02 sampling quantization interpolation

Advanced Digital Image Processing

Spring ’14

National University of Sciences and Technology (NUST)

School of Electrical Engineering and Computer Science (SEECS)

Khawar Khurshid, 20121

Sampling Quantization Interpolation


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)

Sampling and QuantizationSampling and QuantizationSampling and QuantizationSampling and Quantization


SamplingSamplingSamplingSampling


SamplingSamplingSamplingSampling


Quantization Quantization Quantization Quantization –––– (Example)(Example)(Example)(Example)


Return change using only these notes

6



0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40 45 50

Actual change

Chan

ge y

ou r

etu

rn

For Rs. 3 Return 5

For Rs. 7 Return 5

For Rs. 9 Return 10

For Rs. 12 Return 10

For Rs. 23 Return 25

… …

7



Continuous color input

Dis

crete

col

or o

utpu

t

Continuous colors mapped to a Continuous colors mapped to a finite, discrete set of colors.

8



16 m

illion

col

ors

16 c

olor

s

9

Pixel Grid

SampledReal Image Quantized Sampled & Quantized









256x256256 gray levels

128x128 64x64 32x32

Binary Image8 gray levels 4 gray levels16 gray levels

13

InterpolationInterpolationInterpolationInterpolation


Why?

• Changing Perspective

• Rotation

• Resizing

• Warping



Simply Replicate the value from neighboring pixels.

Nearest Neighbor ReplacementNearest Neighbor ReplacementNearest Neighbor ReplacementNearest Neighbor Replacement


1 0 1

1 1 0

1 0 1

1 0 1

1 1 0

1 0 1

16

Adds blocky effect.

Nearest Neighbor ReplacementNearest Neighbor ReplacementNearest Neighbor ReplacementNearest Neighbor Replacement


1 0 1

1 1 0

1 0 1

1 1 0 0 0 1 1

1 1 0 0 0 1 1

1 1 1 1 1 0 0

1 1 1 1 1 0 0

1 1 1 1 1 0 0

1 1 0 0 0 1 1

1 1 0 0 0 1 1

17

Interpolation works by using known data to estimate values at unknown points.

1111----D InterpolationD InterpolationD InterpolationD Interpolation


Linear Interpolation: Fit a linear function piecewise between the points.



Quadratic: Fit a Quadratic Polynomial between the three points.



Fitting Polynomial to data points.





Degree Polynomial How it looks Req. Pts.

Zero

Polynomialf(x) = 0 None

0 f(x) = a0

horizontal line

with y-intercept

a0

1

1 (Linear) f(x) = a0 + a1x

an oblique line

with y-intercept

a0 and slope a1

2

2

(Quadratic)f(x) = a0 + a1x + a2x

2Parabola

3

3 (Cubic)f(x) = a0 + a1x + a2x

2+

a3x3

Spline

4

• 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 value22

Linear InterpolationLinear InterpolationLinear InterpolationLinear Interpolation


• Equation of a line

• Interpolation using Linear Polynomial– y = a0 + a1x

– Requires two unknowns

( )00

01

01yxx

xx

yyy +−

−

−=

23

BiBiBiBi----linear Interpolationlinear Interpolationlinear Interpolationlinear Interpolation


• Key Points:

– Q11 = (x1, y1), Q12 = (x1, y2),

– Q21 = (x2, y1), Q22 = (x2, y2).

• Linear interpolation in the x-direction followed by linear interpolation in y-direction

24



Nearest Neighbor

Bi-Linear

25



215

128

26



Interpolation Interpolation Interpolation Interpolation –––– (Comparison)(Comparison)(Comparison)(Comparison)


We’ll enlarge this image by a factor of 4 …We’ll enlarge this image by a factor of 4 …

… via bilinear interpolation and compare it to a nearest neighbor enlargement.

… via bilinear interpolation and compare it to a nearest neighbor enlargement.

28



To better see what happens, we’ll look at the parrot’s eye. To better see what happens, we’ll look at the parrot’s eye.

Original ImageOriginal Image

29



Pixel replication Pixel replication Bilinear interpolationBilinear interpolation

30



Pixel replication Pixel replication Bilinear interpolationBilinear interpolation

31

Interpolation Interpolation Interpolation Interpolation –––– (Non Integer)(Non Integer)(Non Integer)(Non Integer)


Zoom in on a section for a closer look at the process

Zoom in on a section for a closer look at the process

Example: resize to 3/7 of the original

Example: resize to 3/7 of the original

32



Zoom in for a better lookZoom in for a better look

3/7 resize 3/7 resize

33



Outlined in blue: 7x7 pixel squares

Outlined in blue: 7x7 pixel squares


34



In yellow: 3 pixels for every 7 rows, 3 pixels for every 7 cols.

In yellow: 3 pixels for every 7 rows, 3 pixels for every 7 cols.


35



Keep the highlighted pixels…

Keep the highlighted pixels…


36



… don’t keep the others.… don’t keep the others.


37




Copy them into a new image.Copy them into a new image.

38




Copy them into a new image.Copy them into a new image.

39



3/7 times the linear dimensions of the original

3/7 times the linear dimensions of the original

40



Resize to 3/7 of the original dims.Resize to 3/7 of the original dims.

Original image

Original image

Detail of resized imageDetail of resized image

41



Original image

Original image

Pixels spread out for a 7/3 resize …Pixels spread out for a 7/3 resize …

… then filled in.… then filled in.


42




Detail Detail

Each 3x3 block of pixels from here …

Each 3x3 block of pixels from here …

… is spread out over a 7x7 block here… is spread out over a 7x7 block here

Original image

Original image

43



3x3 blocks distributed over 7x7 blocks

3x3 blocks distributed over 7x7 blocks


44



Empty pixels filled with color from non-empty pixel



45






46



Original image

Original image

7/3 resized 7/3 resized

47



• 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

• Adaptive algorithms

– Many of these apply a different version of their algorithm (on a pixel-by-pixel 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.

48

EndSampling, Quantization,

Interpolation
