Digital Image Processing

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Digital Image Processing

• Scenes Digital Images Image Resolution Criteria
• Mathematical Preliminaries
• Image Transform
• Image Restoration
• Image Digitization
• Image Quantization
• Image Reconstruction
• Image Enhancement
• Image Processing in Frequency Domain
• Image Compression
• Image Representation Description

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Fundamentals of DIP

• Ultimate Goal To help better
• understand, interpret the
• content of the image

3
Scenes and Images

• Optical image of the scene produced by a lens
• a scene -gt an image (via a lens)
• an image illumination pattern (recorded by
  sensors)
• sensor response varies with color (light
  wavelength)

light rays
Object
lens
image
u
v
1/u 1/v 1/f (f focal length)
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Digital Images

• An Image ??A Picture
• Any 2D function that bears information
• Denoted f(x,y) or f(x), where f() is the
  brightness or gray values in BW image, or RGB
  values in color image
• f(x,y) y
• 
  x

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• Properties of Brightness
• Real
• Non-negative
• Bounded (due to finite field of view)
• Image Examples
• x-ray absorption, proton density distribution of
  MR images, radar images, temperature profile,
  luminance of a scene, drawings

A tumor at choroid plexus region (2124s4.11)
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• Digital Image Processing
• To process a picture using digital techniques,
  i.e., computers
• A typical image processing system
• Digitizer To sample and quantize an image into
  digital form of discrete picture elements (pixel
  , pel)

Image Maker

Scene
Image
Digitizer
Users
Display
CPUs
Storage
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• Sampling and quantization
• Yellow dots are not integer samples
• round-off or truncation is needed
• larger dynamic range may alleviate this problem

gray levels
dynamic range
quantization domain

samples
sampling domain
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• Image Resolution Criteria minimal
• Resolution ...
• as d gets smaller and smaller, two peaks merge
  into one, i.e., d metric resolution is lost

ideal case
real case
d
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• Human eye under visible light and viewing
  distance can resolve separation resolution is
  tied to grain size of the silver halides
  crystals.
• Examples
• A film in electron microscopy is 520 ?m
• 300 dpi (dot per inch) corresponds to 85 ?m
• 10 ?m resolution of a Xenopus laevis embryo
  (microscopic MRI)

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Mathematical Preliminaries

• f(x,y) y
• Digital Picture Function
• analytically well-behaved, i.e., bounded,
  integrable, have Fourier transform pairs, etc.

x
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• Operations on pixels
• point by point operations, e.g., difference image
• local operations, e.g., edge detection
• geometric operation, e.g., image translation
• Problems with pixels on the border
• Assume equal to zero
• Define a sub-picture excluding those
  borderingosed, repetition, (problem oriented)

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• Arithmetic Operations
• Addition p q, Subtraction p - q
• Multiplication p q, Division p q

q (2124s4.3)
p (2124s4.2)
(p-q) 0.5 128
p - q
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• Logic Operations
• AND pANDq (also, p q)
• OR pORq (also, p q)
• COMPLEMENT NOTq (also, )
• EXCLUSIVE OR pXORq
• Example of COMPLEMENT
• before after
• Examples of logic operations on binary images
  (from RCG fig. 2.14)

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Image Digitization

• Uniform Sampling and Quantization
• f(x,y) an image f(i,j) an image element, pixel,
  or pel
• Sampling partitioning the image as an ordered
  pairs of elements (a,b), with a and b being
  integers. a 0 .. N-1, b 0 .. M-1
• Quantization Assigning a real value to the
  sampled image elements (pixels). In black and
  white images, this value is called the gray level
• For an NxM digital image with 2L gray levels, it
  requires NxMxL bits to represent it

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• Digitization Sampling Quantization
• To better represent the picture, M, N, and L
  should be large. Nothing is gained, however, by
  increasing M, N, and L beyond the resolution
  capabilities of the receiver.
• Question How to choose M, N and L for a fixed
  data size (bytes)?

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• Image digitization

f(x,y)
fs(x,y)
u(m,n)
Sampler
Quantizer
Digital Computer
D/A
Display
Converter
u(m,n)

• Sampler performs 2D spatial sampling on input
  image, f(x,y), e.g., sequential sampling
  (scan-in, scan-out digitizer), array sampling
  (CCD camera).
• Quantizer quantizes each pixel of the sampled
  image into a predefined set of values. This range
  is called the dynamic range.

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• Example of a digitized image

• A tumor at choroid plexus region (2124s4.11)

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• Image Sampling
• One dimensional sampling function
• Let f(t) denote the 1D signal and T be the
  sampling period

t
0
T 2T ... nT ...
-T
0
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• Sampling and Reconstruction from sampled data

f(t)
F(w)
-fc
fc
fs(t)
Fs(w)
-fc
fc
-1/T
1/T
f(t)
F(w)
-fc
fc
-1/T
1/T
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• Two problems associated with the reconstruction
  of the original signal from its samples
• 1. If f(t) is not a bandlimited signal, i.e.,
• F(w) ? 0, -8 w 8 or fc 8
• 2. If 1/T is not sufficiently distanced.
• Both cases result overlapped spectrum, a
  phenomenon called aliasing.
• For bandlimited signal, the sufficient condition
  to reconstruct the original signal back from its
  sampled signal is
• 1/T 2 fc
• where fc is the bandwidth of the original signal.
  The lower bound is called the Nyquist rates or
  Nyquist frequency.

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• Two dimensional Sampling
• Ideal sampling function

y
x
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Image Enhancement

• Purposes To make an image better appealing and
  easier to deal with than the original image
• Three categories
• 1. Spatial domain methods operate on the images
  itself, examples as
• Point processing, e.g., image averaging logic
  operation contrast stretching ...
• Mask processing, e.g., filtering or mask
  operation, (blurring, median

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• 2. Frequency domain methods work on the Fourier
  transformed output of the image, examples from
  the convolution theory
• g(x,y) f(x,y) ??h(x,y)
• gt G(u,v) F(u,v) H(u,v)
• gt certain properties of F(u,v) can be
• emphasized into G(u,v)
• gt spatial domain g(x,y) F-1G(u,v)
• 3. Combination of the above two categories

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• Spatial Domain Methods
• Point processing enhancement
• Image intensity transformation
• Negative image

L-1
T(r)
s
0
L-1
r
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• Contrast stretching to increase the dynamic
  range of the gray levels in the image

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• Dynamic range compression
• linear scaling
• input range R, output range L
• output gray level s (r-I0)L/R
• I0 lower bound of the input
• logarithm scaling s c log(1 r)
• useful when the input range is very large, e.g.,
  106, and the brightest parts are dominating (from
  RC Gonzalez)

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• Histogram Processing
• A histogram is a plot of the number of gray
  level, rk, its occurrencesk, 0kL-1, versus the
  range of gray levels normalized to the total
  number of pixels, n
• Histogram of a dark image
• Histogram Equalization to equalize the histogram
  according to its probability density function

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• Example of histogram equalization (from RC
  Gonzalez)

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• Image subtraction
• g(x,y) f(x,y) - h(x,y)

Image subtraction enhancement (a) mask image (b)
image with mask subtracted out (after
injection of dye into the bloodstream) (from RC
Gonzalez)
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• Image averaging consider a noisy image

M 1
M 2
M 16
M 8
M 32
M 128
(from RC Gonzalez)
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• Spatial Filtering (Mask processing)
• lowpass filtering eliminating high frequency
  components gt image blurring (from RC Gonzalez)

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• highpass filtering sharpening edges or fine
  details in an image gt deblurring (from RC
  Gonzalez)

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• Derivative filters
• averaging ? integration gt blurring
• difference ? differentiation gt sharpening
• The most common method of differentiation in
  image processing applications is the gradient
  operator

In discrete case
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• Consider the digital image

At point z5
?f(z5-z4)2(z5-z2) 21/2
z5-z4z5-z2
Laplacian operator
Digital Laplacian has the effect of increasing
the ramp steepness, and of increasing the
contrast at the edges
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• High-emphasis filtering
• differentiation enhances high spatial frequencies
• integration weakens high frequencies
• the effect of subtracting a Laplacian from an
  image itself

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Original (2124s4.1)
Laplace(3x3)
Laplace-(3x3)
High Emphasis
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• Various image processing effects

Original
Smooth
1 1 1 1 1 1 1 1 1
Shadow
Sharpen
-2 -1 0 -1 1 1 0 1 2
-1 -1 -1 -1 -12 -1 -1 -1 -1
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Original
Trace edge
1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 -1 -1 -1
-1 0 1 -1 0 1 -1 0 1
Dither (Floyd-Steinberg)
Reduce Noise (median filter)
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Grad-NW (3x3)
Grad-N (3x3)
Laplace (5x5)
Laplace (9x9)
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Grad-W (7x7)
High-Emphasis (3x3)
Hat (5x5)
Hat (13x13)
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Mean (5x5)
Mean (15x15)
Gauss (15x15)
Gauss (5x5)
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Original
Subtract Background
Enhance Contrast
Equalize
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Original Binary
Erode A?BxA?(B)x
Open AoB(A?B) ??
Dilate A?Bx(B)x?A??
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Close
Original Binary
Skeletonize
Outline
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• Two pictures are worth more than ten thousand
  words