02/05/2002 (C) University of Wisconsin 2002, CS 559 Last Time Color Quantization Mach Banding –Humans exaggerate sharp boundaries, but not fuzzy ones

02/05/2002 (C) University of Wisconsin 2002, CS 559

Last Time

• Color Quantization

• Mach Banding– Humans exaggerate sharp boundaries, but not fuzzy ones


Today

• Homework 1 is due– Box up front

• Dithering (Digital Halftoning)


Dithering (Digital Halftoning)

• Mach bands can be removed by adding noise along the boundary lines

• General perceptive principle: replaced structured errors with noisy ones and people complain less

• Old industry dating to the late 1800’s– (Can’t say since the turn of the century!)

– Methods for producing grayscale images in newspapers and books


Dithering to Black-and-White

• Black-and-white is still the preferred way of displaying images in many areas– Black ink is cheaper than color

– Printing with black ink is simpler and hence cheaper

– Paper for black inks is not special

• To get color to black and white, first turn into grayscale: I=0.299R+0.587G+0.114B– This formula reflects the fact that green is more representative of

perceived brightness than blue is


Sample Images


Threshold Dithering

• For every pixel: If the intensity < 0.5, replace with black, else replace with white– 0.5 is the threshold– This is the naïve version of the algorithm

• To keep the overall image brightness the same, you should:– Compute the average intensity over the image– Use a threshold that gives that average– For example, if the average intensity is 0.6, use a threshold that is higher

than 40% of the pixels, and lower than the remaining 60%

• For all dithering we will assume that the image is gray and that intensities are represented as 0-1.0– If you have a 0-255 image, you can scale all the thresholds (multiply by

255)


Naïve Threshold Algorithm


Random Modulation

• Add a random amount to each pixel before thresholding– Typically add uniformly random amount from [-a,a]

• Pure addition of noise to the image– As stated above, we are adding uniformly random noise

– For better results, add better quality noise

– For instance, use Gaussian noise

• Should use same procedure as before for choosing threshold

• Not good for black and white, but OK for more colors– Add a small random color to each pixel before finding the closest

color in the table


Random Modulation


Ordered Dithering

• Break the image into small blocks

• Define a threshold matrix– Use a different threshold for

each pixel of the block

– Compare each pixel to its own threshold

• The thresholds can be clustered, which looks like newsprint

• The thresholds can be “random” which looks better

Threshold matrix

59812

151144

711610

133162

16

1

25.025.025.025.0

25.05.05.05.0

25.05.075.075.0

25.05.075.01

0000

0101

0010

0101


Clustered Dithering

18 5 17325

15 6 2 112174912

18 19 13 24 16

1420102322


Dot Dispersion

111 7610

4 14 1559812

133162


Pattern Dithering

• Compute the intensity of each sub-block and index a pattern

• NOT the same as before– Here, each sub-block has one of a fixed number of patterns – pixel

is determined only by average intensity of sub-block

– In ordered dithering, each pixel is checked against the dithering matrix before being turned on


Floyd-Steinberg Dithering

• Start at one corner and work through image pixel by pixel– Usually scan top to bottom in a zig-zag

• Threshold each pixel• Compute the error at that pixel: The difference between

what should be there and what you did put there• Propagate error to neighbors by adding some proportion of

the error to each unprocessed neighbor

e

3/16 5/16 1/16

7/16


Floyd-Steinberg Dithering


Color Dithering

• All the same techniques can be applied, with some modification– Below is Floyd-Steinberg: Error is difference from nearest color in

the color table, error propagation is the same


Images as Signals

• We can view images as a signal - a value that changes over space

• Spatial domain: signal is given as values at points in space

• Frequency Domain: signal is given as values of frequency components


Why Signal Processing?

• Reconstruction:– What happens when a monitor converts a set of points into a

continuous picture?– How do we enlarge images?

• Sampling:– How do we know how many pixels are required?

• Jaggies, Aliasing– How do we avoid jagged, pixelated lines

• Digital Signal Processing provides the tools


A Striking Example

domain same )sin( 22 ,yx


Periodic Functions

• Fact: Any periodic signal can be represented as a sum of sine and cosine waves with harmonic frequencies– If one function has frequency f, then its harmonics are function with

frequency nf for integer n

• Sequence of boxes is:

xxx

k

xkxS

k

kboxes

5cos5

13cos

3

1cos

2

2

1

12

)12cos()1(

2

2

1)(

1

1


Boxes as Sum of Cosines

-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

-0. 25

-0. 05

0. 15

0. 35

0. 55

0. 75

0. 95

1. 15

-0. 25

-0. 05

0. 15

0. 35

0. 55

0. 75

0. 95

1. 15


General Functions

• A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies:

• F() is the spectrum of the function f(x)– The spectrum is how much of each frequency is present in the

function

– We’re talking about functions, not colors, but the idea is the same

deFxf xi)(

2

1)(

xixe xi sincos


Fourier Transform

• F() is computed from f(x) by the Fourier Transform:

dxexfF xi )()(


Example: Box Function

21 02

1 1)(

x

xxf

f

ff

fF

sinc2

sin

)(


Box Function and Its Transform

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5


Cosine and Its Transform

-1.5

-1

-0.5

0

0.5

1

1.5

1-1

If f(x) is even, so is F()


Sine and Its Transform

-1. 5

-1

-0. 5

0

0. 5

1

1. 5

1

-1

-

If f(x) is odd, so is F()


Constant Function and Its Transform

The constant function only contains the 0th frequency – it has no wiggles


Delta Function and Its Transform

Fourier transform and inverse Fourier transform arequalitatively the same, so knowing one direction gives you the other


Shah Function and Its Transform


Gaussian and Its Transform (the same)

-0. 02

0. 03

0. 08

0. 13

0. 18

-0. 02

0. 03

0. 08

0. 13

0. 182

2

2

1 x

e


• The spectrum of a functions tells us the relative amounts of high and low frequencies– Sharp edges give high frequencies

– Smooth variations give low frequencies

• A function is bandlimited if its spectrum has no frequencies above a maximum limit– sin, cos are band limited

– Box, Gaussian, etc are not

Qualitative Properties


Functions to Images

• Images are 2D, discrete functions

• 2D Fourier transform uses product of sin’s and cos’s (things carry over naturally)

• Fourier transform of a discrete, quantized function will only contain discrete frequencies in quantized amounts– In particular, we can store the Fourier transform of a discrete image

in the same amount of space as we can store the image

• Numerical algorithm: Fast Fourier Transform (FFT) computes discrete Fourier transforms

Documents

02/05/2002 (C) University of Wisconsin 2002, CS 559 Last Time Color Quantization Mach Banding –Humans exaggerate sharp boundaries, but not fuzzy ones