values (0, . . . , 255) into
“discrete” intervals Rt = [at, bt ], t = 1, . . . , T , where T is the total number
of segments.
• Each Rt is
typically obtained as a range of pixel values that correspond
to a hill of hA(l).
• “Label”
the pixels with pixel values within each Rt via a point function.
• Main Assumption: Each object
pixels with similar pixel values.
°
is assumed to be composed of
Limitations
• Histogram based
segmentation operates on each image pixel independently. As mentioned earlier, the main assumption is that objects
must be composed of pixels with similar pixel values.
• This
independent processing ignores a second important property:
Pixels within an object should be spatially connected.
For example,
B3, B4, B5 group spatially disconnected objects/regions into the same
segment.
•
In practice, one would use histogram based segmentation in tandem
with other algorithms that make sure that computed objects/regions
are spatially connected.
°
Histogram Equalization
• For
a given image A, we will now design a special point function gAe (l)
which is called the histogram equalizing point function for A.
• If B(i, j) = gAe (A(i, j)), then
possible irrespective of hA(l)
• Histogram
our aim is to make hB (l) as uniform/flat as
equalization will help us:
– Stretch/Compress
an image such that:
∗ Pixel values that occur frequently in A occupy a bigger dynamic range in B,
i.e., get stretched and become more visible.
∗ Pixel values that occur infrequently in A occupy a smaller dynamic range in B,
i.e., get compressed and become less visible.
–
• The
Compare images by “mapping” their histograms into a standard
histogram and sometimes “undo” the effects of some unknown
processing.
techniques we are going to use to get gAe (l) are also applicable
in
histogram modification/specification.
°
Histogram Equalizing Point Function
• Let g1(l) =
Pl
• Image A ⇒
. Note that g1(l) ∈ [0, 1].
k=0 pA (k)
“equalize image” ⇒ B(i, j) = ge (A(i,
j)).
A
•in
general pB (l) will not be a uniform probability mass function but
hopefully it will be close.
• In matlab >> help filter
to construct geA(A(i, j)) fast.
• Assuming you gAe is an array that contains the computed g e (l)A, you
can use >> B = gAe(A + 1); to obtain the equalized image.
Stretching and Compression
°
• gAe (l)
stretches the range of pixel values that occur frequently in A.
• gAe (l)
A.
compresses the range of pixel values that occur infrequently in
Example
°
Comparison/“Undoing”
Instead of comparing A and C, compare their equalized versions.
°
Comparison/“Undoing” - contd.
1
Histogram Equalization
• gl (l) =
Pl
k=0 pA (k)
= gl (l) — gl (l — 1) = pA(l) =
• gAe (l) = round(255gl (l))
image A.
hA (l)
NM
(l = 1, . . . , 255).
is the histogram equalizing point function for the
• B(i, j) = g e A(A(i, j))
is the histogram equalized version of A.
• In general, histogram equalization stretches/compresses an image such
that:
– Pixel values that occur frequently in A occupy a bigger dynamic range in B, i.e., get
stretched and become more visible.
– Pixel values that occur infrequently in A occupy a smaller dynamic range in B, i.e., get
compressed and become less visible.
• Histogram
equalization is not ideal, i.e., in general B will have a
“flatter” histogram than A, but pB (l) is not guaranteed to be uniform
(flat).
Random Images - Images of White Noise
• A single outcome of a
χ ∈ [0, 1] in matlab: >>
continuous amplitude uniform random variable
x = rand(1, 1);
• An N × M
matrix of outcomes of a continuous amplitude uniform
random variable χ ∈ [0, 1] in matlab: >> X = rand(N, M );
• An N × M
image matrix of outcomes of a discrete amplitude uniform
random variable Θ ∈ {0, 1, . . . 255} in matlab: >> A = round(255 ∗ X );
• A single outcome of a continuous amplitude
χ (µ = 0, σ 2 = 1) in matlab: >>
x = randn(1, 1);
gaussian random variable
• An N × M
matrix of outcomes of a continuous amplitude gaussian
random variable χ (µ = 0, σ2 = 1) in matlab: >>
X = randn(N, M );
image matrix of outcomes of a discrete amplitude “gaussian”
random variable Θ ∈ {0, 1, . . . 255}: A(i, j) = g s X (X (i, j)).
• An N ×M
Example
Warning
• Remember,
tograms.
two totally different images may have very similar his-
Histogram Matching Specification
• Given images A
and B, using point processing we would
like to generate an image C from A such that hC (l) ∼ hB
(l), (l = 0, . . . , 255).
• More generally, given an image A and a histogram hB
(l) (or sample proba- bility mass function pB (l)), we would like to
generate an image C such that hC (l) ∼ hB (l), (l = 0, . . . ,
255).
•
Histogram matching/specification enables us to
“match” the grayscale distribution in one image to
the grayscale distribution in another im- age.
UNIT II
2 Marks
1. Specify the objective of image enhancement technique.
2. Explain the 2 categories of image enhancement.
3. What is contrast stretching?
4. What is grey level slicing?
5. Define image subtraction.
6. What is meant by masking?
7.Define Histogram.
8.What is meant by Histogram Equilisation
9. Differentiate linear spatial filter and non-linear spatial filter.
10. What is meant by laplacian filter?
11. Write the application of sharpening filters?
16 Marks
1. Explain the types of gray level transformation used for image enhancement.
# Linear (Negative and Identity)
# Logarithmic( Log and Inverse Log)
# Power_law (nth root and nth power)
# Piecewise_linear (Constrast Stretching, Gray level Slicing, Bit plane Slicing)
2. What is histogram? Explain histogram equalization.
# P(rk) = nk/n
# Ps(s) = l means histogram is arranged uniformly.
3. Discuss the image smoothing filter with its model in the spatial domain.
# LPF-blurring
# Median filter – noise reduction & for sharpening image
4. What are image sharpening filters. Explain the various types of it.
# used for highlighting fine details
# HPF-output gets sharpen and background becomes darker
# High boost- output gets sharpen but background remains unchanged
# Derivative- First and Second order derivatives
Appl:
# Medical image
# electronic printing
# industrial inspection
5. Explain spatial filtering in image enhancement.
# Basics
# Smoothing filters
# Sharpening filters
6. Explain image enhancement in the frequency domain.
# Smoothing filters
# Sharpening filters
# Homomorphic filtering
7. Explain Homomorphic filtering in detail.
# f(x, y) = i(x, y) . r(x, y)
# Calculate the enhanced image g(x,y)
UNIT III IMAGE RESTORATION
Model of image degradation/restoration process – Noise models – Inverse
filtering – Least mean square filtering – Constrained least mean square filtering –
Blind imager– Pseudo inverse – Singular value decomposition.
Image restoration
• Restoration is an objective process that attempts to
recover an image that has been degraded
– A priori knowledge of the degradation phenomenon
– Restoration techniques generally oriented toward
modeling the degradation
– Application of the inverse process to “recover” the original
image
– Involves formulating some criterion (criteria) of
“goodness” that is used to measure the desired result
Image restoration (continued)
• Removal of blur by a deblurring function is an
example restoration technique
• We will consider the problem only from where a
degraded digital image is given
– Degradation source will not be considered here
• Restoration techniques may be formulated in the
– Frequency domain
– Spatial domain
Image degradation/restoration process
• Given g(x,y), some knowledge about H, and some
knowledge about the noise term, the objective is to produce
an estimate of the original image.
– The more that is known about H and the noise term the closer the
estimate can be
• Various types of restoration filters are used to accomplish
this.
g ( x, y)
f ( x, y) Degradation
Function
H
+
Noise
η ( x, y)
Restoration
filter(s)
fˆ ( x, y)
Image degradation/restoration process
• If H is a linear, position invariant process, then the
degraded image can be described as the
convolution of h and f with an added noise term:
g ( x, y) = h(x, y) ∗ f ( x, y) + η ( x, y)
• h(x,y) is the spatial domain representation of the
degradation function.
• In the frequency domain, the representation is:
G(u, v) = H (u, v)F (u, v) + N (u, v)
• Each term in this expression is the Fourier
transform of the of the corresponding terms in the
equation above.
Noise models
• Common sources of noise
– Acquisition
• Environmental conditions (heat, light), imaging sensor quality
– Transmission
• Noise in transmission channel
• Spatial and frequency properties of noise
– Frequency properties of noise refer to the frequency content of noise
in the Fourier sense
– For example, if the Fourier spectrum of the noise is constant, the
noise is usually called white noise
• A carry over from the fact that white light contains nearly all frequencies
in the visible spectrum in basically equal proportions
– Excepting spatially periodic noise, we will assume that noise is
independent of spatial coordinates and uncorrelated to the image
Noise probability density functions
• With respect to the spatial noise term, we will be concerned
with the statistical behavior of the intensity values.
• May be treated as random variables characterized by a
probability density function (PDF)
• Common PDFs used will describe:
–
–
–
–
–
–
Gaussian noise
Rayleigh noise
Erlang (Gamma) noise
Exponential noise
Uniform noise
Impulse (salt-and-pepper) noise
Gaussian noise
• Gaussian (normal) noise
models are simple to
consider
• The PDF of a Gaussian
random variable, z, is given
to the right as:
• In this case, approximately
70% of the values of z will
be within within one
standard deviation
• Approximately 95% of the
values of z will be within
within two standard
deviations
p( z) =
1
2π σ
e −( z − z )
2
/ 2σ 2
where
z represents intensity
z represents the mean (average) value of z
σ is the standard deviation
σ 2 is the variance of z
Rayleigh noise
• The PDF of Rayleigh noise is
given as:
⎧2
⎪ ( z − a)e −( z − a )
p( z) = ⎨ b
⎪
⎩0
2
/ b
for z ≥ a
for z < a
where
z represents intensity
z = a + πb / 4
σ2 =
b(4 − π )
4
• Note the displacement, by a,
from the origin
• The basic shape of this PDF is
skewed to the right
– Can be useful in approximating
skewed histograms
Erlang (Gamma) noise
• The PDF of Erlang noise is
given as:
⎧ a b z b −1 − az
⎪
e
p( z) = ⎨ (b − 1)!
0
for z ≥ 0
for z < 0
where
z represents intensity
z =b/a
σ 2 = b / a2
• a > 0, b is a positive integer
Exponential noise
• The PDF of exponential
noise is given as:
⎧ae − az
p( z) = ⎨
⎩0
where
for
for zz ≥
< 00
z represents intensity
z = 1/ a
σ 2 = 1/ a2
• a>0
• This PDF is a special case
of the Erlang PDF with b=1
Uniform noise
• The PDF of uniform noise is
given as:
⎧ 1
⎪
if a ≤ z ≤ b
p( z) = ⎨ b − a
0
otherwise
where
z represents intensity
z=
σ2
a +b
2
(b − a) 2
=
12
Impulse (salt-and-pepper) noise
• The PDF of (bipolar)
impulse noise is given as:
⎧ Pa for z = a
⎪
p( z) = ⎨ Pb for z = b
⎪
⎩ 0 otherwise
• If b>a then any pixel with
intensity b will appear as a
light dot in the image
• Pixels with intensity a will
appear as a dark dot
Example noisy images
Example noisy images (continued)
Sample periodic images and their spectra
50
50
100
100
150
150
200
200
250
250
50
100
150
200
250
50
100
150
200
250
122
124
126
128
130
132
134
136
105
110
115
120
125
130
135
140
145
150
Estimation of noise parameters
• Noise parameters can often be estimated by
observing the Fourier spectrum of the image
– Periodic noise tends to produce frequency spikes
• Parameters of noise PDFs may be known (partially)
from sensor specification
– Can still estimate them for a particular imaging setup
– One method
• Capture a set of “flat” images from a known setup (i.e. a uniform
gray surface under uniform illumination)
• Study characteristics of resulting image(s) to develop an indicator
of system noise
Estimation of noise parameters (continued)
• If only a set of images already generated by a sensor are
available, estimate the PDF function of the noise from small
strips of reasonably constant background intensity
• Consider a subimage (S) and let
ps(zi), i=0,1,2,…L-1
• denote the probability estimates of the intensities of the
pixels in S.
• L is the number of possible intensities in the image
• The mean and the variance of the pixels in S are given by:
L −1
z = ∑ zi p ( z ) and σ
=0
L −1
2
= ∑ ( z i − z ) 2 p ( zi )
i =0
s
s
i
i
Estimation of noise parameters (continued)
• The shape of the noise histogram identifies the
closest PDF match
– If the shape is Gaussian, then the mean and variance are
all that is needed to construct a model for the noise (i.e.
the mean and the variance completely define the
Gaussian PDF)
– If the shape is Rayleigh, then the Rayleigh shape
parameters (a and b) can be calculated using the mean
and variance
– If the noise is impulse, then a constant (with the exception
of the noise) area of the image is needed to calculate Pa
and Pb probabilities for the impulse PDF
Histograms from noisy strips of an area of an
image
Restoration in the presence of noise only –spatial
filtering
• When only additive random noise is present, spatial
filtering is commonly used to restore images
• Common types
– Mean filters
– Order-Statistic filters
– Adaptive filters
Mean filters (arithmetic)
• Arithmetic mean filter
– Computes the average value of a corrupted image g(x,y)
in the area defined by a window (neighborhood)
ˆf ( x, y) = 1
∑ g (s, t )
mn ( s ,t )∈S xy
– The operation is generally implemented using a spatial
filter of size m*n in which all coefficients have value 1/mn
– A mean filter smoothes local variations in an image
– Noise is reduced as a result of blurring
Mean filters (geometric)
• Geometric mean filter
– A restored pixel is given by the product of the pixels in an
area defined by a window (neighborhood), raised to the
power 1/mn
1
⎡
⎤ mn
fˆ ( x, y) = ⎢ ∏ g (s, t )⎥
⎢⎣( s ,t )∈S xy
– Achieves smoothing comparable to the arithmetic mean
filter, but tends to loose less detail in the process
Arithmetic and geometric mean filter examples
Mean filters (harmonic)
• Harmonic mean filter
– A restored pixel is given by the expression
fˆ ( x, y) =
mn
∑
( s ,t )∈S xy
1
g (s, t )
– Works well for salt noise (fails for pepper noise)
– Works well for Gaussian noise also
Mean filters (contraharmonic)
• Contraharmonic mean filter
– A restored pixel is given by the expression
fˆ ( x, y) =
∑ g (s, t )
Q +1
( s ,t )∈S xy
∑ g (s, t )
Q
( s ,t )∈S xy
–
–
–
–
–
Q is the order of the filter
Works well for salt and pepper noise (cannot do both simultaneously)
+Q eliminates pepper noise, -Q eliminates salt noise
Q=0 → arithmetic mean filter
Q=-1 → harmonic mean filter
Contraharmonic mean filter examples
Contraharmonic mean filter examples
UNIT III
2 Marks
1. What is meant by Image Restoration?
2. What are the two properties in Linear Operator?
3. Explain additivity property in Linear Operator?
4. How a degradation process is modeled?
5. Explain homogenity property in Linear Operator?
8. Define circulant matrix?
10. What are the two methods of algebraic approach?
11. Define Gray-level interpolation?
12. What is pseudo inverse filter?
13.What is meant by least mean square filter?
14. Write the properties of Singular value Decomposition(SVD)?
16 Marks
1. Explain the algebra approach in image restoration.
# Unconstrained
# Constrained
2. What is the use of wiener filter in image restoration. Explain.
# Calculate f^
# Calculate F^(u, v)
3. What is meant by Inverse filtering? Explain.
# Recovering i/p from its o/p
# Calculate f^(x, y)
4. Explain singular value decomposition and specify its properties.
# U= m=l∑rψ√λm mφ T
This equation is called as singular value decomposition of an image.
# Properties
•The SVD transform varies drastically from image to image.
•The SVD transform gives best energy packing efficiency for any given
image.
•The SVD transform is useful in the design of filters finding least
square,minimum solution of linear equation and finding rank of large
matrices.
5. Explain image degradation model /restoration process in detail.
# Image degradation model /restoration process diagram
# Degradation model for Continuous function
# Degradation model for Discrete function – l_D and 2_D
6. What are the two approaches for blind image restoration? Explain in detail.
> Direct measurement
> Indirect estimation
UNIT IV IMAGE COMPRESSION
Lossless compression: variable length coding – LZW coding – Bit plane
coding– Predictive coding– DPCM. Lossy Compression: Transform coding –
Wavelet coding – Basics of image compression standards – JPEG, MPEG, basics of
vector quantization.
Objectives
At the end of this lesson, the students should be able to:
1. Explain the need for standardization in image transmission and reception.
2. Name the coding standards for fax and bi-level images and state their
characteristics.
3. Present the block diagrams of JPEG encoder and decoder.
4. Describe the baseline JPEG approach.
5. Describe the progressive JPEG approach through spectral selection.
6. Describe
the
progressive
JPEG
approach
through
successive
approximation.
7. Describe the hierarchical JPEG approach.
8. Describe the lossless JPEG approach.
9. Convert YUV images from RGB.
10. Illustrate the interleaved and non-interleaved ordering for color images.
Introduction
With the rapid developments of imaging technology, image compression and coding
tools and techniques, it is necessary to evolve coding standards so that there is
compatibility and interoperability between the image communication and storage
products manufactured by different vendors. Without the availability of standards,
encoders and decoders can not communicate with each other; the service providers
will have to support a variety of formats to meet the needs of the customers and the
customers will have to install a number of decoders to handle a large number of data
formats. Towards the objective of setting up coding
standards, the
international standardization
agencies,
such
as
International Standards Organization (ISO), International Telecommunications Union
(ITU), International Electro-technical Commission (IEC) etc. have formed expert
groups and solicited proposals from industries, universities and research laboratories.
This has resulted in establishing standards for bi-level (facsimile) images and
continuous tone (gray scale) images. In this lesson, we are going to discuss the
highlighting features of these standards. These standards use the coding and
compression techniques – both lossless and lossy which we have already studied in the
previous lessons.
The first part of this lesson is devoted to the standards for bi-level image coding.
Modified Huffman (MH) and Modified Relative Element Address Designate
(MREAD) standards are used for text-based documents, but more recent
standards like JBIG1 and JBIG2, proposed by the Joint bi-level experts’ group (JBIG)
can efficiently encode handwritten characters and binary halftone images. The latter part
of this lesson is devoted to the standards for continuous tone images. We are going to
discuss in details about the Joint Photographic Experts Group (JPEG) standard and its
different modes, such as baseline (sequential), progressive, hierarchical and lossless.
Coding Standards for Fax and Bi-level Images
Consider an A4-sized (8.5 in x 11 in) scanned page having 200 dots/in. An
uncompressed image would require transmission of 3,740,000 bits for this
scanned page. It is however seen that most of the information on the scanned page is
highly correlated along the scan lines, which proceed in the direction of left to right in
top to bottom order and also in between the scan lines. The coding standards have
exploited this redundancy to compress bi-level images. The coding standards
proposed for bi-level images are:
(a) Modified Huffman (MH): This algorithm performs one-dimensional run
length coding of scan lines, along with special end-of-line (EOL), end-of- page
(EOP) and synchronization codes. The MH algorithm on an average achieves a
compression ratio of 20:1 on simple text documents.
(b) Modified Relative Element Address
Designate
(MREAD): This
algorithm uses a two-dimensional run length coding to take advantage of
vertical spatial redundancy, along with horizontal spatial redundancy. It uses
the previous scan line as a reference when coding the current line. The position
of each black-to-white or white-to-black transition is coded relative to a
reference element in the current scan line. The compression ratio is improved to
25:1 for this algorithm.
(c) JBIG1: The earlier two algorithms just mentioned work well for printed texts
but are inadequate for handwritten texts or binary halftone images (continuous
images converted to dot patterns). The JBIG1 standard, proposed by the
Joint Bi-level Experts Group uses a larger region of support for coding the
pixels. Binary pixel values are directly fed into an arithmetic coder, which
utilizes a sequential template of nine adjacent and previously coded pixels plus
one adaptive pixel to form a 10-bit context. Other than the sequential mode
just described, JBIG1 also supports progressive mode in which a reduced
resolution starting layer image is followed by the transmission of progressively
higher resolution layers. The compression ratios of JBIG1 standard is
slightly better than that of MREAD for text images but has an
improvement of 8-to-1 for binary halftone images.
(d) JBIG2: This is a more recent standard proposed by the Joint bi-level Experts
Group. It uses a soft pattern matching approach to provide a solution to the
problem of substitution errors in which an imperfectly scanned symbol is
wrongly matched to a different symbol, as frequently observed in Optical
Character Recognition (OCR). JBIG2 codes the bit- map of each mark, rather
than its matched class index. In case a good match cannot be found for the
current mark, it becomes a token for a new class. This new token is then coded
using JBIG1 with a fixed template of previous pixels around the current mark.
The JBIG2 standard is seen to be 20% more efficient than the JBIG1 standard
for lossless compression.
Continuous tone still image coding standards
A different set of standards had to be created for compressing and coding
continuous tone monochrome and color images of any size and sampling rate. Of these,
the Joint Photographic Expert Group (JPEG)’s first standard, known as JPEG is the
most widely used one. Only in recent times, the new standard JPEG-2000 has its
implementations in still image coding systems. JPEG is a very simple and easy to
use standard that is based on the Discrete Cosine Transform (DCT).
JPEG Encoder
Figure shows the block diagram of a JPEG encoder, which has the following
components:
(a) Forward Discrete Cosine Transform (FDCT): The still images are first
partitioned into non-overlapping blocks of size 8x8 and the image samples
are shifted from unsigned integers with range [0,2 p − 1] to signed integers
with range [− 2 p −1 ,2 p −1 ], where p is the number of bits (here, p = 8 ). The
theory of the DCT has been already discussed in lesson-8 and will not be
repeated here. It should however be mentioned that to preserve freedom for
innovation and customization within implementations, JPEG neither specifies
any unique FDCT algorithm, nor any unique IDCT algorithms.
The implementations may therefore differ in precision and JPEG has
specified an accuracy test as a part of the compliance test.
(b) Quantization: Each of the 64 coefficients from the FDCT outputs of a block
is uniformly quantized according to a quantization table. Since the aim is to
compress the images without visible artifacts, each step-size should be chosen
as the perceptual threshold or for “just noticeable distortion”. Psycho-visual
experiments have led to a set of quantization tables and these appear in ISOJPEG standard as a matter of information, but not a requirement.
The quantized coefficients are zig-zag scanned, as described in lesson-8. The DC
coefficient is encoded as a difference from the DC coefficient of the previous
block and the 63 AC coefficients are encoded into (run, level) pair.
(c) Entropy Coder: This is the final processing step of the JPEG encoder.
The JPEG standard specifies two entropy coding methods – Huffman and
arithmetic coding. The baseline sequential JPEG uses Huffman only, but codecs
with both methods are specified for the other modes of operation. Huffman
coding requires that one or more sets of coding tables are specified by the
application. The same table used for compression is used needed to decompress
it. The baseline JPEG uses only two sets of Huffman tables – one for DC and
the other for AC.
JPEG Decoder
Figure shows the block diagram of the JPEG decoder. It performs the inverse operation
of the JPEG encoder.
Modes of Operation in JPEG
The JPEG standard supports the following four modes of operation:
• Baseline or sequential encoding
•
Progressive
encoding
(includes
spectral
selection
and
successive
approximation approaches).
•
Hierarchical encoding
•
Lossless encoding
Baseline Encoding: Baseline sequential coding is for images with 8-bit samples and
uses Huffman coding only. In baseline encoding, each block is encoded in a single
left-to-right and top-to-bottom scan. It encodes and decodes complete 8x8 blocks with
full precision one at a time and supports interleaving of color components. The FDCT,
quantization, DC difference and zig-zag ordering proceeds. In order to claim JPEG
compatibility of a product it must include the support for at least the baseline encoding
system.
Progressive Encoding: Unlike baseline encoding, each block in progressive
encoding is encoded in multiple scans, rather than a single one. Each scan follows the
zig zag ordering, quantization and entropy coding, as done in baseline encoding, but
takes much less time to encode and decode, as compared to the single scan of
baseline encoding, since each scan contains only a part of the complete information.
With the first scan, a crude form of image can be reconstructed at the decoder and with
successive scans, the quality of the image is refined. You must have experienced this
while downloading web pages containing images. It is very convenient for browsing
applications, where crude reconstruction quality at the early scans may be sufficient for
quick browsing of a page.
There are two forms of progressive encoding: (a) spectral selection approach and (b)
successive approximation approach. Each of these approaches is described below.
Progressive scanning through spectral selection: In this approach, the first scan
sends some specified low frequency DCT coefficients within each block. The
corresponding reconstructed image obtained at the decoder from the first scan therefore
appears blurred as the details in the forms of high frequency components are missing.
In subsequent scans, bands of coefficients, which are higher in frequency than the
previous scan, are encoded and therefore the reconstructed image gets richer with
details. This procedure is called spectral selection, because each band typically
contains coefficients which occupy a lower or higher part of the frequency spectrum
for that 8x8 block.
The spectral select on approach. Here all the 64 DCT coefficients in a block are of
8-bit resolution and successive blocks are stacked
one after the other in the scanning order. The spectral selection approach performs
the slicing of coefficients horizontally and picks up a band of coefficients,
starting with low frequency and encodes them to full resolution.
Progressive scanning through successive approximation: This is also a multiple
scan approach. Here, each scan encodes all the coefficients within a block, but not to
their full quantized accuracy. In the first scan, only the N most significant bits of each
coefficient are encoded (N is specifiable) and in successive scans, the next lower
significant bits of the coefficients are added and so on until all the bits are sent. The
resulting reconstruction quality is good even from the early scans, as the high
frequency coefficients are present from the initial scans.
The successive approximation approach. The organization of the DCT coefficients and
the stacking of the blocks are same as before. The successive approximation approach
performs the slicing operation vertically and picks up a group pf bits, starting with the
most significant ones and progressively considering the lower frequency ones.
Hierarchical encoding: The hierarchical encoding is also known as the pyramidal
encoding in which the image to be encoded is organized in a pyramidal structure
of multiple resolutions, with the original, that is, the finest resolution image on the
lowermost layer and reduced resolution images on the successive upper layers. Each
layer decreases its resolution with respect to its adjacent lower layer by a factor of two
in either the horizontal or the vertical direction or both. Hierarchical encoding may be
regarded as a special case of progressive encoding with increasing spatial resolution
between the progressive stages.
The steps involved in hierarchical encoding may be summarized below:
•
Obtain the reduced resolution images starting with the original and for each,
reduce the resolution by a factor of two, as described above.
•
Encode the reduced resolution image from the topmost layer of the
pyramid .
Decode the above reduced resolution image. Interpolate and up-sample it by a
factor of two horizontally and/or vertically, using the identical interpolation
filter which the decoder must use. Use this interpolated and up-sampled image
as a predicted image for encoding the next lower layer (finer resolution) of the
pyramid.
•
•
Encode the difference between the image in the next lower layer and the
predicted image using baseline, progressive or lossless encoding.
•
Repeat the steps of encoding and decoding until the lowermost layer
(finest resolution) of the pyramid is encoded.
Hierarchical encoding (Pyramid structure)
Figure illustrates the hierarchical encoding process. In hierarchical encoding, the image
quality at low bit rates surpass the other JPEG encoding methods, but at the cost of
increased number of bits at the full resolution. Hierarchical encoding is used for
applications in which a high-resolution image should be accessed by a low resolution
display device. For example, the image may be printed by a high-resolution printer,
while it is being displayed on a low resolution monitor.
Lossless encoding: The lossless mode of encoding in JPEG follows a simple
predictive coding mechanism, rather than having FDCT + Entropy coder for encoding
and Entropy decoder + IDCT for decoding. Theoretically, it should have been possible
to achieve lossless encoding by eliminating the quantization block, but because of finite
precision representation of the cosine kernels, IDCT can not exactly recover what the
image was before IDCT. This led to a modified and simpler mechanism of predictive
coding.
In lossless encoding, the 8x8 block structure is not used and each pixel is predicted
based on three adjacent pixels, as illustrated in figure using one of the eight possible
predictor modes listed here.
Predictive coding for lossless JPEG
An entropy encoder is then used to encode the predicted pixel obtained from the lossless
encoder. Lossless codecs typically produce around 2:1 compression for color images
with moderately complex scenes. Lossless JPEG encoding finds applications in
transmission and storage of medical images.
Selection
Value
0
1
2
3
4
5
6
7
Prediction
None
A
B
C
A+B-C
A+(B-C)/2
B+(A-C)/2
(A+B)/2
Color image formats and interleaving
The most commonly used color image representation format is RGB, the
encoding of which may be regarded as three independent gray scale image
encoding. However, from efficient encoding considerations, RGB is not the best format.
Color spaces such as YUV, CIELUV, CIELAB and others represent the chromatic
(color) information in two components and the luminance (intensity) information in
one component. These formats are more efficient from image compression
considerations, since our eyes are relatively insensitive to the high frequency
information from the chrominance channels and thus the chrominance components can
be represented at a reduced resolution as compared to the luminance components for
which full resolution representation is necessary.
It is possible to convert an RGB image into YUV, using the following relations:
Y = 0.3R + 0.6G + 0.1B
U=
B −Y
+ 0.5
2
V=
R −Y
+ 0.5
1.6
YUV representation of an example 4x4 image
Figure illustrates the YUV representation by considering an example of a 4x4 image.
The Y components are shown as Y1, Y2, ……, Y16. The U and the V components are
sub-sampled by a factor of two in both horizontal and vertical directions and are
therefore of 2x2 size. The three components may be transmitted in either a noninterleaved manner or an interleaved manner.
The non-interleaved ordering can be shown as
Scan-1: Y1,Y2,Y3,……,Y15,Y16.
Scan-2: U1,U2,U3,U4. Scan-3: 1,V2,V3,V4.
The interleaved ordering encodes in a single scan and proceeds like
Y1, Y2, Y3, Y4, U1, V1, Y5, Y6, Y7, Y8, U2, V2, ………
Interleaving requires minimum of buffering to decode the image at the decoder.
JPEG Performance
Considering color images having 8-bits/sample luminance components and 8- bits/sample for each of
the two chrominance components U and V, each pixel requires 16-bits for representation, if both U and
V are sub-sampled by a factor of two in either of the directions. Using JPEG compression on a wide
variety of such color images, the following image qualities were measured subjectively:
Bits/pixel
Quality
≥2
1.5
0.75
0.5
0.25
Indistinguishable
Excellent
Very good
Good
Fair
Compression
Ratio
8:1
10.7:1
21.4:1
32:1
64:1
A more advanced still image compression standard JPEG-2000 has evolved in recent times. This will
be our topic in the next lesson.
UNIT IV
2 Marks
1. What is image compression?
2. What is Data Compression?
3. What are two main types of Data compression?
4. What is the need for Compression?
5. What are different Compression Methods?
7. Define interpixel redundancy?
8. What is run length coding?
9. Define compression ratio.
10. What are the operations performed by error free compression?
11. What is Variable Length Coding?
12. Define Huffman coding
16 Marks
1. What is data redundancy? Explain three basic data redundancy?
Definition of data redundancy
The 3 basic data redundancy are
> Coding redundancy
> Interpixel redundancy
> Psycho visual redundancy
2. What is image compression? Explain any four variable length coding
compression schemes.
• Definition of image compression
• Variable Length Coding
* Huffman coding
* B2 Code
* Huffman shift
* Huffman Truncated
* Binary Shift
*Arithmetic coding
3. Explain about Image compression model?
• The source Encoder and Decoder
• The channel Encoder and Decoder
4. Explain about Error free Compression?
a. Variable Length coding
i. Huffman coding
ii. Arithmetic coding
b. LZW coding
c. Bit Plane coding
d. Lossless Predictive coding
5. Explain about Lossy compression?
• Lossy predictive coding
• Transform coding
• Wavelet coding
6. Explain the schematics of image compression standard JPEG.
• Lossy baseline coding system
• Extended coding system
• Lossless Independent coding system
7. Explain how compression is achieved in transform coding and explain about DCT
• Block diagram of encoder
• decoder
• Bit allocation
• 1D transform coding
• 2D transform coding, application
• 1D,2D DCT
8. Explain arithmetic coding
• Non-block code
• One example
9. Explain about Image compression standards?
• Binary Image compression standards
• Continuous tone still Image compression standards
• Video compression standards
10. Discuss about MPEG standard and compare with JPEG
• Motion Picture Experts Group
1. MPEG-1
2. MPEG-2
3. MPEG-4
• Block diagram
• I-frame
• p-frame
• B-frame
UNIT V IMAGE SEGMENTATION AND REPRESENTATION
Edge detection – Thresholding – Region based segmentation – Boundary representation –
Chair codes– Polygonal approximation – Boundary segments – Boundary descriptors – Simple
descriptors– Fourier descriptors – Regional descriptors – Simple descriptors– Texture.
What is thresholding segmentation?
Thresholding segmentation is a method, which separates an image into two meaningful regions: foreground and background,
through a selected threshold value T. If the image is a grey image, T is an integer in the range of [0..K], where K is the maximum
intensity value. For example, if the image is an 8-bit gray image, K takes the value of 255 and T is in the range of [0..255].
Whenever the value of T is decided, the segmentation procedure is indicated by the following equation:
GB x, y
1 , if G x, y
T
0 , if G x, y
T
(1)
In equation (1), G(x,y) indicates the intensity value of pixel (x,y) in the grey image G. GB is the segmentation result. Actually it
forms a binary image, in which each value of GB(x,y) gives the category (foreground or background) that the corresponding pixel
belongs to. If GB(x,y) = 1, then pixel (x,y) in the image G is classified as a foreground pixel, otherwise it is classified as a
background pixel.
Equation (1) is formulated under the assumption that foreground pixels in the image G have relatively high intensity values and
background pixels take low intensity values. Of course you can reverse the equation when you need to set the low intensity region
as the foreground.
How to select the value of T
The major problem of thresholding segmentation is how to select the optimal value of threshold T. Usually in order to get the
optimal value of T, we need to statically analyze the so-called “histogram” (or “intensity histogram”) of the input gray image G.
Before talking about the algorithm, first we list all the notions and statistic definitions that relate to histogram as follows.
Basic notations
G(x,y): The input gray image that we want to segment.
GB(x,y): The segmentation result of G. It is a binary image, the value of GB(x,y) is either 0 or 1 indicating the corresponding pixel
(x,y) in G belongs to background or foreground respectively.
K:
T:
N:
The maximum possible intensity value defined by G. If G is an 8-bit gray image, then K takes the value 255.
The thresholding value. It is an integer within the range [0..K].
The total number of pixels in G. If G has width = w, and height = h, then of course N = w h.
Definition of histogram and normalized histogram
HG is the intensity histogram of image G and it maps from each intensity value to an integer. The value of HG(i) indicates the
number of pixels in G that takes the intensity value, i, where i [0..K], K is the maximum intensity value as mentioned above (for
example K = 255 for 8-bit grey images). Obviously HG(i) is an integer value within range [0..N], N is the total number of pixels in
G as mentioned above. Based on the definition of histogram HG, we can get the so-called normalized histogram PG. It is defined
as follows:
PG(i) = HG(i) / N
(2)
In equation (2), the value of each PG(i) indicates the percentage of pixels in G that takes the intensity value i. Clearly PG(i) is a
real value within the range [0..1]. The main reason of introducing the normalized histogram is that sometime we need to compare
two histogram from two images that contains different number of pixels. And the definition of normalized histogram makes this
kind of comparison meaningful.
Statistic analysis of normalized histogram
Assume the current thresholding value is T, which separates the input image G into two regions: foreground and background
according the equation (1). The frequency of background, B(T) and the frequency of foreground, F(T) are defined as follows:
T
B
K
PG i ,
T
T
F
(3)
PG i
i T 1
i 0
Of course the frequency of the entire image G is calculated as = B(T) + F(T) = 1, no matter what value T takes. The mean
intensity values of background and foreground, B(T) and F(T) are calculated as:
T
B
T
K
i PG i
B
,
T
F
i PG i
i 0
(4)
F
i T 1
The mean intensity value of the entire image can be calculated as: = B(T) B(T) + F(T)
T takes, keeps the same. The intensity variances of background and foreground, 2B(T) and
i
T
2
B
T
i 0
2
T
B
P i
,
T
B
i
K
2
F
T
i T 1
2
T
F
F
F(T).
2
F(T)
P i
T
Clearly no matter what value
are defined as:
(5)
Having the definition of variances of the background and the foreground, it is the time to define the so-called “within-class”
variance, 2within:
2
within
T
B
T
B
T
F
T
F
T
(6)
Also we can define the so-called “between-class” variance,
2
between
2
T
In the above equation,
K
2
2
within
i
2
T
B
T
F
T
B
2
between:
T
F
2
T
(7)
indicates the intensity variance of the entire image G and it is calculated as:
2
P i
(8)
i 0
Obviously given the image G,
2
is a constant value independent to the selection of the value of T.
Otsu’s algorithm
The Otsu’s algorithm is simple. We let T try all the intensity values from 0 to K and choose the one that gives the minimum
“within-class” variance 2within as the optimal thresholding value. Formally speaking:
Optimal value of T = TOpt, where
2
within(TOpt)
= min
0 T K
2
within
(9)
T
As we said before, 2 = 2within(T)+ 2between(T), and 2 is independent of the selection of T, therefore, minimization of
means maximization of 2between. So the optimal value of T can also be taken as:
Optimal value of T = TOpt, where
2
between(TOpt)
= max
0 T K
2
between
T
2
within
(10)
In fact equation (10) is the usual way that we use to find the optimal thresholding value. It is because that for each T, the
calculation of 2between only needs the calculations of B, F, B and F according to equation (7). And these values can be updated
iteratively:
Initially, T = 0:
Calculate the mean intensity of the entire image, .;
B (0)
= PG(0); F(0) = 1 B(0) = 1 PG(0);
(0)
=
0; F(0) = / F(0), be careful if F(0) =0, then F(0) = 0.
B
Iteratively, T = T+1:
B (T+1) = B(T) + PG(T+1); F(T+1) = 1
B(T+1);
If B(T+1) = 0, then B(T+1) = 0;
ELSE B(T+1) = [ B(T)
PG(T+1)] / B(T+1);
B(T) + (T+1)
If F(T+1) = 0; then F(T+1) = 0;
ELSE F(T+1) = [
B(T+1)
B(T+1)] / F(T+1);
Part2: Connect Object Recognition
After obtaining the binary image GB, it is the time to count the number of connected objects in the foreground region (or in the
background region if you want). Assume here we interest in the foreground (for background region, you just need to reverse the
equation 1). The way we used to count the connected objects in the foreground region is based on the warshell’s algorithm. The
entire algorithm of connect object counting is described as follows:
Step 1:
Create a label image GL with the same size (width and height) as GB, and initially set each GL(x,y) = –1. Create a variable,
current_label, for recording the current available label and initially current_label = 0.
Step 2:
Scan the binary image GB sequentially and update the label image GL as follows:
FOR each y FROM 0 TO height – 1
FOR each x FROM 0 TO width –1
IF pixel (x,y) is a foreground, which means GB(x,y) = 1,
THEN
Check the 8 neighborhood pixels around the pixel (x,y);
FOR each neighborhood pixel (x’,y’)
IF it has a nonnegative label in the label image GL, which means GL(x’,y’)>=0,
THEN
SET GL(x,y) = GL(x’,y’);
ELSE
SET GL(x,y) = current_label;
SET current_label = current_label +1;
END IF
END FOR
END IF
END FOR
END FOR
Step 3:
Build the 2D transit matrix M_T. Initially create an empty matrix:
M_T[0..current_label–1][0..current_label–1].
Clearly it is an 2D array with size=(current_label) (current_label), and each element in the matrix is first set to 0. Then assign 1
to each element of the matrix that locates at the diagonal. It means: set M_T[i][i]=1, where i is from 0 to current_label–1.
After the above initialization, we need to update the matrix M_T according to the label image GL and let M_T become the transit
matrix of GL. The update procedure is given as follows:
FOR each y FROM 0 TO height – 1
FOR each x FROM 0 TO width –1
IF pixel (x,y) is a foreground pixel, which means GB(x,y) = 1,
THEN
Check the 8 neighborhood pixels around pixel (x,y).
FOR each neighborhood pixel (x’,y’)
IF its label is nonnegative, which means GL(x’,y’)>=0
THEN
SET M_T[GL(x,y)][GL(x’,y’)] = 1;
SET M_T[GL(x’,y’)][GL(x,y)] = 1;
END IF
END FOR
END IF
END FOR
END FOR
After the above procedure, you get the updated matrix M_T, which represents the transit relationship in GL.
Step 4:
Calculate the transit closure matrix, M_TC, of M_T. This calculation of transit closure matrix is based on the warshell’s algorithm
which is an iteration procedure described as follows:
(1) Initially create two temporary matrixes M_0 and M_1, which have the same size of the matrix M_T. Copy each value in
M_T into the corresponding position in M_0 and M_1, which is:
FOR i FROM 0 TO current_label–1
FOR j FROM 0 TO current_label–1
SET M_0[i][j] = M_1[i][j] = M_T[i][j];
(2) Update the matrix M_1 using the transitivity law, which is:
FOR i FROM 0 TO current_label–1
FOR j FROM 0 TO current_label–1
FOR k FROM 0 TO current_label–1
IF M_1[i][j] = 1 AND M_1[j][k] = 1
THEN
SET M_1[i][k] = 1;
SET M_1[k][i] = 1;
END IF
END FOR
END FOR
END FOR
(3) Compare M_1 and M_0 to see if they are exactly the same, which means each element from M_1 has the same value as
the corresponding element from M_0. If so, set M_TC = M_1, which means we got the transit closure matrix. If not,
copy M_1 to M_0 and go back to (2).
Step 5:
Count the number of connected objects (or the number of equivalent classes) in the transit closure matrix M_TC. It is not hard to
see that the number connected objects equals to the number of distinct rows (or columns) in matrix M_TC. Assume M_TC[i] and
M_TC[j] are the two rows of matrix M_TC, where i j. We say these two rows are distinct if and only if there exists at least one
k [0.. current_label–1] that M_TC[i][k] M_TC[j][k].
Step 6:
Output the number of connected objects (or the number of distinct rows of M_TC) and return.
One example of transit closure calculation (step 4)
Assume after step 3, we got the transit matrix M_T. In step 4 (1), we initially set two temporary matrixes M_0 and M_1, and copy
M_T to these two matrixes, as shown in Figure 1. Then we do some operations on the matrix M_1 as described in the Step 4 (2),
we get an updated M_1 as shown in Figure 2:
Figure 1: Copy M_T to M_0 and M_1.
Figure 2: Updated M_1 by applying Step 4 (2).
Then according to Step 4 (3), we compare matrix M_1 (Figure 2) with matrix M_0 (Figure 1). We find M_1 is different from
M_0. Therefore we copy M_1 to M_0, as shown in Figure 3. Then go back to (2) do the same operations on M_1 again, and get
M_1 updated again, as shown in Figure 4.
Figure 3: Copy M_1 to M_0.
Figure 4: M_1 is updated again.
Then we compare M_1 (Figure 4) and M_0 (Figure 3). Again we find that they are different. So we need to copy M_1 to M_0 (as
shown in Figure 5) and go back to (2) to get M_1 updated again (as shown in Figure 6).
Figure 5: Copy M_1 to M_0.
Figure 6: M_1 is updated again.
This time we find that M_1 did not change, which means M_0 (Figure 5) and M_1 (Figure 6) are identical. So we say the current
M_1 is the transit closure matrix that we want, and set M_TC = M_1. Furthermore we discover there are two distinct rows in the
transit closure matrix: (1111100) and (0000011). It means that there are two connected objects.
 Detection of Discontinuities
 Edge Linking and Boundary Detection
 Thresholding
 Region-Based Segmentation
 Segmentation by Morphological Watersheds
 The Use of Motion Segmentation
UNIT V
2 Marks
1. What is segmentation?
2. Write the applications of segmentation.
3. What are the three types of discontinuity in digital image?
4. How the derivatives are obtained in edge detection during formulation?
5. Write about linking edge points.
6. What are the two properties used for establishing similarity of edge pixels?
7. Define Gradient Operator?
8. Define region growing?
9. Define compactness
16 Marks
1. What is image segmentation. Explain in detail.
• Definition - image segmentation
• Discontinity – Point, Line, Edge
• Similarity – Thresholding, Region Growing, Splitting and
merging
2. Explain Edge Detection in details?
* Basic formation.
* Gradient Operators
* Laplacian Operators
3. Define Thresholding and explain the various methods of thresholding in detail?
• Foundation
• The role of illumination
• Basic adaptive thresholding
• Basic adaptive thresholding
• Optimal global & adaptive thresholding.
4. Discuss about region based image segmentation techniques. Compare
threshold region based techniques.
* Region Growing
* Region splitting and merging
* Comparison
5. Define and explain the various representation approaches?
• chain codes
• Polygon approximations
• Signature
• Boundary segments
• Skeletons.
6. Explain Boundary descriptors.
• Simple descriptors.
• Fourier descriptors.
7. Explain regional descriptors
• Simple descriptors
• Texture
i. Statistical approach
ii. Structural approach
iii. Spectral approach
8. Explain the two techniques of region representation.
• Chain codes
• Polygonol approximation
9. Explain the segmentation techniques that are based on finding the regions
directly.
• Edge detection line detection
• Region growing
• Region splitting
• region merging
10. How is line detected? Explain through the operators
• Types of line masks
1. horizontal
2. vertical
3. +45˚,-45˚