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


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


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For Rs. 3 Return 5

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Continuous color input

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

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Pixel Grid

SampledReal Image Quantized Sampled & Quantized





256x256256 gray levels

128x128 64x64 32x32

Binary Image8 gray levels 4 gray levels16 gray levels

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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

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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

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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

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

an oblique line

with y-intercept

a0 and slope a1

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

2Parabola

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

2+

a3x3

Spline

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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

( ) 0001

01 yxxxx

yyy +

=

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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

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Nearest Neighbor

Bi-Linear

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128

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


Well enlarge this image by a factor of 4 Well 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.

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To better see what happens, well look at the parrots eye. To better see what happens, well look at the parrots eye.

Original ImageOriginal Image

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Pixel replication Pixel replication Bilinear interpolationBilinear interpolation

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Pixel replication Pixel replication Bilinear interpolationBilinear interpolation

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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

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Zoom in for a better lookZoom in for a better look

3/7 resize 3/7 resize

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Outlined in blue: 7x7 pixel squares

Outlined in blue: 7x7 pixel squares


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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.


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Keep the highlighted pixels

Keep the highlighted pixels


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dont keep the others. dont keep the others.


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

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

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3/7 times the linear dimensions of the original

3/7 times the linear dimensions of the original

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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

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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.


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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

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3x3 blocks distributed over 7x7 blocks

3x3 blocks distributed over 7x7 blocks


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Empty pixels filled with color from non-empty pixel



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Original image

Original image

7/3 resized 7/3 resized

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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

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.

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EndSampling, Quantization,

Interpolation


02 sampling quantization interpolation

Documents

quantizationkhawar khurshid

gray level quantization

return change

linear interpolation

unknown points

discrete set of colors

linear function

quadratic polynomial