Classical image compression
Paul Cockshott
Faraday partnership

We will look at some or  the currently well established methods of compressing images.
These are used for digital TV and for digital cameras

Summary of this section
 Lossless compression
Quadtree compression
Lossy compression
Reduce to 8 bits
Frequency domain compression systems
Orthogonal bases
Representation of blocks using orthogonal bases

Quadtree compression
Lossless compression system
Image is successively divided into 4 until all pixels in a square are the same

Quad Tree Flattening

Reduction to 8bit colour

 Frequency domain compression systems
Further compression is possible by making use of the way that our perception is biased towards not noticing slight changes in brightness or very fine details.
This is what various compression systems working in the frequency domain  such as DCT and wavelet compressors take advantage of.

Orthogonal Bases
The key to these systems is the concept of an orthogonal basis or coordinate system in which areas of a picture can be represented.
In what follows let us initially consider just monochrome images.

In 2 D Geometry
A vector is a sequence of numbers. [4,3] for example is vector with 2 elements.
We are all familiar with how these can be used to represent lines from the origin in 2d geometry.

Vectors at right angles
An orthogonal pair of vectors is a pair of vectors at right angles.
For example the vectors along the X and Y axes [1,0], [0,1] are at right angles and thus orthogonal.

Vectors at right angles
Similarly the 45degree vectors   [1,1], [1,-1] are at right angles and thus orthogonal.

Vector Lengths

Orthonormal vectors
If in addition to being at right angles the vectors are of length 1, then they are orthonormal.
For example the pairs of vectors ([0,1],[1,0]) and

Change of basis
Any pair of orthonormal vectors can substitute for the x and y axes in defining the position of a point. The new axes serve as a ‘basis’ for defining points.

Extension to n dimensions

Inner product operations
We can measure how far along an axis provided by one of the basis vectors a given v is by using the inner product operator.
Let v be defined in the x,y basis and let p,q be and alternative orthonormal basis. The point v in terms of p,q is given by the vector [p.v,q.v]

Transforming to and From a Basis
 Given v of dimension n and orthonormal basis vectors b1,b2,..,bn we can express v in this new basis as w=v [b1,b2,..,bn ] = [v.b1,v.b2,.., ] without loss of information

Representation of Blocks Using Orthogonal Bases
Haar wavelets
DCT coefficients

Consider the block of 4 pixels a,b,c,d
These can be considered a 4d vector
[a,b,c,d] expressed in the basis Space

Haar wavelets
An alternative basis to express the picture block would be Haar wavelets
[1,1,1,1], [1,1,-1,-1], [1,-1,0,0], [0,0,1,-1]

Harr wavelets

DCT coefficients
The Discrete Cosine Transform, uses a set of discrete cosine waves of different frequency combinations to act as basis vectors

Use of blocks in JPEG
Divide each channel (Y, U, V) into 8x8 blocks which are then 64 dimension vectors
Multiply each block by each of 64 DCT basis vectors.
Each 8x8 block of pixel values becomes an 8x8 block of coefficients.

Zig Zag
The block of coefficients is read out in zig zag order so that low frequency coefficients come before high frequency ones

Clearly if we encode a coefficient to less precision we need fewer bits to do so.
A suitable technique for this is to divide the coefficients by an integer and ignore the remainder on encoding and do the reverse on decoding.

Quantization continued
Suppose we have the coefficient vector [ 11, 93, 102, 77]
We divide by a constant, (14 in this instance) to obtain [0.78, 6.64,7.24,5.5].
Ignore remainder and we have [0, 6,7,5].
Multiply by 14 again and we get [0,84,98,70] which is close to the original vector

JPEG quantization
Different quantization divisors are used in each position in the block so that the low frequency information is less quantized.

JPEG quantization
This is done using quantization matrix

Run length encoding
The zig zag ordering plus quantization causes means that most of the last set of coefficients are zero. The transmission of an EOB, (end of block) code, indicates that there are no more coefficients in the block.

MPEG is a set of standards for the encoding of moving pictures. It follows on from JPEG with which it shares the basic features of
Zig Zag encoding
Run length encoding
It adds the new features of
Key frames
Motion compensation

Key frames
Since there is substantial sequential redundancy in video ( one frame similar to the next) only Key Frames or Interframes are fully encoded, using JPEG techniques
Typically one frame in 15 is an  Interframe

Motion compensation

Search for best match

Classical compression works by tranfering the image into a vector space that better captures the redundancy of the image
DCT or wavelet coefficients
Motion compensation for video

Where to get more information

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