Digital image processing - OLD

Digital Image Processing
Dr. Chandran Saravanan
Assistant Professor, Dept. of C.S.E.,
NIT Durgapur
dr.cs1973@gmail.com
Dr. C. Saravanan, NIT Durgapur,
India

Digital Image Processing (DIP)
• DIP is the use of computer algorithms to
perform image processing on digital images.
--WIKI
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Digital
• Any data fed into the computer is digital in
the form of 0s and 1s (binary form -
voltage or magnetic polarization)
• The data may be represented using 8 bit /
16 bit / 32 bit / 64 bit
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Image Processing
• Representation of a scene in a paper /
stone / cloth / etc. is picture or image
• In general image is assumed as in digital
form
• Performing a task using various steps is a
processing
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Structure of the Human Eye
•The retina is covered with
light receptors called
cones (6-7 million) and
rods (75-150 million)
•Cones are concentrated
around the fovea and are
very sensitive to colour
•Rods are more spread
out and are sensitive to
low levels of illumination
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Image Formation in the Eye
•Muscles within the eye can be used to
change the shape of the lens allowing us
focus on objects that are near or far away
•An image is focused onto the retina causing
rods and cones to become excited which
ultimately send signals to the brain
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Brightness Adaptation & Discrimination
•The human visual system can perceive
approximately 1010 different light intensity
levels
•However, at any one time we can only
discriminate between a much smaller
number – brightness adaptation
•Similarly, the perceived intensity of a region
is related to the light intensities of the
regions surrounding it
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Brightness Adaptation & Discrimination
(cont…)
An example of Mach bands
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Optical Illusions
•Our visual
systems play lots
of interesting
tricks on us
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Optical Illusions (continues …)
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Light And The Electromagnetic Spectrum
•Light is just a particular part of the
electromagnetic spectrum that can be
sensed by the human eye
•The electromagnetic spectrum is split up
according to the wavelengths of different
forms of energy
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Image Representation
col
row
f (row, col)
•Before we discuss image acquisition recall
that a digital image is composed of M rows
and N columns of pixels
each storing a value
•Pixel values are most
often grey levels in the
range 0-255(black-white)
•We will see later on
that images can easily
be represented as
matrices
Dr. C. Saravanan, NIT Durgapur, India

Image Acquisition
•Images are typically generated by
illuminating a scene and absorbing the
energy reflected by the objects in that scene
– Typical notions of
illumination and scene
can be way off:
• X-rays of a skeleton
• Ultrasound of an
unborn baby
• Electro-microscopic
images of molecules
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Image Sensing
•Incoming energy lands on a sensor material
responsive to that type of energy and this
generates a voltage
•Collections of sensors are arranged to
capture images
Imaging Sensor
Line of Image Sensors
Array of Image SensorsDr. C. Saravanan, NIT Durgapur, India

Image Sensing
Using Sensor Strips and Rings
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Image Sampling
•A digital sensor can only measure a limited
number of samples at a discrete set of
energy levels
•Quantisation is the process of converting a
continuous analogue signal into a digital
representation of this signal
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Image Sampling (continues …)
•Remember that a digital image is always
only an approximation of a real world
scene
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Spatial Resolution
•The spatial resolution of an image is
determined by how sampling was carried out
•Spatial resolution simply refers to the
smallest discernable detail in an image
– Vision specialists will
often talk about pixel
size
– Graphic designers will
talk about dots per
inch (DPI)

Spatial Resolution (continues …)
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Number of Bits
Number of Intensity
Levels
Examples
1 2 0, 1
2 4 00, 01, 10, 11
4 16 0000, 0101, 1111
8 256 00110011, 01010101
16 65,536 1010101010101010
Intensity Level Resolution
Intensity level resolution refers to the number of intensity levels
used to represent the image
• The more intensity levels used, the finer the level of detail
discernable in an image
• Intensity level resolution is usually given in terms of the
number of bits used to store each intensity level

Resolution (continues …)
128 grey levels (7 bpp) 64 grey levels (6 bpp) 32 grey levels (5 bpp)
16 grey levels (4 bpp) 8 grey levels (3 bpp) 4 grey levels (2 bpp) 2 grey levels (1 bpp)
256 grey levels (8 bits per pixel)

Resolution: How Much Is
Enough?
•The big question with resolution is always
how much is enough?
– This all depends on what is in the image and
what you would like to do with it
– Key questions include
• Does the image look aesthetically pleasing?
• Can you see what you need to see within the
image?
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How Much Is Enough? (continues …)
•The picture on the right is fine for counting
the number of cars, but not for reading the
number plate
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Resolution (continues …)
Low Detail Medium Detail High Detail
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Enhancement
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1 Selection 12 Enhancing images
2 Layers 13 Sharpening and softening images
3 Image size alteration 14 Selecting and merging of images
4 Cropping an image 15 Slicing of images
5 Histogram 16 Special effects
6 Noise reduction 17 Stamp Clone Tool
7 Removal of unwanted elements 18 Change color depth
8 Selective color change 19 Contrast change and brightening
9 Image orientation 20 Gamma correction
10 Perspective control & distortion 21 Color adjustments
11 Lens correction 22 Printing / Warping
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Applications

Histogram Equalization
• Histogram equalization is a technique for
adjusting image intensities to enhance contrast.
• Let f be a given image represented as a m by n
matrix of integer pixel intensities ranging from 0
to L − 1.
• L is the number of possible intensity values,
often 256.
• Let p denote the normalized histogram of f with a
bin for each possible intensity.
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HE Formula
• Pn= number of pixels with intensity n i.e.
total number of pixels,
•
• n=0, 1, 2, ... L-1
Histogram Equalized image G = floor ((L-1)
Summation of Pn from n=0 to F(i,j))
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Correlation and Convolution
• Correlation and Convolution are basic
operations that we will perform to extract
information from images.
• They are shift-invariant, and they are
linear.
• Shift-invariant means “same operation
performed at every point in the image”.
• Linear means that we replace every pixel
with a linear combination of its neighbors.
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Correlation
• Performed by Local Averaging
Image I = {5,4,2,3,7,4,6,5,3,6}
Fourth value 3 is replaced with average of 2,
3, 7 (neighbours)
J(4)=(I(3)+I(4)+I(5))/3=(2+3+7)/3=4
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2D correlation
8 3 4 8 8 3 4 4 6.44 5.22 4
7 6 4 8 8 3 4 4 5.77 5.33 4.88
4 5 7 7 7 6 4 4 5.11 5.44 5.77
4 4 5 7 7
4 4 5 7 7
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Convolution
• Convolution is just like correlation, except
that flip over the filter before correlating.
• For example, convolution of a 1D image
with the filter (3,7,5) is exactly the same as
correlation with the filter (5,7,3).
• In the case of 2D convolution flip the filter
both horizontally and vertically.
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Filtering
• Images are often corrupted by random variations
in intensity, illumination, or have poor contrast
and can’t be used directly
• Filtering: transform pixel intensity values to
reveal certain image characteristics
– Enhancement: improves contrast
– Smoothing: remove noises
– Template matching: detects known patterns
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Smoothing Filters
• Additive smoothing / Laplace / Lidstone
• Butterworth filter
• Digital filter
• Kalman filter
• Kernel smoother
• Laplacian smoothing
• Stretched grid method
• Low-pass filter
• Savitzky–Golay smoothing filter based on the least-squares fitting of
polynomials
• Local regression also known as "loess" or "lowess"
• Smoothing spline
• Ramer–Douglas–Peucker algorithm
• Moving average
• Exponential smoothing
• Kolmogorov–Zurbenko filter
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Sharpening Filters
• A high-pass filter can be used to make an
image appear sharper.
• If there is no change in intensity, nothing
happens. But if one pixel is brighter than
its immediate neighbors, it gets boosted.
0 -1/4 0
-1/4 +2 -1/4
0 -1/4 0
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Unsharp Mask
• An unsharp mask is simply another type of
high-pass filter.
• First, a mask is formed by low-pass
filtering an image.
• This mask is then subtracted from the
original image.
• This has the effect of sharpening the
image.
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Image Restoration
• A photograph blurred by a defocused
camera is analyzed to determine the
nature of the blur.
• Degradation comes in many forms such as
motion blur, noise, and camera misfocus.
• It is possible to come up with an very good
estimate of the actual blurring function and
"undo" the blur to restore the original
image
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Image Restoration
• To remove or reduce the degradations.
• The degradations may be due to
– sensor noise
– blur due to camera misfocus
– relative object-camera motion
– random atmospheric turbulence
– others
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Restoration Domain
• Restoration techniques are best in the
spatial domain - scanned old images - not
occured at the time of image acquiring.
• Spatial processing is applicable if the
degradation is additive noise.
• Frequency domain filters too used.
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Restoration Methods
• Deterministic
– work with sample by sample processing
• Stochastic
– work with the statistics of the image
• Non-blind
– The degradation process is known
• Blind
– The degradation process is unknown
• Direct, Iterative, and Recursive
(implementation)
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Model of Degradation/Restoration
Source
Image
s(x,y)
Degradation
Function H
Noise
n(x,y)
+
Degraded
Image
d(x,y)
Restoration
FiltersReproduced
Image
r(x,y)
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Estimation of Restoration
• Objective of restoration is to estimate
r(x,y) using
– Degraded image d(x,y)
– Knowledge of degraded function H
– Knowledge of additive noise n(x,y)
• The estimate should be as close as
possible to the source image s(x,y).
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Enhancement Vs. Restoration
• Reconstruction techniques are oriented
towards modelling the degradation.
• Applying the inverse process for
recovering the original image.
• Restoration - Deblurring - removal of blur.
• Enhancement - Contrast Stretching -
improves the visual quality of the image.
• Enhancement is Subjective and
Restoration is Objective.
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Application Domains
• Scientific explorations
• Legal investigations
• Film making and archival
• Image and video (de-)coding
....
• Consumer photography
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Image Morphing
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• Depict one person turning into another
• Morphing is a special effect in motion
pictures and animations that changes (or
morphs) one image or shape into another
through a seamless transition.
• Achieved through cross-fading techniques
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• Morphing is an image processing
technique used for the metamorphosis
from one image to another.
• Cross-dissolve between them
• The color of each pixel is interpolated over
time from the first image value to the
corresponding second image value.
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 
• Translation x = u + t, u = x - t
• Scaling x = s * u, x / s
• Rotation x = u cos - v sin
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Morphing
• The fundamental techniques behind image
morphing is image warping.
• Let the original image be f(u) and the final
image be g(x). In image warping, we create
g(x) from f(u) by changing its shape.
• In image morphing, we use a combination of
both f(u) and g(x) to create a series of
intermediate images Geometric
Transformation create a series of
intermediate images.
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Image Warping
• Warping may be used for correcting image
distortion and for creative purposes.
• Points are mapped to points without
changing the colors.
• Applications - Cartoon Pictures and
Movies.
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Noise
• Digital image has several different noise
sources, some of them fixed, some
temporal, some are temperature
dependent.
• The noise characteristics of the camera
are analyzed by shooting numerous
frames both in complete darkness (for
dark signal and noise properties) with
several exposure times and on different
exposure levels with flat field target.
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Noise
• Principal source of noise - image
acquisition (digitisation) - transmission.
• Performance of imaging sensors are
affected by environmental conditions and
quality of sensing elements.
• Light levels and sensor temperature -
major factors.
• Transmission - Wireless - Atmospheric
disturbance - Lightning.
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• Based on these measurements, several
parameters describing camera’s noise
characteristics and sensitivity are
calculated.
• These parameters include for example
camera gain constant (electrons/output
digital number), dark signal nonuniformity,
photoresponse nonuniformity and STDs of
temporal noise and fixed pattern noise.
Characterisation of Noise
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Types of Noise
• Noise components divided into three
groups: signal dependent, temperature
dependent and time dependent.
• Signal dependent noise components are
photon shot noise, photoresponse
nonuniformity (PRNU) and amplifier gain
non-uniformity.
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Classification
• Time dependent noise components include
photon shot noise, dark current shot noise, reset
noise and thermal noise.
• Random components include photon shot noise,
dark current shot noise, reset noise and thermal
noise.
• Pattern components are amplifier gain non-
uniformity, PRNU, dark current non-uniformity
and column amplification offset.
• Quantization noise is caused by discrete output
levels of A/D converter.
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Noise Types
• Gaussian noise
• Rayleigh noise - skewed histograms.
• Erland (Gamma) noise
• Exponential noise
• Uniform noise
• Impulse (salt-and-pepper) noise
– bipolar, shot, spike, are known as S&P noise.
– assumption - impulse noise - extreme values -
negative impulse - black - +ve impluse - white.
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Shot Noise
• Shot noise results from the discrete nature of
electronic charge.
• During Current flows, the electrons will fluctuate
slightly and give rise to variation in current flow.
• Both photoelectrons and dark current contribute
to shot noise and are referred to as photon shot
noise and dark current shot noise respectively.
• Shot noise exhibits a Poisson distribution and
has a square root relationship between signal
and noise.
• PRNU is signal dependent and multiplicative in
nature.
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Thermal Noise
• Temperature dependent noise components are thermal
noise and dark current.
• Dark current arises through thermal generation of
electrons (TGEs) in the silicon. The dark current is
usually different from pixel to pixel. The average value of
dark current for all pixels might be referred to as the dark
current bias.
• If this bias is subtracted from every pixel the variation
remaining is referred to as fixed pattern noise (FPN).
• Additionally there is dark current shot noise (shot noise
associated with the portion of current that is dark
current).
• Dark current can be significantly reduced through cooling
of the sensor; the number of TGEs can be halved with
every 8-9 °C of cooling.
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Noise reduction
• Salt and pepper noise
• Gaussian noise
• Shot noise
• Quantization noise (uniform noise)
• Film grain
• Anisotropic noise
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Salt and Pepper Noise
• Also known as Impulse Noise.
• Cause by sharp & sudden disturbance in
the image signal.
• Sparsely occurring white and black pixels.
• Certain amount of the pixels in the image
are either black or white.
• Median Filter
• Morphological Filter
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S & P Noise
• Minimum Filtering
– Current pixel replaced by minimum pixel value
• Maximum Filtering
– Current pixel replaced by maximum pixel value
• Mean Filtering
– Current pixel replaced by mean value of neighbour.
• Median Filtering
– Current pixel replaced by mid value of neighbour.
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Periodic Noise
• Irises from electrical or electromechanical
interference.
• Spatially dependent noise.
• Reduced using frequency domain filtering.
• Peaks in Fourier Transform.
• Applying a Butterworth band-reject filter
(sets the peaks to zero)
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Gaussian Noise
• Karl Friedrich Gauss, an 18th century
German physicist
• Is caused by Random fluctuations
• Also called statistical noise
• Gaussian noise in digital images arise
during acquisition.
• Sensor noise caused by poor illumination
and/or high temperature, and/or
transmission e.g. electronic circuit noise
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Estimating Noise Parameters
• Choose an Optical sensor
• Preoare a solid gray board
• Set up an uniform illumination
• Capture set of images
• Resulting images are indicators of system
noise.
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Spatial Filtering
• Mean Filter
– Arithmetic mean - simple average of m x n.
– Geometric mean - product of pixels - raised to
power 1/mn
– Harmonic mean - mn/(sum(1/pixel)) - works
well for salt noise - fails for pepper noise.
– Contraharmonic mean - sum(pixel, pow(q+1))/
sum(pixel, pow(q))
– q is order of the filter - +ve values of q
eliminates salt noise. if q=0 Arithmetic Mean,
q=-1 Harmonic Mean.
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Order Statistics Filters
• Median Filter - median of the pixel value in the
neighborhood pixels - random, bipolar, unipolar
noise - less blurring.
• Max and Min Filter - used to reduce brightest
and darkest pixels - S&P noise.
• Midpoint Filter - midpoint between max. and min.
pixel values - gaussion noise.
• Alpha-trimmed Mean Filter - delete d/2 lowest
and d/2 highest gray level values - filter formed
by averaging these remaining pixels.
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Adaptive Filters
• Adaptive filters - better performance.
• Adaptive local noise reduction filter
– AF based on pixel value, variance, local mean,
local variance
– mean gives a measure of average gray level
– variance gives a measure of average contrast
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Adaptive Median Filter
• AMF can handle impulse noise with larger
probabilities compared to median filter -
preserves details while smoothing.
• sxy is rectangular working area
• zmin = minimum gray level in sxy
• zmax = maximum gray level in sxy
• zmed = median of gray level in sxy
• zxy = gray level at coordinates (x,y)
• smax = maximum allowed size of sxy.
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AMF Algorithm
Level A: A1 = zmed - zmin
A2 = zmed - zmax
if A1 > 0 and A2 < 0, Goto Level B
else increase the window size
if window size <= smax repeat level A
else output zxy.
Level B: B1 = zxy - zmin
B2 = zxy - zmax
if B1 > 0 and B2 < 0, output zxy
else output zmed.
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Purposes
• The output is single value.
• The algorithm starts from first area location 0,0
• outputs value and
• shifts to the next area
• algorithm reinitializes and applies on the area
• Three main purposes
– removing salt and pepper noise
– provide smoothing
– reduce distortion
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Frequency Domain Filtering
• Bandreject Filters - attenuates those in a
specific range to very low levels -
Butterworth - Gaussian
• Bandpass Filters - attenuates those in a
specific range to very high levels
• Notch Filters - rejects frequencies in
predefined neighborhoods about center
frequency.
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• D(u,v) is the distance from the origin of the
centered frequency rectangle.
• W is the width of the band.
• Do is its radical centre.
• order 'n' is given by the expression.
• br - bandreject, bp - bandpass
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Estimating the Degrading Function
• Observation - degraded image - no
knowledge - gather information from the
image itself - uniform background regions -
strong signal content - estimate the power
spectrum or the probability density
function - studying the mechanism that
caused the degradation.
• Experimentation - equivalent equipment
• Mathematical Modeling
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Inverse Filtering
• If we know of or can create a good model
of the blurring function that corrupted an
image, the quickest and easiest way to
restore that is by inverse filtering.
• The inverse filter is a form of high pass
filter.
• Noise tends to be high frequency.
• Thresholding and an Iterative method.
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Weiner Filtering
• Also called Minimum Mean Square Error
Filter or Least Square Error Filter
• Its a statistical approach.
• Incorporates both the degradation function
and statistical characteristics of noise.
• Noise and image are uncorrelated.
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Error Metrics
B is the number of bits used for
each pixel. (i.e.8 bits)
Peak Signal to Noise Ratio (PSNR) – in decibel (dB)
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• Subjective evaluations used for Image
quality rating
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Encoder and Decoder Model
• The Encoding/decoding model of
communication was first developed by
cultural studies scholar Stuart Hall in 1973,
titled 'Encoding and Decoding in the
Television Discourse,'
• Hall's essay offers a theoretical approach
of how media messages are produced,
disseminated, and interpreted.
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Three different positions
• For example, advertisements can have
multiple layers of meaning, they can be
decoded in various ways and can mean
something different to different people.
• Decoding subject can assume three
different positions: Dominant (hegemonic)
position, negotiated position, and
oppositional position.
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Four-stage model
• Production – This is where the encoding of a
message takes place.
• Circulation – How individuals perceive things:
visual vs. written.
• Use (distribution or consumption) – This is the
decoding/interpreting of a message which
requires active recipients.
• Reproduction – This is the stage after audience
members have interpreted the message in their
own way based on their experiences and beliefs.
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Dominant/hegemonic position
• This position is one where the consumer takes
the actual meaning directly, and decodes it
exactly the way it was encoded.
• The consumer is located within the dominant
point of view, and is fully sharing the texts codes
and accepts and reproduces the intended
meaning.
• Here, there is barely any misunderstanding
because both the sender and receiver have the
same cultural biases.
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Negotiated position
• This position is a mixture of accepting and
rejecting elements.
• Readers are acknowledging the dominant
message, but are not willing to completely
accept it the way the encoder has
intended.
• The reader to a certain extent shares the
texts code and generally accepts the
preferred meaning
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Oppositional position
• In this position a consumer understands
the literal meaning, but due to different
backgrounds each individual has their own
way of decoding messages, while forming
their own interpretations.
• The readers' social situation has placed
them in a directly oppositional relation to
the dominant code.
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Image Compression
• Lossy and lossless compression
• Data compression refers to the process of
reducing the amount of data required to
represent given quantity of information.
• Data redundancy is the central concept in
image compression and can be
mathematically defined.
• IMAGE = INFORMATION + REDUNDANT
DATA
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Lossy Compression
• "Lossy" compression is the class of data
encoding methods that uses inexact
approximations (or partial data discarding) for
representing the content that has been encoded.
• Lossy compression technologies attempt to
eliminate redundant or unnecessary information.
• Most video and audio compression technologies
use a lossy technique.
• Compromise in the accuracy of the
reconstructed image - distortion to a level of
acceptance
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Compression Ratio
• CR=n1 / n2
• CR - Compression Ratio
• n1 -Source image
• n2 - Compressed image
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Types of Redundancy
• Coding redundancy
• Interpixel redundancy
• Psychovisual redundancy
• Coding Redundancy : A code is a system of
symbols (i.e. bytes, bits) that represents
information.
• Interpixel correlations
• Certain information has relatively less
importance for the quality of image perception.
This information is said to be psychovisually
redundant.
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Types of Redundancies
• Psycho-visual Redundancy
– corresponding to eliminating less relative
important information in a visual object.
• Inter-pixel Redundancy
– corresponding to statistical dependencies
among pixels (neighbouring).
• Coding Redundancy
– Fixed length code - variable length code
(Huffman coding and arithmetic coding).
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REDUNDANT DATA
INFORMATION
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Coding Redundancy
• A code is a system of symbols (i.e. bytes,
bits) that represents information.
• Each piece of information is represented
by a set of code symbols.
• The gray level histogram of an image can
be used in construction of codes to reduce
the data used to represent it.
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Interpixel Redundancy
• This type of redundancy is related with the
interpixel correlations within an image.
• Much of the visual contribution of a single
pixel is redundant and can be guessed
from the values of its neighbours.
• Assume a line is represented in an binary
image,
– The line is represented 000000… could be
represented as (0,n). Run Length Coding
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Psychovisual Redundancy
• The improved gray-scale quantization
(IGS) is one of the possible quantization
procedures and summarized in the
following table.
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Average Number of Bits
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Fourier Transform
• The Fourier Transform is used to
decompose an image into its sine and
cosine components.
• The output of the transformation
represents the image in the Fourier or
frequency domain, while the input image is
the spatial domain equivalent.
• In the Fourier domain image, each point
represents a particular frequency
contained in the spatial domain image.
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Discrete Fourier Transform
• Jean Baptiste Joseph Fourier (1768--
1830), a French mathematician and
physicist.
• Fourier transform is linear!
• It possesses the properties of
homogeneity and additivity.
• Time / Frequency Domain.
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DFT
• The DFT is the sampled Fourier Transform
and therefore does not contain all
frequencies forming an image, but only a
set of samples which is large enough to
fully describe the spatial domain image.
• The number of frequencies corresponds to
the number of pixels in the spatial domain
image, i.e. the image in the spatial and
Fourier domain are of the same size.
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DFT
For a square image of size N×N, the two dimensional
DFT is given by:
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The value of each point F(k,l) is obtained by
multiplying the spatial image with the
corresponding base function and summing the
result.
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Inverse DFT







1
0
1
0
)(2
2
),(
1
),(
N
k
N
l
N
lb
N
ka
l
elkF
N
baF

•Using those last two formulas, the spatial domain
image is first transformed into an intermediate image
using N one--dimensional Fourier Transforms.
•This intermediate image is then transformed into the
final image, again using N one--dimensional Fourier
Transforms.
•Expressing the two--dimensional Fourier Transform in
terms of a series of 2N one--dimensional transforms
decreases the number of required computations.
India

Discrete Cosine Transform (DCT)
• The DCT transforms a signal or image from the
spatial domain to the frequency domain.
• The DCT-II is often used in signal and image
processing, especially for lossy data
compression, because it has a strong "energy
compaction" property.
• Modified Discrete Cosine Transform (MDCT)
(based on the DCT-IV), is used in JPEG, AAC,
Vorbis, WMA, MP3, MJPEG, MPEG, DV, Daala,
and Theora.
India

DCT










1
0
)
2
1
(cos
N
n
nk kn
N
xXIIDCT

.1,...,0  Nk










1
0
)12(
2
cos][][
1
)(
N
k
x nk
N
kCkw
N
nxIIIDCT

India

FT vs. DCT
• DCT use Cosine functions and real
coefficients.
• FT use both Sines and Cosines and
complex numbers.
India

67,854 bytes 7,078 bytes 5,072 bytes
3,518 bytes 2,048 bytes 971 bytes
India

Discrete Wavelet Transform
• In mathematics, a wavelet series is a
representation of a square-integrable (real- or
complex-valued) function by a certain
orthonormal series generated by a wavelet.
• Wavelet-transformation contains information
similar to the short-time-Fourier-transformation,
but with additional properties of the wavelets,
which show up at the resolution in time at higher
analysis frequencies of the basis function.
India

DWT
• The discrete wavelet transform (DWT) is a linear
transformation that operates on a data vector
whose length is an integer power of two,
transforming it into a numerically different vector
of the same length.
• It is a tool that separates data into different
frequency components, and then studies each
component with resolution matched to its scale.
India

Burrows-Wheeler-Transformation
(BWT)
• The BWT is named after Michael Burrows
and David J. Wheeler, who had published
this procedure in their article "A Block-
sorting Lossless Data Compression
Algorithm" in 1994.
• The fundamental algorithm had been
developed by David J. Wheeler in 1983.
India

Principle of the algorithm
• A block of original data will be converted
into a certain form and will be sorted
afterwards.
• The result is a sequence of ordered data
that reflect frequently appearing symbol
combinations by repetitions.
India

Example
• Matrix for "abracadabra"
• 0 1 2 3 4 5 6 7 8 9 10
» n-1 right shift
• 0 a b r a c a d a b r a a
• / / / / / / / / / / /
• 1 b r a c a d a b r a a
• 2 r a c a d a b r a a b
• 3 a c a d a b r a a b r
• 4 c a d a b r a a b r a
• 5 a d a b r a a b r a c
• 6 d a b r a a b r a c a
• 7 a b r a a b r a c a d
• 8 b r a a b r a c a d a
• 9 r a a b r a c a d a b
• 10 a a b r a c a d a b r
India

• Afterwards this matrix will be sorted "lexicographically"
• 0 1 2 3 4 5 6 7 8 9 10
• 0 a a b r a c a d a b r
• 1 a b r a a b r a c a d
• 2 a b r a c a d a b r a
• 3 a d a b r a a b r a c
• 4 a c a d a b r a a b r
• 5 b r a a b r a c a d a
• 6 b r a c a d a b r a a
• 7 c a d a b r a a b r a
• 8 d a b r a a b r a c a
• 9 r a a b r a c a d a b
• 10 r a c a d a b r a a b
India

• The Burrows-Wheeler-Transformation
output will be
• r d a c r a a a a b b
• Add1 - Sort - Add2 - Sort - ... Add11 - Sort
• last row will be the desired ouput
India

• BWT is a transformation algorithm
• BWT transforms a set of symbols to a
large blocks
• the transformed large blocks will have
longer runs of identical symbols
• compression rate will be higher
• BWT implemented using pointer
India

IGS Quantization Procedure
• Add the Least Significant Nibble (4 bits) of
the previous sum to the current 8-bit pixel
value.
• If the MSN of a given 8-bit pixel is 1111
then add zero instead.
• Declare the Most Significant Nibble of the
sum to be the 4-bit IGS code.
India

Huffman Coding
• Set of symbols and probability are
collected.
• Binary tree is formed using probability.
• The shortest codeword is given for the
symbol/pixel with the highest probability.
• The longest codeword is given for the
symbol/pixel with the lowest probability.
India

Example
Source Symbols A, B, C, D, E
Frequency 24, 12, 10, 8, 8
A B C D E
0
0 0
0
1
1 1
1
India

Example continues ...
Symbol Frequency Code Code total
Length Length
A 24 0 1 24
B 12 100 3 36
C 10 101 3 30
D 8 110 3 24
E 8 111 3 24
Total 62
Normal code 62 * 8 = 496, Huffman Code = 138
India

Arithmetic Coding
• A form of entropy encoding used in
lossless data compression.
• Normally, a string of characters such as
the words "hello there" is represented
using a fixed number of bits per character,
as in the ASCII code.
• Arithmetic coding encodes the entire
message into a single number, a fraction n
where (0.0 ≤ n < 1.0).
India

Arithmetic Encoding
• Probability is calculated for each unique symbol.
• Based on the probability value range is
calculated for each unique symbol.
Step 1: Initially, L=0 and R=1.
Step 2: Read Symbol (S)
Step 3: R=L+(R-L) X CIV
Step 4: L=L+(R-L) X OIV
Step 5: Repeat steps 2-4 until symbols.
Step 6: Any value between L and R is encoded
value (EV) normally L is referred.
India

Arithmetic Decoding
Step 1: Read the encoded value (EV)
2: Find the matching range value
3: Popup the corresponding symbol
4: EV=EV-OIV
5: EV=EV/(CIV-OIV)
6: Repeat steps 2-5 until EV reaches 0
India

• The shortest code words are assigned to
the most frequent (high probability) gray
levels.
• The longest code words are assigned to
the least frequent (low probability) gray
levels.
• Data compression is achieved by
assigning fewer bits to more probable gray
levels than the less probable gray levels.
India

Lossy Predictive Coding
• Local, Global, Adaptive methods
• Encoder
• Decoder
• Predictor
– pixel value sent to the predictor
– based on past inputs a output value is
predicted
– the predicted value is converted to integer
India

nnn ffe ˆ






 
m
i
in iyxfroundyxf
1
),(),(ˆ 
• prediction is linear combination of m
• m is the order of the linear predictor
• αi is prediction coefficient - root mean square
criterion - autocorrelation criterion.
• prediction error
India

First order
 )1,(),(ˆ  yxfroundyxf 
Input
Image
Predictor
Nearest
Integer
Symbol
Encoder
Compressed
Image
Symbol
Decoder
Predictor
Reproduced
Image
India

Transform Coding
• Reversible, Linear transform such as
Fourier Transform is used.
• Transform coding is used to convert
spatial image pixel values to transform
coefficient values.
• Since this is a linear process and no
information is lost, the number of
coefficients produced is equal to the
number of pixels transformed.
India

• The desired effect is that most of the
energy in the image will be contained in a
few large transform coefficients.
• If it is generally the same few coefficients
that contain most of the energy in most
pictures, then the coefficients may be
further coded by lossless entropy coding.
India

• In addition, it is likely that the smaller
coefficients can be coarsely quantized or
deleted (lossy coding) without doing visible
damage to the reproduced image.
Input Image
N x N
Forward
Transform
Quantizer
Symbol
Encoder
Construct
Subimage
n x n
Compressed
Image
Inverse
Transform
Symbol
Decoder
Merge
n x n
subimages
Decompressed
Image
India

• Fourier Transform
• Karhonen-Loeve Transform (KLT)
• Walsh-Hadamard Transform
• Lapped orthogonal Transform
• Discrete Cosine Transform (DCT)
• Wavelets Transform
India

LZW coding
• Lempel-Ziv-Welch, 1978, 1984
• assigns fixed length code words to
variable length sequences of source
symbols.
• no priori knowledge of the probability of
source symbols
• GIF, PDF, TIFF
India

• Codebook / dictionary is constructed
• Encoder: As the input is processed
develop a dictionary and transmit the
index of strings found in the dictionary.
• Decoder: As the code is processed
reconstruct the dictionary to invert the
process of encoding
India

LZW Compression Algorithm
• w = NIL;
• while ( read a character k )
• {
• if wk exists in the dictionary
• w = wk;
• else
• add wk to the dictionary;
• output the code for w;
• w = k;
• } Dr. C. Saravanan, NIT Durgapur,
India

Example
Input string is "^WED^WE^WEE^WEB^WET".
• w k output index symbol
• -----------------------------------------
• NIL ^
• ^ W ^ 256 ^W
• W E W 257 WE
• E D E 258 ED
• D ^ D 259 D^
• ^ W
• ^W E 256 260 ^WE
• E ^ E 261 E^
• ^ W
• ^W E
• ^WE E 260 262 ^WEE
• E ^
• E^ W 261 263 E^W
• W E
• WE B 257 264 WEB
• B ^ B 265 B^
• ^ W
• ^W E
• ^WE T 260 266 ^WET
India

LZW Decompression Algorithm
• read a character k;
• output k;
• w = k;
• while ( read a character k )
• /* k could be a character or a code. */
• {
• entry = dictionary entry for k;
• output entry;
• add w + entry[0] to dictionary;
• w = entry;
• }
India

Example
• Input string is "^WED<256>E<260><261><257>B<260>T".
• w k output index symbol
• -------------------------------------------
• ^ ^
• ^ W W 256 ^W
• W E E 257 WE
• E D D 258 ED
• D <256> ^W 259 D^
• <256> E E 260 ^WE
• E <260> ^WE 261 E^
• <260> <261> E^ 262 ^WEE
• <261> <257> WE 263 E^W
• <257> B B 264 WEB
• B <260> ^WE 265 B^
• <260> T T 266 ^WET
India

Problems
• Too many bits,
• everyone needs a dictionary,
• dictionary space
• Creates entries in the dictionary that may
never be used.
India

Run Length Coding
• Stored as a single data value and count,
rather than as the original run
• Simple graphic images such as icons, line
drawings, and animations
B BBB BB
12W1B12W3B24W2B14W
India

Example
• consider a screen containing plain
black text on a solid white
background
• There will be many long runs of white
pixels in the blank space, and many
short runs of black pixels within the
text.
India

2D Run Length Coding
• scan data one row after another
• find the largest rectangle of uniform color
India

Procedure
• Make a duplicate image matrix
• quantize the colour levels
– reduce the colour levels
– from 256 to 128 / 64 / 32 / 16
– depends on the application reduction level is
maintained
– uniform colour level regions are identified
– different colour level regions are labeled and
mapped
India

Applications
• Texture Analysis
• Compression
• Segmentation
• Medical Image Analysis
• Layout Analysis
• Text finding, word / line spacings
India

Entropy Coding
• Entropy - smallest number of bits needed,
on the average, to represent a symbol
• creates and assigns a unique prefix-free
code to each unique symbol that occurs in
the input.
• compress data by replacing each fixed-
length input symbol with the corresponding
variable-length prefix-free output
codeword
India

Entropy Coding Techniques


i
iin ppppH 21 log)...(
• Huffman coding
• Arithmetic coding
• Lempel-Ziv coding
• Entropy of a set of elements e1,…,en
with probabilities p1,…,pn is:
India

• Entropy of e1,…en is maximized when
• 2k symbols may be represented by k bits
• Entropy of p1,…pn is minimized when
0),...,(0...,1 121  nn eeHppp
Max and Min Entropy
neeH
n
ppp nn 2121 log),...,(
1
... 
India

Entropy Example
• Example 1
– A, B pA=0.5 pB=0.5
– H(A,B) = -pA log2 pA -pB log2 pB
– = -0.5 log2 0.5 -0.5 log2 0.5 =1
• Example 2
– A, B pA=0.8 pB=0.2
– H(A,B) = -pA log2 pA -pB log2 pB
– = -0.8 log2 0.8 - 0.2 log2 0.2 = 0.7219
India

Digital image processing - OLD

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