Medical Bruce

K. 1. Ho,

PhD

#{149} Johnny

Chao,

BS

a

Ping

Zhu,

MS

Design and Implementation of Full-Frame, Bit-Allocation Image-Compression Hardware Work

terms:

Picture

Images.

archiving

(PACS) diographv.

I

storage

and

retrieval

communication

a

system

Ra-

computer-assisted

#{149}

digital

Radiology

1991;

From

ences,

and

Radiography.

.

the

179:563-567

Department

Division

of

Medical

Center

for

Health

fornia, tember

Los 25,

Angeles. CA 1990; revision

Sciences,

of Radiological Imaging, University

SciAR-277, of Cali-

90024. Received Seprequested November

13; revision received December 17; accepted December 26. Supported by National Cancer Institute grant ROl CA 40456. Address reprint

requests (

RSNA,

H.

K.

Huang,

DSc

Module

in Progress1

A hardware module was designed and built to implement the fullframe, bit-allocation image-cornpression algorithm in a clinical setting. The algorithm transforms an entire image without prepartitioning into small subimages. This adaptation eliminates block artifacts at subirnage borders that can mimic relevant pathologic conditions. The quality of 1,024- and 2,048-pixel irnages compressed at a rate up to 10:1 with a custom-designed processor board (which contains four digital signal processors that transform and quantize separate rows and columns of an image independently with a two-pass cosine transform) and a 16Mbyte frame buffer was found to be diagnostically acceptable in prelimmary receiver operating characteristic studies. The module can cornpress a 1,024-pixel image in 4 seconds in a general-purpose computer system; images can be compressed in 1 second with the addition of a custom-designed data transporter. Copies of the compression module are being installed in the authors’ department and in collaborating hospitals for laboratory and clinical evaluation. Index

a

Physics

to B.K.I.H. 1991

C

achievements in picture archiving and communication systems (PACSs) have cast a bright outlook for realizing digital radiobogy in the very near future (1). The effect of image digitization on diagnostic accuracy and radiologic practice is still a major source of anxiety to the radiologists involved in the forefront of this transition. A streamlined PACS must provide rapid and concurrent access to many years worth of a patient’s image history at multiple viewing sites. At 8 Mbyte per computed radiographic (2) image and, typically, 15-30 Mbyte per wholebody computed tomographic (CT) study, the data volume generated at our hospital has been estimated at 2 Tbyte a year. The various imaging modalities and viewing rooms for different specialities are spread out over two buildings in our hospital: the Center for Health Sciences and the Medical Plaza. These buildings are approximately 2,000 feet apart, and the digital communication link between them consists of fiberoptic cables about 1.5 km long (1). Given these extraordinary storage and communication requirements, the implementation of a totally digital radiobogy department is clearly beyond the capabilities of today’s computer systems. Various image-compression techniques exist for reducing the volume of data to a more manageable level (3,4). It has been demonstrated that a computed radiographic image, which normally requires an average of 40 seconds to transmit over the Ethernet, can be sent in merely 2.5 seconds after a iO:i compression. The gain in transfer speed is much greater than a factor of 10 because the frequency of network traffic collisions is also drastically reduced. Although recoverabbe compression methods exist, they only provide a reduction of 2-3 times for medical images (5). Compression techniques with use of the cosine URRENT

transform are well industry pression

followed by established in and typically ratios of 10:1

quantization the aerospace achieve comwith no notice-

able loss in quality (3). On the basis of the same principles, the full-frame bit-allocation technique developed at our laboratory makes a major adaptation by transforming an entire image without prepartitioning into small (typically 8 X 8-bit) subimages. This important adaptation eliminates the so-called block artifacts (6) at subimage borders that can mimic rebevant pathologic conditions. A high degree of image fidelity was immediately obvious from visual inspections during the early phase of algorithmic development (7-9). A pilot receiver operating characteristic (ROC) study with use of hard-copy chest radiographs that depicted pulmonary nodules and septal lines, as well as normal structures, showed that equivalent diagnostic accuracy was found when original images were compared with images reconstructed from i5:i compression (10). Similar pilot ROC studies on CT and magnetic resonance (MR) images have also been performed (1 i). Recently, a more extensive ROC study confirmed that no significant difference was found in observer diagnostic performance for detecting small pulmonary abnormalities (12). A comprehensive

ROC

study

of chest

and

graphs as well as MR and in various disease categories rently under way. On the this compression algorithm,

we developed past

4 years,

and we

have

bone

radio-

CT

images is curbasis of which

refined designed

during

the and

Abbreviations:

CPU central processing discrete cosine transform, DFT discrete Fourier transform, DSP digital signal processor, FF1 fast Fourier transform, PACS = picture archiving and communication system, QDSP = quad digital signal processor, ROC = receiver operating characteristic. VME = Versa Module European. unit,

DCT

=

563

constructed implement cabby.

a hardware module image compression

to clini-

Sun 3/5

I

Design

AND

METHODS

I

]

cu

I I

DISPLAY

I I

CNTLR]

[MTLRJ

:_

Criterion

Because

cosine

transforms

of large

16MB

ma-

QUAD

FRAME BUFFER

DSP MODULE

Figure 1. Block diagram of the compression module integrated into a host computer (3/E; Sun Microsystems). Enclosed in dashed lines are the three system components of the host computer: the central processing unit (CPU), the display controller, and the Ethernet/SCSI. Together, these

rnentation,

make

a large

number

of images

are

archived on optical disks over a network. Currently, may take several minutes

and transferred both tasks to complete.

Compression

the

will

reduce

Future

dude (15).

PACS

the Very

archiving

implementations

use of soft-copy fast decompression in

radiologists’ the speed

acceptance of soft copies is of image retrieval. A 10:1 corndigitized

case,

in-

display stations is especially

important

pressed

this

may

since

a criterion

radiograph

can

be

in

read

from a conventional magnetic disk in less than 1 second. A decompression time of 4 seconds will then yield a total retrieval time of under 5 seconds. We feel that this would be an acceptable waiting time for one image. Our goal in the hardware module design is to implement high-speed image compression for the various types of dinical modalities including computed radiography, angiography, CT, and MR imaging, with matrix sizes of 2,048 pixels, 1,024 pixels, 512 pixels, and 256 pixels, respectively. Pixel depths of 12 bits (CT, MR), 10 bits (computed radiography), and 8 bits (ultrasound) must all be processed by one module. Compression speeds of 4 seconds

for

for

criteria

2,048-pixel

1,024-pixel for

the

images

images highest

and

1 sec-

are set as the

speed

applica-

tions. Finally, the module must be massproducible, cost-effective, and portable for moving to each archiving and display station anywhere in the hospital.

up

a basic,

Module

Design

Our engineering approach was to custorn build a special processor board and integrate it into a host computer (Sun Microsysterns, Sunnyvale, Calif) via the Versa Module European (VME) backplane, along with other system components necessary to perform the complex cosine transform and subsequent quanti-

a

Radiology

general-purpose

buffer

dedicated

CNTLR

controller, Mbyte, MPU

worksta-

MB

to compression

FPU math

=

=

zation

functions.

processor processor

unit, unit.

The

operating system, deperipheral control, backplane standard, and overall developmental environment supplied by the computer manufacturer allowed us to concentrate on the compression specialty. steps.

bugging

software,

Our development pose workstation tems), consists

system, a general-pur(3/E, Sun Microsysof three essential compo-

shown in Figure 1: a central processing unit (CPU), a video display controller for the console, and a controller board for connecting to Ethernet and small computer system interface (SCSI) nents,

devices

(eg,

Also cessor

shown board

that

make

compression gray-scale

a hard

disk

to show both text and Our custom-designed consists

or a tape

drive).

are a custom-designed and a 16-Mbyte frame up the main components hardware. monitor for

of four

digital

We also the system

signal

tween

cosine

transform (Appendix).

and Fourier The four DSPs

and quantize separate rows and of an image independently. Because an image is two-dimensional, its cosine transform requires two passes (horizontal and vertical), shown in Figure 2. Mathematically, the two passes are summations of the pixel values multiplied by the transformation kernel taken in x and

y

directions.

the

cosine

After

coefficients

the

cosine

transform,

are quantized

2.

Parallel

processing

hardware

makes high-speed compression possible. During the first pass, four separate rows of image data are distributed among four independent processors to be cosine transformed. The processor generates more bits per pixel during the computation step, resuiting temporarily in an increase in data volume. During the second pass, the image data (now partially transformed) are redistributed by columns for concurrent processing. After another stage of cosine transform, a quantization step follows in which zero and insignificant bits are stripped from the cosine coefficients, resulting in a drastic reduction

of

data

volume.

(having the lower bits rounded off) and result in a drastically reduced data volume. Each of the four DSPs is coupled with enough cache memory to process a 2,048pixel line pair of image data. The cache memory is designed out of static random access memory, which has an access speed 10 times that of dynamic random access Figure 3 is a photograph of the QDSP board. The four prominent squares are the DSPs. The three components between each processor and the board edge are the

processors

columns

Figure

memory

board

transform

TWO

Transform & Quant izat Ion

buffer of our

chose a console

images. processor

PASS Cosine

pro-

(DSPs) (DSP56001; Motorola, Phoenix) and is thus named the quad digital signal processor (QDSP) board. Its high speed in the cosine transform computation is due to parallel processing architecture and the optimization of the DSPs (16,17) for linear fast Fourier transform (FF1). Our algorithm took advantage of this optimization by exploiting the close analogy betransform

Hardware

Module

Cosipression

lion. New components can be attached to this basic system by way of the VME bus. Shown attached here are the quad digital signal processor (DSP) module and a frame

and transfer time per image to just a few seconds. The compression or decompression process itself should therefore be equably fast or faster. Efficiency in image decompression for the purpose of printing hard copies has a direct impact on our ability to conduct large-scale ROC studies of compression effects on image quality.

564

Processors

I I

scsi

tnices require enormous computing power, running our compression algorithm on a general-purpose computer such as a VAX 1 1/750 (Digital Equipment, Manard, Mass) requires many hours to complete. A dedicated hardware module that can cornpress and reconstruct an image within seconds is required to implement this technique in a clinical setting (13,14). In the preliminary phase of PACS irnple-

ond

Parallel

I

FiM(RNCT

[NSOLE

I

888IFPU 4(18

I MATERIALS

1

ionu I

Four

System

associated lustrated

but

also

costs

cache memory are two VME

many

times

chips.

more.

Also

il-

connectors for attachment to the backplane and four lightemitting diodes, which are programmed to flash during the computation cycles of the processors. Between the two passes of the transform, the direction of data flow changes from horizontal to vertical (or from the x to the y direction) within the data matrix. Regardless of the original pixel depth, the halfway transformed data at this stage carry 24 bits of accuracy (the inherent data width of the DSP). This increased pixel depth causes a temporary growth in data volume and must be stored in a very large intermediate buffer (16 Mbyte in the case of a 2,048 X 2,048-pixel image). In our development system, we implemented this scratch memory with an external frame buffer (model MM-6326; Micro Memory, Chatsworth, Calif), shown in Figure 1. However, systems with at least

32 Mbyte

of system

memory,

such

May

as

1991

tutorial

guide

for

first-time

users.

The

software-user interface and documentation are designed for wide distribution among both expert and novice computer users who will participate in the venification of image quality and clinical feasibility of the compression module.

RESULTS

Figure 3. A photograph of the parallel processor board with four processor chips and their associated cache memories. The two long connectors on the bottom attach this board to a VME bus backplane. Communication with the system host and other boards such as the frame buffer is made possible through this backplane.

data from the boards (running taneously.

Figure

uration

feature High-speed

compression

(VSB),

at 1 sec-

ond for a 1,024-pixel image is achieved by this four-board configuration with one frame buffer, two processor boards, and a data transporter. The transporter is responsible for fast data transfers between the processons and the frame buffer, which serves as a large intermediate storage between the first

and

the

second

Sun

passes

4 series,

frame

of

do

need

An

electronic

a single chose

linking

system the

external

tance

as the

dard. options

Its

or “bus,” multiple

bus

preeminent

of

fashion.

because

We

stan-

allows us many a suitable host

jon nodes the display manufacturers

In the speed

to any Sun which make

Sun

3/E

of 4 seconds

was

achieved.

sion

speed,

system, for

ed a data

transporter

Volume

179

a

Number

ma-

as well imaging

a 1,024-pixel

designed

boost and

that

can

2

VME

backplane.

as

image

compresimplement-

drive

the

the

a

data

without

and

other

ly.

processing Two

for

a given

beis

software

ular

to work

running

the

software

Unix

package

“decompress,” programs intended

individual

images

entered

compression image.

our

modified

The

parameters

the

demon-

for switching Sun systems

computer system.

to

as to Cur-

systems and on 1131, Motorola) The necessary

that

be easily

capa-

is portable

been

It is expected

consists of “compress,” and “imagesubtract” for

be

as well system. has

to

we are cona constraint of image

module

software modification tween the Motorola

any

of the

4 computers computer

on our many Sun system (System in our laboratory.

bus

through

Because

compatibility

use

transporter going

nently,

operating

is the

1

subsystem

of the data transporter, that speed will not clinical implementation

can

speed of A special

VME

strated another available

control

a compression

To further we

the

on

more CPU,

Microsystems up all the

of our PACS system stations of many (15).

buffer

package

frame buffer, or other board-level system components. Most important, the adoption of the VME bus makes our QDSP board portable workstations,

frame

minimal.

of its accep-

industry

popularity in selecting

is

boards

in a coherent

VME

links

all Sun 3 and any VME-based

this

backplane, for

which

config-

combination,

transporter the

compression. The compression

buffer.

responsible

data bus,

the

bility fident in the

computation.

not

of this express

QDSP simul-

the

four-board

reaches a compression for 1,024-pixel images.

of an

4.

4 illustrates

of this

which second

Figure

frame buffer to two eight processors)

The

manual-

by

ratio

the

user

obtained

imagesubtract

program provides the option of contrast enhancement to emphasize any fine graylevel

differences

and

decompressed

between images.

the

original

A “show”

pro-

gram with a graphic menu is also included for examining images on the Sun console. A manual has been written with detailed

installation

procedures

and

a

We have fabricated the QDSP module in a printed circuit board form suitable for mass production. Our laboratory and staff are set up to do small-quantity manufacturing of this module for limited distribution to collaborating institutions. Although the exact cost of production, including percentage of personnel time, recovery of research and development investment, and overheads from utility and administrative support, is very difficult to estimate in our academic environment, the cost of components was calculated to be approximately $1,500 per board. The completed modules have been tested in four types of Sun workstations: 3/E, 4/330, 4/370, and 4/490. We now have the capability to install the compression modules in our hospital and in other institutions as needed for quality evaluation. The Table shows how fast the 3/E and 4/370 systems compressed images of two sizes: 1,024 X 1,024 X 8 pixels and 2,048 X 2,048 X 8 pixels. In the case of the 3/E, boost performance provided by the data transporter is also given. No noticeable difference existed between the performances of the two system configurations with these image sizes. Howeven, systems without an external frame buffer will undenperform when processing 2,048 X 2,048 X 12pixel images because the system must use virtual memory, thus slowing down overall performance. This may be improved with future operating systems that allow the user to dedicate memory required for one panticprocess. A bone image puted radiography Philips Medical

generated system Systems

by a corn(PCR 7000; North

America, Shelton, Conn) was chosen to demonstrate image fidelity with use of the hardware module. The original, reconstructed, and subtracted images, each in a 2,048 X 2,048 X 10-pixel format, are shown in Figure 5. This image format was converted from the original system format of 2,048 X 2,510 X 10 pixels to exactly 2,048 X 2,048 pixels (by cropping in the 2,510-pixel direction and pixel filling in the other) to meet the de-

Radiology

a

565

b.

a. Figure

5.

(obtained discernible

Three

2,048

X 2,048

with the hardware in the subtraction

X 10-pixel module). image,

bone

C.

images

generated

(c) Contrast-enhanced indicating high fidelity

with

computed

subtraction of diagnostic

radiography.

image. The information

compression in the

(a) Original ratio reconstructed

image.

was

(b) Reconstructed

11:1. No image.

anatomic

image

features

are

1(n)

I

a. x(n)

Tod

b. partmental PACS standard. A graylevel value close to those found near the edges of the original image was used in filling the two sides. The compression ratio used here was 11:1. Some care should be exercised in interpreting this ratio. Because of the byte-aligned data structure used in conventional computers, a 10-bit pixel was usually zero filled to occupy 16 bits of data space. In calculating the compression ratio, we used an original file size of 5 Mbyte, which was determined by allowing bit packing across byte boundaries.

borders in the subtraction image (Fig 5c). Since bone images exhibit the sharpest anatomic edges found on radiographs, this image category is expected to show the most noticeable degradation in edge definition. The test image set in Figure 5 shows that our compression algorithm is refined to the point that even sharp bone edges are preserved without discernible visual degradation at the chosen ratio.

In practice, the original file takes up 8 Mbyte of disk space, and it can be argued that the compression ratio achieved was actually 18:1. The subtracted image was contrast enhanced by a factor of 10 to emphasize fine gray-level differences otherwise not easily visible. An inherent characteristic in our compression technique was a preferential loss of high frequency content in an image. This property is clearly illustrated by the visible artifacts around the sharp artificial edges of the labels and image

The issue of image fidelity is crucial to the clinical acceptance of our compression technique. A 3-year, large-scale ROC study has been planned to scientifically determine the effect, if any, of compression noise on diagnostic accuracy. Hundreds of images of various modality sources, formats, disease categories, and compression ratios are planned for the large-scale ROC study. It would be practically impossible to prepare all the compressed images needed with a general-purpose corn-

566

a

Radiology

DISCUSSION

y(n)

:

T C.

Figure

6.

(a) A finite

discrete

sequence

of

signals 1(n), sometimes referred to as a finite impulse train. (b) An infinite periodic sequence x(n) is generated from (n) by nepetition. The original finite sequence represents one period in the infinite sequence. (c) Another infinite sequence y(n) is generated by repeating (n) and its mirror image.

puten. Our hardware implementation of the compression module makes it possible for the first time to execute such a massive quality study. Because clinical conditions, practide habits, sectioning of specialties, methods of PACS implementation, and image quality standards may vary in different hospitals and differ-

ent lishing

countries, clinical

our

strategy

viability

for

estab-

is to incorpo-

May

1991

rate the participation collaborating hospitals tuners in the United

of a number of and manufacStates, Europe,

and Japan. The first exported compression module that uses a Sun 3/E system was installed at Hitachi Maxebb (Tsukuba, Tokyo) in June 1990. A second module that uses the 4/490 system was installed at the Hospital Cantonal Universitaire de Geneve (Switzerland) in August 1990. Internally, a third module was installed in the genitourinary section of our department as part of a 2,048-line display station. Two major imaging manufacturers have requested that more modules be installed in 1991. Independent image quality studies will be conducted at these various research and development facilities. Additional feedback is expected in the areas of user interface, speed, compatibility with different PACS configurations, and general user concerns with the concept of image compression.

APPENDIX The

primary

hardware is in

challenge

module

the

in

for

efficient

designing

image

the

for

a direct

of discrete

calculation

method

to compute

the

Fourier

microelectronics

cluding

the

research, Fourier

is available. coefficients

signal

is straightforward,

the

and

used

between

one

in

our

finite

them.

into

an

where k is greater less than or equal This

179

a

Number

2

clearly

defined

than

or

to 2N

takes

equal

-

the

in reference

1)k,

I

to 0 but

1.

form

of a DCT

as

20: 11.

C(k)

2c(k)

=

x(n) cos or(2n+

i)k, 12.

(2) where

c(k) equals

for

k

0 and

1 for

k=1,2,...N-1. Then,

another

step

of simplification

can be obtained by generating another sequence x’(n) out of the original y(n) by taking even-numbered data followed by odd-numbered data in reverse order as explained in references 21 and 22. We would then reach the final result: C(k)

infinite

2c(k)Re[e12’t4N

=

13.

14.

f(n)e_12afl].

15. The

summation

term

is just the OFT of x’(n) and can be

the permuted sequence calculated very efficiently with use of the FFT technique. Equation (3) represents the algorithm that we actually implemented with the signal processors (16,17). The complete derivations of the previous equations are given in references 22 and 23.

16.

17.

U 18.

References 1.

2.

3.

4.

se-

quence x(n) and performing a continuous Fourier transform on it (19) (Fig 6a, 6b). Because of the large discontinuity between every two periods, some large

Volume

cos T(2n+

‘

DCT

sequence

x(n)

e’24N2

=

Huang HK, Kangarloo H, Cho PS, Iaira RK, Ho BKT, Chan KK. Planning a totally digital radiology department. AJR 1990; 154:635-639. Huang HK, Hideyuki H, Kumagai M, et al. Dedicated digital projectional radiographic system (abstr). Radiology 1988; 169(P):358. lasto M, Wintz PA. Image coding by adaptive block quantization. IEEE Trans Commun Technol 1971; 19:957-972. Rhodes ML, Quinn JF, Silvester J. Local optimal run-length compression applied to CT

5.

images.

IEEE

Trans

Med

Imaging

1985; 4:84-90. Huang HK, Lo S-CB, Ho BKT, Lou SL. Radiological image compression using nor-free and irreversible two-dimensional direct-cosine-transform coding techniques. I Opt Soc Am 1987; 4:984-992.

19.

20.

21.

22.

en-

full-frame

of blockEng 1984; image

bit-allocation

technique. Radiology 1985; 155:811-817. Lo S-C, Huang HK. Compression of radiological images with 512, 1,024, and 2,048 matrices. Radiology 1986; 161:519525.

‘1’

understands

The DFT of a finite, discrete sequence of signals (n) (such as one row or column in a digital image), where n takes the values from 0 to N, is derived by repeating the

8.

in-

The conversion of to cosine coefficients once

analogy

DFT

industry, processors

Y(k)

Reeve HC III, Lim JS. Reduction ing effects in image coding. Opt 23:34-37. Lo S-C, Huang HK. Radiological compression:

(3)

transform, called the fast Fourier transform (FFT), is highly optimized, and a wide selection of supporting products from

7.

9.

of DCT

are available only for limited applications (eg, small matrix sizes such as 8 X 8 pixels). Discrete Fourier transform (DFT), on the other hand, has long been a ubiquitous tool in engineering and science. An efficient

6.

compression

computation

cosine transform (DCT). Some innovative methods have been proposed by various investigators (18). However, specialized processors

high-frequency components are bound to arise from the transform. For image cornpression, high-frequency coefficients are detrimental. An elegant way to eliminate them is to generate the infinite sequence y(n) with the original sequence and its mirror image (Fig 6c). It is shown that Y(k), the Fourier transform of y(n), can be rewritten in terms of x(n) as follows:

Chan KK, Lou S-L, Huang HK. Radiological image compression using full-frame cosine transform with adaptive bit-allocation. Comput Med Imaging Graphics 1989; 13:153-159. Aberle DR. Fiske RA, Brown K, Batra P. Lou S-L, Huang HK. The effect of digital image compression on diagnostic accuracy of pulmonary abnormalities: an ROC study of observer performance (abstr). Radiology 1987; 165(P):391. Chan KK, Lou S-L, Huang HK. Fullframe transform compression of CT and MR images. Radiology 1989; 171:847-851. Hayrapetian A, Kangarloo H, Chan KK, Ho BKT, Huang HK. ROC comparison of compressed images to original analog film and digital hardcopy. In: Schneider RH, Dwyer SJ III, eds. Medical imaging II. Proc SPIE (in press). Ho BKT, Chan K, Ishimitsu Y, Lo S-C, Huang HK. Expandable image compression system: a modular approach. In: Schneider RH, Dwyer SJ III, eds. Medical imaging. Proc SPIE 1987; 767:286-289. Ho BKT, Chan KK, Ishimitsu Y, Stewart BK, Lo S-C, Huang HK. High speed image compression system, prototype and final configuration. In: Schneider RH, Dwyer SJ III, eds. Medical imaging II: image data management and display. Proc SPIE 1988; 914:786-791. Arenson RL, Chakraborty DP, Seshadni SB, Kundel HL. Digital imaging workstation. Radiology 1990; 176:303-315. Ho BKT, Huang HK. Specialized module for full frame radiological image compression. Opt Eng (in press). Ho BKT, Chao 1’ Wu CS, Huang HK. Full frame cosine transform image compression for medical and industrial applications. Machine Vision Applications (in press). Chen WH, Smith CH, Fralick SC. A fast computational algorithm for the discrete cosine transform. IEEE Trans Commun Technol 1977; 25:1004-1009. Oppenheim AV, Schafer RW. Effects of finite register length in digital signal processing. In: Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1975; 453. Ahmed N, Natarajan T, Rao KR. Discrete cosine transform. IEEE Trans Comput 1974; 25:90-93. Narashima MJ, Peterson AM. On the computation of the discrete cosine transform. IEEE Trans Commun Technol 1978; 26:934-936. Makhoul J. A fast cosine transform in one and two dimensions. IEEE Trans Acoustics Speech Signal Processing 1980; 28:27-34.

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