Unit V. Image Compression Two mark Questions 1. What is the need for image compression?

In terms of storage, the capacity of a storage device can be effectively

increased with methods that compress a body of data on its way to a storage device and decompresses it when it is retrieved. In terms of communications, the bandwidth of a digital communication link can be effectively increased by compressing data at the sending end and decompressing data at the receiving end.

At any given time, the ability of the Internet to transfer data is fixed. Thus, if data can effectively be compressed wherever possible, significant improvements of data throughput can be achieved. Many files can be combined into one compressed document making sending easier.

2. What is run length coding?

Run-length Encoding, or RLE is a technique used to reduce the size of a

repeating string of characters. This repeating string is called a run; typically RLE encodes a run of symbols into two bytes, a count and a symbol. RLE can compress any type of data regardless of its information content, but the content of data to be compressed affects the compression ratio. Compression is normally measured with the compression ratio.

3. What are the different compression methods?

The different compression methods are,

i. Run Length Encoding (RLE)

ii. Arithmetic coding

iii. Huffman coding and

iv. Transform coding

4. Define compression ratio.

Compression ratio is defined as the ratio of original size of the image to compressed size of the image.

It is given as Compression Ratio = original size / compressed size: 1

5. What are the basic steps in JPEG?

The Major Steps in JPEG Coding involve:

i. DCT (Discrete Cosine Transformation)

ii. Quantization

iii. Zigzag Scan

iv. DPCM on DC component

v.RLE on AC Components

vi. Entropy Coding

6. What is coding redundancy?

If the gray level of an image is coded in a way that uses more code

words than necessary to represent each gray level, then the resulting image is said to contain coding redundancy.

7. What is interpixel redundancy?

The value of any given pixel can be predicted from the values of its

neighbors. The information carried by is small. Therefore the visual contribution

of a single pixel to an image is redundant. Otherwise called as spatial redundant geometric redundant or interpixel redundant. Eg: Run length coding

8. What is psychovisual redundancy?

In normal visual processing certain information has less importance

than other information. So this information is said to be psycho visual redundant.

9. What is meant by fidelity criteria?

Data loss due to psychovisual redundancy coding may need to be checked. Fidelity criteria are a measure for such loss. •Two kinds of fidelity criteria

1) subjective and 2) objective

10. What is run length coding?

Run-length Encoding, or RLE is a technique used to reduce the size of a

repeating string of characters. This repeating string is called a run; typically RLE encodes a run of symbols into two bytes, a count and a symbol. RLE can compress any type of data regardless of its information content, but the content of data to be compressed affects the compression ratio. Compression is normally measured with the compression ratio.

11. Define source encoder.

Source encoder performs three operations:

1) Mapper -this transforms the input data into non-visual format. It reduces

the interpixel redundancy.

2) Quantizer - It reduces the psycho visual redundancy of the input images.

This step is omitted if the system is error free.

3) Symbol encoder- This reduces the coding redundancy .This is the final

stage of encoding process.

12. Draw the JPEG decoder.

13. What are the types of decoder?

Source decoder- has two components

a) Symbol decoder- This performs inverse operation of symbol

encoder.

b) Inverse mapping- This performs inverse operation of mapper.

Channel decoder-this is omitted if the system is error free.

14. Differentiate between lossy compression and lossless compression

methods.

Lossless compression can recover the exact original data after compression. It is used mainly for compressing database records, spreadsheets or word processing files, where exact replication of the original is essential.

Lossy compression will result in a certain loss of accuracy in exchange for a substantial increase in compression. Lossy compression is more effective when used to compress graphic images and digitized voice where losses outside visual or aural perception can be tolerated.

15. What is meant by wavelet coding?

16. Define channel encoder.

The channel encoder reduces the impact of the channel noise by inserting

redundant bits into the source encoded data.

Eg: Hamming code

17. What is jpeg?

The acronym is expanded as "Joint Photographic Expert Group". It is an international standard in 1992. It perfectly Works with colour and greyscale images, Many applications e.g., satellite, medical,...

18. Differentiate between jpeg and jpeg2000 standards.

jpeg

JPEG is good for photography Compression ratios of 20:1 are easily attained 24-bits per pixel can be used leading to better accuracy Progressive JPEG(interlacing)

jpeg2000

JPEG 2000 is an all encompassing standard Wavelet based image compression standard Lossless and lossy compression Progressive transmission by pixel accuracy and resolution Region-of-Interest Coding Random codestream access and processing Robustness to bit-errors Content-based description Side channel spatial information (transparency)

19. What are the operations performed by error free compression?

1) Devising an alternative representation of the image in which its interpixel redundant are reduced. 2) Coding the representation to eliminate coding redundancy

20. Define Huffman coding.

Huffman coding is a popular technique for removing coding redundancy.

When coding the symbols of an information source the Huffman code yields the smallest possible number of code words, code symbols per source symbol.

21. What is image compression?

Image compression refers to the process of redundancy amount of data

required to represent the given quantity of information for digital image. The

basis of reduction process is removal of redundant data.

(or)

A technique used to reduce the volume of information to be transmitted about an image

22. Define encoder.

Source encoder is responsible for removing the coding and interpixel

redundancy and psycho visual redundancy.

There are two components

A) Source Encoder

B) Channel Encoder

23. What is variable length coding?

Variable Length Coding is the simplest approach to error free

compression. It reduces only the coding redundancy. It assigns the shortest possible codeword to the most probable gray levels.

24. Define arithmetic coding.

In arithmetic coding, one to one corresponds between source symbols and code word doesn’t exist where as the single arithmetic code word assigned for a sequence of source symbols. A code word defines an interval of number between 0 and 1.

25. Draw the block diagram of transform coding system.

Twelve mark Questions 1. Explain various functional block of JPEG standard? Joint Photographic Expert’s Group. International standard for

photographs. It is Lossless/lossy. Based on the facts that :

Humans are more sensitive to lower spatial frequency components.

A large majority of useful image contents change relatively slowly across images.

Steps involved : Image converted to Y,Cb,Cr format Divided into 8x8 blocks Each 8x8 block subject to DCT followed by quantization Zig-zag scan DC coefficients stored using DPCM RLE used for AC coefficients Huffman encoding Frame generation

Functional block diagram of JPEG standard

Block preparation Compute luminance (Y) & chrominance (I & Q) according

to the formulas: Y = 0.3R + 0.59G + 0.11B (0 to 255) I = 0.6R - 0.28G - 0.32B (0 to 255) Q = 0.21R - 0.52G + 0.31B (0 to 255) Separate matrices are constructed for Y,I,Q. Square block of four pixels are averaged in the I & Q (lossy

and compress image by factor of 2). 128 is subtracted form Y,I and Q. Each matrix is divided up into 8X8 blocks

Discrete cosine transformation Output of each DCT is an 8X8 matrix. DCT element (0,0) is the average value of the block. Other elements are difference between original and average

value. Theoretically lossless but sometimes it may be lossy.

Quantization Less important DCT coefficients are wiped out. It is the main lossy step involved in JPEG. It is done by dividing each of the coefficients in the 8X8 matrix

by a weight taken from a table. These weights are not a part of JPEG std.

Differential quantization It reduces the(0,0) value of each block by replacing it with the

amount it differs from the corresponding element in the previous block.

Since these elements are the average value of their respective blocks ,they should change slowly.

Run length encoding It linearizes the 64 elements and applies run length encoding

to the list.

Statistical output encoding JPEG uses Huffman encoding for this purpose. It often produces a 20:1 compression or better. For decoding we have to run the algorithm backward. JPEG is roughly symmetric: Decoding takes as long as

encoding. Advantages and Disadvantages:-

Advantages Disadvantages

Compression ratios of 20:1 are easily attained.

Doesn’t support transparency.

24-bits per pixel can be used leading to better accuracy.

Doesn’t work well with sharp edges.

Progressive JPEG(interlacing) Almost always lossy and No target bit rate

Another Block Diagram

JPEG 2000 STANDARD:- Wavelet based image compression standard Encoding Decompose source image into components Decompose image and its components into rectangular tiles Apply wavelet transform on each tile Quantize and collect subbands of coefficients into rectangular arrays of

“code-blocks” Encode so that certain ROI’s can be coded in a higher quality Add markers in the bitstream to allow error resilience

Advantages: Lossless and lossy compression. Progressive transmission by pixel accuracy and resolution. Region-of-Interest Coding. Random codestream access and processing. Robustness to bit-errors. Content-based description. Side channel spatial information (transparency).

2. Explain (i) one-dimensional run length coding (ii) two-dimensional run

length coding.

RLE stands for Run Length Encoding. It is a lossless algorithm that only offers decent compression ratios in specific types of data.

• Pre-processing method, good when one symbol occurs with high probability or when symbols are dependent

• Count how many repeated symbol occur • Source ’symbol’ = length of run

(i) one-dimensional run length coding

• Used for binary images

• Length of the sequences of “ones” & “zeroes” are detected.

• Assume that each row begins with a white(1) run.

• Additional compression is achieved by variable length-coding

(Huffman coding) the run-lengths.

An m-bit gray scale image can be converted into m binary images

by bit-plane slicing. These individual images are then encoded

using run-length coding.

However, a small difference in the gray level of adjacent pixels can

cause a disruption of the run of zeroes or ones.

Example: Let us say one pixel has a gray level of 127 and the next

pixel has a gray level of 128.

In binary: 127 = 01111111 & 128 = 10000000

Therefore a small change in gray level has decreased the run-

lengths in all the bit-planes.

(ii) two-dimensional run length coding.

Developed in 1950s and has become, along with its 2-D extensions, the standard approach in facsimile (FAX) coding. Two dimensional array of pixel values Spatial redundancy and temporal redundancy Human eye is less sensitive to chrominance signal than to

luminance signal (U and V can be coarsely coded) Human eye is less sensitive to the higher spatial frequency

components Human eye is less sensitive to quantizing distortion at high

luminance levels Source image as 2-D matrix of pixel values R, G, B format requires three matrices, one each for R, G, B

quantized values In Y, U, V representation, the U and V matrices can be half as

small as the Y matrix Source image matrix is divided into blocks of 8X8 submatrices Smaller block size helps DCT computation and individual blocks

are sequentially fed to the DCT which transforms each block separately

Advantages and disadvantages

This algorithm is very easy to implement and does not require much CPU horsepower. RLE compression is only efficient with files that contain lots of repetitive data. These can be text files if they contain lots of spaces for indenting but line-art images that contain large white or black areas are far more suitable. Computer generated colour images (e.g. architectural drawings) can also give fair compression ratios.

3. Explain variable length coding and Huffman coding.

Variable length coding: Assigning fewer bits to the more probable gray levels than to the less probable ones achieves data compression. This is called variable length coding.

Variable length code whose length is inversely proportional to that

character’s frequency.

Must satisfy non-prefix property to be uniquely decodable.

two pass algorithm

First pass accumulates the character frequency and generate

codebook.

Second pass does compression with the codebook.

Huffman codes require an enormous number of computations. For N source symbols, N-2 source reductions (sorting operations) and N-2 code assignments must be made. Sometimes we sacrifice coding efficiency for reducing the number of computations.

Create codes by constructing a binary tree

1. Consider all characters as free nodes

2. Assign two free nodes with lowest frequency to a parent node

with weights equal to sum of their frequencies

3. Remove the two free nodes and add the newly created parent

node to the list of free nodes

4. Repeat step2 and 3 until there is one free node left. It becomes

the root of tree

Table: Variable-Length Codes

Huffman Coding

This coding reduces average number of bits/pixel.

It assigns variable length bits to different symbols.

Achieves compression in 2 steps.

Source reduction

Code assignment

Steps

1. Find the gray level probabilities from the image histogram. 2. Arrange probabilities in reverse order, highest at top. 3. Combine the smallest two by addition, always keep sum in reverse

order. 4. Repeat step 3 until only two probabilities are left. 5. By working backward along the tree, generate code by alternating

assignment of 0 & 1.

Fig: Huffman Source Reductions

Fig : Huffman code assignment procedure

Extra Notes:

4. Explain arithmetic coding and LZW coding.

Arithmetic coding Arithmetic compression is an alternative to Huffman compression, it

enables characters to be represented as fractional bit lengths. Unlike for Huffman compression, where fractional code lengths are not possible and the allocation of shorter codewords for more frequently occurring characters needs at least one-bit codeword no matter how high its frequency.

Arithmetic coding works by representing a number by an interval of real numbers greater or equal to zero, but less than one. As a message becomes longer, the interval needed to represent it becomes smaller

and smaller, and the number of bits needed to specify it increases. Entire sequence of source symbol (message) is assigned a single

arithmetic code word. There is no one to one coding like Huffman The code word is within interval [0, 1] As the number of symbols in the message increases, the interval used to

represent it becomes smaller and the number of information units (bits) required to represent the interval becomes larger

Ex. More bits are required to represent 0.003 than 0.1

Steps: Arithmetic Coding The basic algorithm for encoding a file using arithmetic coding works

conceptually as follows: (1) Begin with current range [L,H) initialized to [0,1). Note : We denote brackets [0,1) in such a way to show that it is equal to

or greater than 0 but less than 1. (2) For each symbol of the file, we perform two steps : a) Subdivide the current interval into subintervals, one for each

alphabet symbol. b) Select the subinterval corresponding to the symbol that actually occurs next in the file and make it the new current interval.

(3) Output enough bits to distinguish the current interval from all other possible interval.

Example: Encode the message: a1 a2 a3 a4

Table : Arithmetic Coding example

Fig : Arithmetic coding procedure

So, any number in the interval [0.06752, 0.0688) , for example 0.068 can be used to represent the message. Here 3 decimal digits are used to represent the 5 symbol source message. This translates into 3/5 or 0.6 decimal digits per source symbol and compares favorably with the entropy of -(3x0.2log100.2+0.4log100.4) = 0.5786 digits per symbol

As the length of the sequence increases, the resulting arithmetic code approaches the bound set by entropy. In practice, the length fails to reach the lower bound, because: • The addition of the end of message indicator that is needed to separate one

message from another • The use of finite precision arithmetic

LZW (Lempel-Ziv-Welch) coding

LZW (Lempel-Ziv-Welch) coding, assigns fixed-length code words

to variable length sequences of source symbols, but requires no a

priori knowledge of the probability of the source symbols. LZW was

formulated in 1984

The nth extension of a source can be coded with fewer average bits

per symbol than the original source.

LZW is used in:

• Tagged Image file format (TIFF)

• Graphic interchange format (GIF)

Portable document format (PDF)

The Algorithm:

• A codebook or “dictionary” containing the source symbols is

constructed.

• For 8-bit monochrome images, the first 256 words of the dictionary are

assigned to the gray levels 0-255

• Remaining part of the dictionary is filled with sequences of the gray

levels

Example: 39 39 126 126 39 39 126 126 39 39 126 126 39 39 126 126

Table : LZW Coding example Compression ratio = (8 x 16) / (10 x 9 ) = 64 / 45 = 1.4 Important features of LZW: • The dictionary is created while the data are being encoded. So encoding

can be done “on the fly”

• The dictionary need not be transmitted. Dictionary can be built up at

receiving end “on the fly”

• If the dictionary “overflows” then we have to reinitialize the dictionary

and add a bit to each one of the code words.

• Choosing a large dictionary size avoids overflow, but spoils

compressions

Decoding LZW:

Let the bit stream received be:

39 39 126 126 256 258 260 259 257 126

In LZW, the dictionary which was used for encoding need not be sent with

the image. A separate dictionary is built by the decoder, on the “fly”, as it

reads the received code words.

Recognized Encoded value

pixels Dic. address

Dic. Entry

39 39

39 39 39 256 39-39

39 126 126 257 39-126

126 126 126 258 126-126

126 256 39-39 259 126-39

256 258 126-126 260 39-39-126

258 260 39-39-126

261 126-126-39

260 259 126-39 262 39-39-126-126

259 257 39-126 263 126-39-39

257 126 126 264 39-126-126

5. Explain wavelet based image compression. In contrast to image compression using discrete cosine transform (DCT) which is proved to be poor in frequency localization due to the inadequate basis window, discrete wavelet transform (DWT) has a better way to resolve the problem by trading off spatial or time resolution for frequency resolution. Exploiting the structures between coefficients for removing redundancy Wavelet Coding

Fig : Wavelet coding system ( encoder)

Fig : Wavelet coding system ( decoder)

Advantages: Lossless and lossy compression. Progressive transmission by pixel accuracy and resolution. Region-of-Interest Coding. Random code stream access and processing. Robustness to bit-errors. Content-based description. Side channel spatial information (transparency).

6. Explain arithmetic coding and Huffman coding.

Arithmetic coding Arithmetic compression is an alternative to Huffman compression, it

enables characters to be represented as fractional bit lengths. Unlike for Huffman compression, where fractional code lengths are not possible and the allocation of shorter code words for more frequently occurring characters needs at least one-bit codeword no matter how high its frequency.

Arithmetic coding works by representing a number by an interval of real numbers greater or equal to zero, but less than one. As a message becomes longer, the interval needed to represent it becomes smaller

and smaller, and the number of bits needed to specify it increases. Entire sequence of source symbol (message) is assigned a single

arithmetic code word. There is no one to one coding like Huffman The code word is within interval [0, 1] As the number of symbols in the message increases, the interval used to

represent it becomes smaller and the number of information units (bits) required to represent the interval becomes larger

Ex. More bits are required to represent 0.003 than 0.1

Steps: Arithmetic Coding The basic algorithm for encoding a file using arithmetic coding works

conceptually as follows: (1) Begin with current range [L,H) initialized to [0,1). Note : We denote brackets [0,1) in such a way to show that it is equal to

or greater than 0 but less than 1. (2) For each symbol of the file, we perform two steps : a) Subdivide the current interval into subintervals, one for each

alphabet symbol. b) Select the subinterval corresponding to the symbol that actually occurs next in the file and make it the new current interval.

(3) Output enough bits to distinguish the current interval from all other possible interval.

Example: Encode the message: a1 a2 a3 a4

Fig : Arithmetic Coding example

Fig : Arithmetic coding procedure

Huffman Coding

This coding reduces average number of bits/pixel.

It assigns variable length bits to different symbols.

Achieves compression in 2 steps.

Source reduction

Code assignment

Steps

6. Find the gray level probabilities from the image histogram.

7. Arrange probabilities in reverse order, highest at top.

8. Combine the smallest two by addition, always keep sum in reverse

order.

9. Repeat step 3 until only two probabilities are left.

10. By working backward along the tree, generate code by alternating

assignment of 0 & 1.

Fig : Huffman Source Reductions

Fig : Huffman code assignment procedure

7. Explain how compression is achieved in transform coding and explain

the DCT.

Transform Coding Three steps:

Divide a data sequence into blocks of size N and transform each block using a reversible mapping

Quantize the transformed sequence Encode the quantized values

Benefits - transform co efficiently, relatively uncorrelated - energy is highly compacted - reasonable robust relative to channel errors.

• DCT is similar to DFT, but can provide a better approximation with fewer coefficients

• The coefficients of DCT are real valued instead of complex valued in DFT. The discrete cosine transform (DCT) is the basis for many image compression algorithms. One clear advantage of the DCT over the DFT is that there is no need to manipulate complex numbers. The equation for a forward DCT is

and for the reverse DCT

Where,

DCT in Terms of Basis Functions The basis functions or basis images for DCT is given by:

Where,

1 0

2 1,2,..., 1

u for uN

u for u NN

2 1 2 1

, , , cos cos2 2

x u y vg x y u v u v

N N

N is the block size of the image (normally N=8)

Matrix of Discrete Cosine Transform (DCT)

Zig-zag Scan DCT Blocks

• To group low frequency coefficients in top of vector. • Maps 8 x 8 to a 1 x 64 vector.

8. Explain any two basic data redundancies in digital image compression.

Data Redundancy

Various amount of data may be used to represent the same information. Data which either do not provide necessary information or provide the

same information again are called redundant data. Removing redundant data from the image reduces the size.

Redundancies In Image

In image compression 3 basic data redundancies can be identified. 1. Coding redundancy (CR) 2. Interpixel redundancy (IR) 3. Psychovisual redundancy (PVR)

Data compression is achieved when one or more of these redundancies are reduced or eliminated

Coding redundancy

A natural m-bit coding method assigns m-bit to each gray level

without considering the probability that gray level occurs with: Very

likely to contain coding redundancy

Basic concept:

Utilize the probability of occurrence of each gray level (histogram) to

determine length of code representing that particular gray level:

variable-length coding.

Assign shorter code words to the gray levels that occur most frequently

or vice versa.

Fig : Graphical representation of fundamental basis of data compression

Interpixel Redundancy

Caused by High Interpixel Correlations within an image, i.e., gray level

of any given pixel can be reasonably predicted from the value of its

neighbors (information carried by individual pixels is relatively small)

spatial redundancy, geometric redundancy, interframe redundancy (in

general, interpixel redundancy )

Interpixel redundancy occurs because adjacent pixels tend to be highly

correlated.

Adjacent pixel values tend to be close to each other.

The value of a given pixel can be predicated from the value of

its neighbors.

Visual contribution of a single pixel to an image is redundant.

To reduce inter pixel redundancy image is transformed in to

more efficient format.

For Ex. Difference between adjacent pixels can be used to store

an image.

This transformation process is called mapping

Reverse of that is called inverse mapping

We can detect the presence of correlation between pixels (or

interpixel redundancy) by computing the auto-correlation coefficients

along a row of pixels

( )( )

(0)A n

nA

11( ) ( , ) ( , )0

where

N nA n f x y f x y n

N n y

Maximum possible value of γ(∆n) is 1 and this value is approached for this

image, both for adjacent pixels and also for pixels which are separated by 45

pixels (or multiples of 45).

Psychovisual Redundancy

Psychovisual redundancy refers to the fact that some information is

more important to the human visual system than other types of

information.

Use of less no. of gray levels reduces the size of image.

Elimination of psychovisually redundant data from an image results in

a loss of quantitative information.

This process is not reversible

The key in image compression algorithm development is to determine

the minimal data required to retain the necessary information.

This is achieve by taking advantage of the redundancy that exists in the

image.

Any redundant information that is not required can be

eliminated to reduce the amount of data used to represent the

image

The eye does not respond with equal sensitivity to all visual information.

Certain information has less relative importance than other information in

normal visual processing psychovisually redundant (which can be eliminated

without significantly impairing the quality of image perception).

The elimination of psychovisually redundant data results in a loss of

quantitative information lossy data compression method.

Image compression methods based on the elimination of psychovisually

redundant data (usually called quantization) are usually applied to commercial

broadcast TV and similar applications for human visualization.

9. Explain Huffman coding Algorithm giving a numerical example?

Huffman Coding

This coding reduces average number of bits/pixel.

It assigns variable length bits to different symbols.

Achieves compression in 2 steps.

Source reduction

Code assignment

Steps

1. Find the gray level probabilities from the image histogram.

2. Arrange probabilities in reverse order, highest at top.

3. Combine the smallest two by addition, always keep sum in reverse

order.

4. Repeat step 3 until only two probabilities are left.

5. By working backward along the tree, generate code by alternating

assignment of 0 & 1.

Fig : Huffman Source Reductions

Fig : Huffman code assignment procedure

Calculating Lavg & Entropy

Lavg= 2.2 bits/pixel Entropy = 2.14 bits/pixel Efficiency of Huffman code = 2.14/2.2 = 0.973 Constraint : symbol be coded one at a time Uniquely code able & decodable

Encoding

Decoding

10. Explain the Constrained Least Square filtering?

Constrained Least Squares Filtering

Only the mean and variance of the noise is required The degradation model in vector-matrix form

1 1 1MN MNMN MN MN

g H f η

The objective function

21 1 2min [ ( , )]0 0

2 2

M NC f x y

x y

subject to

g Hf η

The solution

*( , )ˆ ( , ) ( , )2

( , ) ( , )

H u vF u v G u v

H u v P u v

0 1 0

( , ) 1 4 1

0 1 0

p x y

In that case we seek for a solution that minimizes the function

( )M 2

f y Hf

A necessary condition for )(fM to have a minimum is that its gradient

with respect to f is equal to zero. This gradient is given below

( )( ) 2(

MM

f T Tf H y H Hf)

ff

And by using the steepest descent type of optimization we can formulate an iterative rule as follows:

Tf H y0

( )( ) ( )

M

fT T Tkf f f H y Hf H y I H H f

k k k kfk

k 1

Constrained least squares iteration In this method we attempt to solve the problem of constrained restoration iteratively. As already mentioned the following functional is minimized

2 2( , )M f y Hf Cf

The necessary condition for a minimum is that the gradient of ),( fM is

equal to zero. That gradient is

( ) ( , ) 2[( ) ]M T T Tf f H H C C f H yf

The initial estimate and the updating rule for obtaining the restored image are now given by

Tf H y0

[ ( ) ]

T T Tf f H y H H C C fk kk 1

It can be proved that the above iteration (known as Iterative CLS or Tikhonov-Miller Method) converges if

20max

where max is the maximum eigenvalue of the matrix,

( )T TH H C C

If the matrices H and C are block-circulant the iteration can be implemented in the frequency domain.