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Image Compression: Mathematical Means

The Importance of Compression at NASA

Abstract

Image/video data compression is a very critical technology for numerous operations at NASA. At

NASA, image compression is done for three main reasons. First, compression saves space. NASA

receives millions of information/data each day that requires a huge storage facility, so

compression saves enormous amounts of space on the hard drive. Secondly, image/video

compression saves transmission time. Due to the massive distance between Mars and Earth, for

example, sending back pictures and data for the NASA Mars Rover project can take years if the

data is uncompressed. Lastly, compression saves money simply by saving hard drive space and

time.

NASA identified various lunar/Mars mission requirements that involve transmission of

image/video. These can be categorized into several types: high rate video, edited high rate

video, low rate video, science imaging data, and telerobotic video. Some of this image/video

data can benefit greatly from compression because it would take up less space and can

therefore be transmitted faster. Other image/video data such as scientific/medical data and

telerobotics videos are very valuable and irreplaceable, so NASA is reluctant to consider any

type of compression on these.1

A study of the basic fundamentals of image compression was conducted in an effort to inform

NASA about the most efficient means of compression. NASA uses images to reveal information,

data, and evidence concerning astronomical research. For this reason each NASA image must

have the best quality and adequate dimensions. The purpose of this project was to compare the

compression ratios resulting from compression by three variations of pixel matrices, and to

understand the difference between arithmetic mean and geometric mean compression

methods. For this task, a total of 50 planetary images were selected from the NASA website.

With the aid of the Mathematica software, the team created a program to compress the images.

The team then recorded its findings from the experiment in the form of graphs; visual

representation helped to understand the resulting trends.

1

1 + 2+...+

= =

=1

Formula for Geometric Mean:

=

= 1 2 ...

2x2

To find the most efficient means of compression, the team compared two different compression

mechanisms: arithmetic and geometric mean compression. Within each broad compression

mechanism, we looked at images from five different categories Venus, Earth, Mars, Jupiter,

and Saturn (all images were taken from the NASA website). We compressed each photo in each

category in three different ways: by a 1x4 matrix, 2x2 matrix, and a 2x3 matrix.

To carry out the experiment, we used a

computational software program called

Mathematica. The image was first separated

into three distinct color channels: red, green,

and blue.

R

After compressing all of the

images, the team compared the

compression ratios. A

compression ratio is defined by

the original file size divided by

the file size of the compressed

image.

G

Each separate image is then converted into

a matrix; that is the only way the computer

can understand and manipulate the image.

We then set a variation (i.e., 2x2, 2x3, 1x4

matrix). Infinite zeros are then added to the

right and below the image to account for

compression by a matrix which does not

divide the dimensions of the original image.

B

Following this, each of the three images (R,

G, B color channels) is compressed by the

desired variation. The compressed images

are the combined, yielding a compressed

version of the original image.

The teams hypothesis is tri-fold:

1. The geometric mean compression will save less space than arithmetic mean compression.

2. An image that is compressed by a 2x3 matrix saves more space than one that is compressed

by fewer pixels (i.e., a 1x4 and 2x2 matrix). An image compressed by a 2x2 matrix and 1x4

matrix will save the same amount of space. In other words, the orientation of the matrix does

not matter as long as it has the same number of pixels.

3. The compression ratio is not affected by the different categories (Venus, Earth, Mars, Jupiter,

Saturn).

Conclusion and Future Work

The original image files with the lowest and highest file sizes render a lower compression

ratio while the middle range file sizes has a higher compression ratio. This is the reason why

Mars has a larger compression ratio.

Based on the arithmetic mean sampling technique, the larger the matrix size, the redder the

image becomes.

Based on the geometric mean sampling technique, the larger the matrix size, the bluer the

image becomes.

The geometric and arithmetic mean methods indicate that the higher the sample matrix

size, the greater the compression ratio.

Compression by the same number of pixels (for example 2x2 and 1x4) saves roughly the

same amount of space (for both the arithmetic and geometric means).

In the future, the team hopes to hopes to study other algorithms that will explain the change of

color associated with the arithmetic and geometric mean compressions. Why does compression

by the arithmetic mean produce red images while compression by the geometric mean produce

blue images?

Sponsors and Contributors

Sponsors:

National Aeronautics and Space Administration

(NASA)

NASA Goddard Space Flight Center (GSFC)

NASA Goddard Institute for Space Studies

(GISS)

CUNY Hostos CC & City Tech

USDE

NSF

Prof. Tanvir Prince (faculty mentor)

With special thanks to Prof. Nieves Angulo.

www.PosterPresentations.com

EARTH

MARS

JUPITER

SATURN

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Arithmetic Mean

Geometric Mean

VENUS

EARTH

MARS

JUPITER

SATURN

1x4

22

20

18

16

14

12

10

8

6

4

2

0

Arithmetic Mean

Geometric Mean

VENUS

EARTH

MARS

JUPITER

SATURN

Variations (2x2, 1x4, 2x3) vs. Compression Ratio:

How does changing the matrix size affect the compression ratio for both the arithmetic and

geometric means?

Geometric Mean

Arithmetic Mean

30

27

24

21

2X2

2X3

1X4

18

15

12

9

6

3

0

VENUS

Contributors:

Karina Shah (high school student)

William Ashong (undergraduate student)

Ildefonso Salva (high school teacher)

RESEARCH POSTER PRESENTATION DESIGN 2012

VENUS

As technology is rapidly advancing, computerized images must maintain their quality, even

after compression. Compression is really just a mathematical algorithm that is applied to

an images content, and as such, altering these algorithms could significantly improve

image compression.

=1

Procedure

Arithmetic Mean

Geometric Mean

On the other hand, lossy compression reduces a file by permanently eliminating certain

information, especially redundant information. After the file is compressed, only a part of

the original information remains, although the user may not notice it. Lossy compression is

generally used for video and sound, where a certain amount of information loss will not be

detected by most users and is of no consequence.

Hypotheses

Average Compression Ratio

Image compression is the reduction of image pixels (bytes) in order to minimize the

memory an image saves on a computers hard drive. Image compression has a wide

variety of multi-media application: digital cameras, computers, smartphones, in addition to

several peripheral devices. Images can be compressed in two distinct ways by lossless

and lossy compression. In a lossless compression process, no data is lost in the

compression. This technique is generally used for text or spreadsheet files, where losing

words or financial data often poses a problem.

22

20

18

16

14

12

10

8

6

4

2

0

2x3

EARTH

MARS

JUPITER

SATURN

Average Compression Ratio

Arithmetic vs. Geometric Mean:

Does compression by arithmetic mean save more space than compression by geometric

mean, or vice versa? Is there any difference at all between the amount of space each

compression mechanism saves?

Average Compression Ratio

Formula for Arithmetic Mean:

Results

Average Comression Ratio

Arithmetic and Geometric Mean Formulas

Stefany Franco, Ildefonso Salva, Charlie Windolf, Tanvir Prince, Mathematics Behind Image Compression, (New York: Journal of Student Research), p. 1.

Applying the principles of matrices to examine the difference between compression by

arithmetic mean and compression by geometric mean, specifically understanding which

method saves more space.

Using hands-on experience on the Mathematica software to determine how changing matrix

size affects the compression ratio for both geometric and arithmetic mean compression

methods.

Average Compression Ratio

1.

Methods and Materials

33

30

27

24

21

18

15

12

9

6

3

0

VENUS

2X2

2X3

1X4

EARTH

MARS

JUPITER

SATURN

Color Changes for Arithmetic and Geometric Mean Compression

Methods