National University of Sciences and
Technology (NUST)
School of Electrical Engineering and Computer Science
(SEECS)
Digital Image Processing
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Khawar Khurshid
Sampling
Quantization
Interpolation
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Sampling and Quantization
Sampling:
Digitization of the spatial coordinates (x,y)
Quantization:
Digitization in amplitude (also known as gray
level quantization)
• 8 bit quantization: 28 =256 gray levels
(0: black, 255: white)
• 1 bit quantization: 21 = 2 gray levels
(0: black, 1: white)
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Sampling
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Sampling
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Quantization – (Example)
Return change using only
these notes
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Quantization – (Example)
Return 5
For Rs. 7
Return 5
For Rs. 9
Return 10
For Rs. 12
Return 10
For Rs. 23 Return 25
…
…
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Change you return
For Rs. 3
50
40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
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Actual change
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Discrete color output
Quantization – (Example)
Continuous colors mapped to a
finite, discrete set of colors.
Continuous color input
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16 colors
16 million colors
Quantization – (Example)
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Sampling and Quantization
Pixel Grid
Real Image
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Sampled
Quantized
Sampled &
Quantized
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Sampling and Quantization
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Sampling and Quantization
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Sampling and Quantization
256x256
256 gray levels
128x128
64x64
16 gray levels
8 gray levels
4 gray levels
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32x32
Binary Image
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Interpolation
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Interpolation
Why?
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Changing Perspective
Rotation
Resizing
Warping
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Nearest Neighbor Replacement
Simply Replicate the value from neighboring
pixels.
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1
1
16
0
1
0
1
0
1
1
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0
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0
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Nearest Neighbor Replacement
Adds blocky effect.
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0
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0
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0
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1-D Interpolation
Interpolation works by using known data to
estimate values at unknown points.
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1-D Interpolation
Linear Interpolation: Fit a linear function piecewise
between the points.
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1-D Interpolation
Quadratic: Fit a Quadratic Polynomial between the three
points.
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1-D Interpolation
Fitting Polynomial to data points.
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1-D Interpolation
Degree
Polynomial
Zero
Polynomial
f(x) = 0
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Req. Pts.
None
f(x) = a0
horizontal line
with y-intercept
a0
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1 (Linear)
f(x) = a0 + a1x
an oblique line
with y-intercept
a0 and slope a1
2
2
(Quadratic)
f(x) = a0 + a1x + a2x2
3 (Cubic)
f(x) = a0 + a1x + a2x2+
a3x3
0
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How it looks
Parabola
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Spline
No. of unknowns in a polynomial of degree n-1 = n
Select n nearest known neighbours to generate n equations
Solve the linear system of equations to estimate the constants
Simply put the unknown x value and interpolate the y value
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Linear Interpolation
• Equation of a line
y1 − y0
(x − x0 ) + y0
y=
x1 − x0
• Interpolation using Linear Polynomial
–
–
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y = a0 + a1x
Requires two unknowns
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Bi-linear Interpolation
• Key Points:
– Q11 = (x1, y1), Q12 = (x1, y2),
– Q21 = (x2, y1), Q22 = (x2, y2).
• Linear interpolation in the xdirection followed by linear
interpolation in y-direction
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Interpolation
Nearest Neighbor
Bi-Linear
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Interpolation
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128
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Interpolation
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Interpolation – (Comparison)
We’ll enlarge this image
by a factor of 4 …
… via bilinear interpolation
and compare it to a nearest
neighbor enlargement.
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Interpolation – (Comparison)
Original
Image
To better see what happens, we’ll look at the parrot’s eye.
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Interpolation – (Comparison)
Pixel replication
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Bilinear interpolation
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Interpolation – (Comparison)
Pixel replication
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Bilinear interpolation
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Interpolation – (Non Integer)
Zoom in on a
section for a
closer look at
the process
Example:
resize to 3/7
of the original
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Interpolation – (Non Integer)
3/7 resize
Zoom in for a
better look
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Interpolation – (Non Integer)
3/7 resize
Outlined in
blue: 7x7
pixel squares
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Interpolation – (Non Integer)
3/7 resize
In yellow: 3 pixels
for every 7 rows,
3 pixels for every
7 cols.
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Interpolation – (Non Integer)
3/7 resize
Keep the
highlighted
pixels…
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Interpolation – (Non Integer)
3/7 resize
… don’t keep
the others.
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Interpolation – (Non Integer)
3/7 resize
Copy them into
a new image.
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Interpolation – (Non Integer)
3/7 resize
Copy them into
a new image.
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Interpolation – (Non Integer)
3/7 times the
linear dimensions
of the original
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Interpolation – (Non Integer)
Original
image
Detail of
resized image
Resize to 3/7 of
the original dims.
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Interpolation – (Non Integer)
7/3 resize
Original
image
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Pixels spread out
for a 7/3 resize …
… then filled in.
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Interpolation – (Non Integer)
7/3 resize
Each 3x3 block
of pixels from
here …
Original
image
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… is spread out over
a 7x7 block here
Detail
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Interpolation – (Non Integer)
7/3 resize
3x3 blocks
distributed over
7x7 blocks
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Interpolation – (Non Integer)
7/3 resize
Empty pixels filled
with color from nonempty pixel
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Interpolation – (Non Integer)
7/3 resize
Empty pixels filled
with color from nonempty pixel
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Interpolation – (Non Integer)
Original
image
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7/3 resized
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Interpolation
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Non-adaptive algorithms
– Examples
• nearest neighbour, bilinear, bicubic, spline etc.
– Depending on their complexity, these use anywhere from 0 to 256 (or more)
adjacent pixels when interpolating.
– The more adjacent pixels they include, the more accurate they can become,
but this comes at the expense of much longer processing time.
– These algorithms can be used to both distort and resize a photo
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Adaptive algorithms
– Many of these apply a different version of their algorithm (on a pixel-bypixel basis) when they detect the presence of an edge-- aiming to minimize
unsightly interpolation artifacts in regions where they are most apparent.
– These algorithms are primarily designed to maximize artifact-free detail in
enlarged photos, so some cannot be used to distort or rotate an image.
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End
Sampling, Quantization,
Interpolation
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