FIT 2011 Blok Sort

8/3/2019 FIT 2011 Blok Sort

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by Basar KOC & Ziya ARNAVUT



What are digital colors? Human beings perceive colors by the nature of the light

reflected from an object!

Achromatic Color: means literally ³without color´

Only intensities (amount of light)

Gray levels as seen on black/white TV-monitor

Ranges from black to white

Chromatic Color: All colors other than neutral colors (white, black,

and the pure grays), are chromatic.

The word ³color´ in ordinary language is often used to refer

exclusively to chromatic colors, e.g., color vs. black-and-white

television.



RGB Color Theory RGB stands for Red, Green, Blue RGB color is in use in many devices,

Televisions, computer screens, digital cameras.

When the visible color spectrum wavelength (400-700 nm) is

broken into thirds,

RGB colors are the predominant colors.

Three types of cone cells exist in human eye.

Each cell is more sensitive to either short ( S ), medium( M ), or long( L ) wavelength light.

These curves are often also referred as the ³tri-stimulus functions´.



RGB Color Space A single pixel consists of three components each in range of

[0,255].

Each pixel is a triplet

128 251 60 =

Pixel vectorin memory Final pixel inthe image



Size consideration in an RGB image

The lowest resolution for a monitor

Displaying a Windows desktop is 640 x 480 pixels.

In a bitmap of this resolution, then, there would be 3 bytes per

pixel, For a total of 640 x 480 x 3, or about 900 kilobytes.



Bit-Depth = Color-Depth Number of Colors = 2^(Bit-depth) Bit-depth is the number of bits.

It is also called ³Color resolution´.

Bit depthBit depth Color resolutionColor resolution CalculationCalculation1-bit 2 colors 2^1 = 2

2-bits 4 colors 2^2 = 4



8-bits 256 colors 2^8 = 256

16-bits 65,536 colors 2^16 = 65536

24-bits 16,777,215 colors 2^24 = 16.7 million



Color Palettes Images look best

If they¶re stored with as RGB images with 16,777,216 colors.

However, file seizes may be very large.

For practical purposes,

we would like to reduce the number of colors from 16,777,216 to

256 colors.

Such files are referred to as using palette-color.

The colors in a palette-color file are derived from a potential paletteof 16,777,216 colors, but no more than 256 of them can be used in

any one image.



Graphics Interchange Format (GIF)

The GIF is one of the most commonly used graphic file formats,

especially on the Internet.

an indexed color image format.

The color of the image is indexed in a palette. (a color-table)

The GIF is only capable of supporting a maximum of 256 colors.

Uses lossless compression algorithm.



Structure of a BMP

Color-Palette Image

Information ± 56 bytes

Size

Dimensions Width, height

Bit per pixel 8, 4, 2, 1

Color Table 3 x 256 bytes at most

The color-mapped table of an image

Index R G B

0 28 0 1

1 19 2 5

2 34 1 1

3 39 2 3

4 44 0 2

.. .. .. ..

254 193 211 223

255 206 212 222



Euclidean Distance Metric

Let E[a,b] be the Euclidean distance between two color indices

a and b.

(0 a, b 255)

where,

E[a,b]: Distance between index a and index b. eR: Difference between R values of a and b indices.

eG: Difference between G values of a and b indices.

eB: Difference between B values of a and b indices.



Euclidean and Pseudo Distance

matrices

Pseudo-distance matrix

Distance matrix

created from the color palette

converted

Index R G B

0 28 0 1

1 19 2 5

2 34 1 1

3 39 2 3

4 44 0 2

.. .. .. ..

254 193 211 223255 206 212 222

Color-map table



Encoding

Reference pixel x and its neighbors Ranking with pseudo-distance matrix

X is the index value to be predicted. a, b, c are neighboring indices.

Using (a, x) we determine ranking value, or error e.

Pseudo-distance matrix



Decoding

In error matrix E,

the index number of the first pixel at the top-left corner, call it a, is

copied to I. Later we apply the following:

Receive the error signal e of the next pixel from the encoded image.

In row a of the pseudo-distance matrix search the error signal value e.

Emit the corresponding column value x as the original index value of

the image.

Let x be a and repeat the process until we reach to the end of file.

Clearly, in decoding process, we obtain original index valueswithout any loss.



a1 x1 x2e11

e12

e13

e14

e15

..

e21

e22

e23

e24

..

..

ei1

..

.. x12 .. x13 ..

.. .. .. .. .. ..

a1 .. e12 .. .. ..

.. .. .. .. ..

a2 .. .. .. e13 ..

«. .. .. .. .. ..

Copy e11 to I(1,1)

Let a1 e11

Determine x12 by searching e12 in row a1.

I(1,2)

x12Let a2 x12

Determine x13 by searching e13 in row a2.

I(1,3) x13

We repeat the process similarly.

1

2

3

Decoding (cont.)

E: Encoded image P: Pseudo-distance matrix I: Image



Why do we use the pseudo-distance

transformation?

0%

10%

20%

30%

40%

50%

60%70%

80%

90%

100%

01

2

3Sunset Serrano Sea_dusk Yahoo

0%

10%

20%

30%

40%

50%

60%70%

80%

90%

100%

01

2

3Sunset Serrano Sea_dusk Yahoo

Before transformation After transformation



Distribution of Music.bmp (n = 8) before and after transformation

Distribution of Serrano.bmp (n = 256) before and after transformation

Why do we use the pseudo-distance

transformation?



Block-sorting Transformations

After the pseudo-distance technique, we apply block

sorting transformations on the data.

Linear Ordering Transformation (LOT)

Context-level one

Burrows-Wheeler Transformation (BWT)

Context-level n



Linear Ordering Transformation For example, let w = [1, 3, 1, 3, 2]

Original l l -1

1 2 3 4 5 1 4 2 5 3 1 3 5 2 4

1 3 1 3 2 1 3 1 3 2 1 1 2 3 3

Note that, the inverse permutation of LOT l isl -1 = [1 ,3, 5, 2, 4]. l -1 is called the Canonical SortingPermutation of w.

Also, elements of w is sorted in non-decreasing order by l -1 and consists of m-blocks of different sizes.

Sorted data can be encoded cheaply.



Burrows-Wheeler Transformation

Let w = [3,1,3,1,2] be a data string. Construct

3 1 3 1 2

M= 1 3 1 2 33 1 2 3 1

1 2 3 1 3

2 3 1 3 1by forming the successive rows of M , which are

consecutive cyclic left-shifts of w.



Note that the original data string w is the 5throw of M ¶ .

Given the I = 5 (row index) of w in M ¶ and

L¶ = [3, 3, 1, 1, 2] we can recover w .

1 3

1 3

M ¶

= 2 13 1

3 _ _ _ 2

Burrows-Wheeler Transformation



Binary Arithmetic Coder

We observed that after the pseudo-distance metric is applied to

indices of various color-mapped images, on some images

percentage of 0s varies from 72-90.

Hence, we applied context-adaptive binary arithmetic coder which includes run length coding and context modeling.

This yields better compression gain than Huffman coder, which

was originally proposed by Kuroki et al.



Images #Colors Size GIF PDT+SAC

PDT+LIN+BAC

PDT+BWT+BAC

Benjerry 48 28,326 1.239 1.218 1.255 1.182

Books 7 29,338 3.046 3.447 3.121 2.757

Clegg 256 719,158 3.623 2.808 2.474 1.972

Cwheel 256 481,078 1.492 0.984 0.957 0.919

Descent 122 64,542 2.928 2.765 2.731 2.716

Fractal 256 389,190 6.923 5.335 5.355 5.354

Frymire 256 1,238,678 1.485 1.373 1.426 1.276

Gate 84 61,302 2.939 2.471 2.418 2.376

Ghouse 256 481,078 3.713 3.219 3.169 3.128

Music 8 6,302 2.482 2.305 2.356 2.037

Netscape 32 61,382 2.113 1.922 1.880 1.825

Party8 12 75,606 0.854 0.778 0.772 0.699

Pc 6 1,721,858 1.694 1.723 1.594 0.977

Sea_dusk 46 157,538 0.323 0.052 0.052 0.048

Serrano 256 502,886 1.629 1.389 1.341 1.211

Sunset 204 308,070 2.601 1.590 1.557 1.550

Winaw 10 148,894 0.997 0.980 0.933 0.856

Yahoo 229 28,110 1.983 1.818 1.855 1.777

Average 361,296 2.337 2.010 1.958 1.814

W. Avg. 2.328 1.974 1.899 1.629

Normalized 1.429 1.212 1.166 1.000

Experimental Results



Thank You!

Questions?

FIT 2011 Blok Sort

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