Compression Principles

I. Don’t allocate space for values that never occur

• The image has values from 0 to 255

• But only 16 levels are occupied

I. Probabilistic extension

Example

Suppose we use

1 to code the value 128

0xxxx to code other values.

If 128 occurs with probability P, these pixels we save 3*P bits;

One extra bit at 1*(1-P) of the pixels.

For this strategy and 4 bits per pixel, the break even point is 3*P - 1*(1-P) = 0 4P – 1 = 0 P = 0.25 128 occurs with high probability Use short symbols to code values that occur a lot.

II. Spatial redundancy

Some percentage of the pixels are set to random intensity levels.

Subject must set the dot intensity to the correct value.

With 1% of the pixels deleted,

observers set the value to its original 80% of the time;

With 40% deleted, they are still correct half the time

Horace Barlow

The intensity histogram

Values are distributed

fairly uniformly across the different levels.

No obvious way to code the intensities efficiently.

The spatial correlation means we can recode to reduce the range Digital value (x) D igi ta l va lue ( x+ 1)

If we code the difference we reduce the variance

Difference is easier to compress:

u = x + y

v = x - y

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De-correlating image data

Do you know why u,v are 9 bits and

x,y are 8 bits?

The counter-intuitive observation is that

the potential range of the transformed data may be larger than the original range;

JPEG-DCT

Best lossless compression factors are around 3-5. People want more.

What to do?

Application: JPEG-DCT

• Compress the stuff people can’t see anyway. They will never know. • This is called lossy compression

(also called perceptually lossless compression).

Discriminating contrast levels

Quantization table

Quantization table

Quantization table

The DCT formula 7 7 0 0 ( ) ( ) 2 1 2 1 ( , ) ( , )cos cos 4 _{x y} 16 16 C u C v x y F u v f x y u

p

p

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The DCT formula 7 7 0 0 ( ) ( ) 2 1 2 1 ( , ) ( , )cos cos 4 _{x y} 16 16 C u C v x y F u v f x y u

p

p

åå

DCT Basis Functions

Multiple resolution and orientation much like human vision models

8 6 6 7 6 5 8 7 7 7 9 9 8 10 12 20 13 12 11 11 12 25 18 19 15 20 29 26 31 30 29 26 28 28 32 36 46 39 32 34 44 35 28 28 40 55 41 44 48 49 52 52 52 31 39 57 61 56 50 60 46 51 52 50

Luminance table (default, q = 75)

Quantization matrix

Number of output levels is ceil(256/value)

Quantization values

Low frequencies High frequencies 128 30 10 7 6 5 8 7 7 80 9 9 8 10 12 20 13 12 11 11 12 25 18 19 15 17 29 26 2 30 5 0 28 28 9 0 0 0 0 0 44 35 5 0 0 0 0 0 48 49 2 0 0 0 0 0 3 0 0 0 0 0 0 0

After scaling and quantization, there are many zeros in the coefficient table

Compressing the JPEG coefficients

Special handling of the DC term

1. DC Term is usually large

2. It is usually similar to the value in the previous block 3. So, the DC term is encoded as the difference from the

previous block

Repeated characters:

Data: AAAABBBAABBBBBCCCCCCC Code: 4A3B2A5B7C

Run length encoding

JPEG: is mostly zeros only in high spatial frequency, so method is not effective

Data: GBCDEAFGAAAAAA Code 1: 1G1B1C1D1E1A1F1G6A

Simple encoder/decoder

If B > 0 repeat next character B times If B < 0 copy next B characters to output If B = 0, stop

100 75

50 20

Artifact visibility depends on image size physical units

Aside – For EE parties:

Phase and amplitude scrambling

Randomize Amplitude Randomize Phase

JPEG – Color

Facts you know about color perception that matter for JPEG

JPEG color - informal

Color profile

Many JPEG files embed an ICC color profile (color space). Commonly used color profiles include sRGB and Adobe RGB. Because these color spaces use a non-linear transformation, the dynamic range of an 8-bit JPEG file is

about 11 stops; see gamma curve. [However, many applications are

not able to deal with JPEG color profiles and simply ignore them.

(emphasis added, text deleted at some point).]

JPEG color space: YCbCr

0.299

0.587

0.114

0

0.1687

0.3313

0.5

128

0.5

0.4187

0.0813

128

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R

Cb

G

Cr

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Default luminance table (q=75)

Y = 0.299R + 0.587G + 0.114B

Cb = -0.1687R - 0.3313G + 0.5B + 128 Cr = 0.5R - 0.4187G - 0.0813B + 128

JPEG Color: YCbCr

Target is an RGB image

Y = 0.299 R + 0.587 G + 0.114 B I = 0.596 R - 0.275 G - .321 B Q = 0.212 R - 0.523 G + .311 B

For transmission in the US, the image was converted to YIQ Transformed to

I Q

Y Q

Multiscale representations

Pyramid

construction

principles

Pyramid

construction

principles

Pyramid

construction

principles

Pyramid

construction

principles

Gaussian and Laplacian Pyramids

Blur and sub-sample

Upsample, error

Pyramids

Gaussian

The pyramids have more coefficients than the original

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Coefficient variance is smaller in Laplacian pyramid

Gaussian pyramid coefficients