The document discusses image compression using the discrete cosine transform (DCT). It explains that the DCT represents an image as a sum of sinusoids, allowing most visual information to be concentrated in DCT coefficients. This allows for lossy compression by removing insignificant coefficients. The document provides an example MATLAB code to compress an image using DCT. It shows the original 33.8kb image and compressed 13kb image, achieving a compression ratio of 2.6 with minimal quality loss.

1. Penn State Harrisburg
Digital Image Processing
Image Compression using DCT
Chad RyanWeiss
4/25/2016

2. Abstract:
Image compressionhasbecome the mostimportant,energyefficiencymethodfortransferringvideos
and imagesviaany communicationsystem. Before some of the veryfirstimage compressiontechniques
were everestablished,itwouldtake hourstoupload,download,file share andtransmitmediafiles
because the file sizeswere toobig. Evenwiththe 4G technologythatwe have today, these tremendous
file sizeswouldstillpose amajorproblemtomemorystorage systems. Since the dawnof the early
digital image processingdays,scientistsandmathematicianshave beencomingupwithcleverwaysto
compressimagesandframes foreasystorage and data transfer. Methodslike the discrete cosine
transformuse highlyadvancedmathematical principlestorepresentimagesthatallow formodification
and cleversize manipulationof the overall image. Byimplementingdigital image processingtechniques
like the discrete cosine transform,one wouldessentiallymake the packetof informationalotlighterfor
transportand storage.

3. Theory:
The discrete cosine transform(DCT) isa methodthatrepresentsanimage asa sum of sinusoidsof
varyingmagnitudesandfrequencies. Itisthisrepresentationof the image asa sumof sinusoidsthat
allowsforthe fact that mostof the visuallysignificantinformationaboutthe image isconcentratedin
the DCT coefficients. Forthisreason,DCT hasbecome a well-known,popularimage compression
standardfor lossyimage compression. Animportantqualityaboutthe DCTisthat it isinvertible,which
isconvenientforthe realmof image compressionbecause itissometimesnecessaryandrecoverthe
original image forthe sake of acquiringthe primarysource of information.
Whencompressinganimage usingDCT,the image inputisfirstdividedinto8by 8 or 16 by16 blocks
and thenthe DCT is computedforeachblockof information. The coefficientsof the DCTtransformare
thenobtained,quantized,codedandtransmitted. Once the transmitterhasessentiallytransferredall of
the informationandthe receiverhaspickedupthe signal,the receiverbeginstodecode the quantized
coefficients,computesthe inverse DCTof eachblockand thenreconstructsthe original image.
The reasonthe DCT iscalleda lossymethodforcompressingimagesisdue tothe fact that many of
these coefficientsare nearlyzeroandholdnot-so-importantinformationinregardstothe bigpicture.
These valuesare simple thrownoutbecause theycanbe andthe image becomessmaller. Usually,the
userhas no ideathatthese have been thrownoutbecause of how small orinsignificantthese partsof
the image were tothe overall whole.
Example Code:
clear all; close all; clc;
I = imread('NASA.jpg');
I=double(I(:,:,1))/255;
T = dctmtx(8);
B = blkproc(I,[8 8],'P1*x*P2',T,T');
mask = [1 1 1 0 0 0 0 0
1 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0];
B2 = blkproc(B,[8 8],'P1.*x',mask);
I2 = blkproc(B2,[8 8],'P1*x*P2',T',T);
imtool(I);
imtool(I2);
The code seenatleftwill compressa
two-dimensional imageusingthe DCT.
We start byreadinginan image using
the imreadfunction. Followingthis
immediate actionwe thenconvertthe
image to data type double andscale it
downto account forthisdata shift.
Next,we create a DCT coefficient
matrix usingthe dctmtx function,which
will be usedinthe precedinglinesof
code to pre andpost multiply the 8-by-
8 blocksof data of ourimage.The
blkprocfunctioncreatesthe 8-by-8
chunksof data andallowsusto throw
inthe pre and postmultiplicationas
part of the function’sinputarguments.
Finallywe decide whichcoefficientsto
keepbyapplyingthe maskandthen
applyingthe inverseDCTto recoverthe
image anddisplaythe outputs.(NEXT)

4. Figure 1 representsthe original image. Aftersavingthisimage tothe desktop,the size of thisoriginal
file wasfoundtocontain 33.8 kb of data. Now we will determinethe compressionratioof this
particularexample byreferringtothe nextimage.
Figure 1: Original Image

5. Figure 2 representsthe compressedimage. Itcan be seenthatsome of the finerdetailshave been
blurredor leftoutentirely. Aftersavingthisfile tothe desktop,the file sizewasfoundtobe only13 kb!
The DCT yieldedacompressionratioof aboutC = 33.8 kb/ 13 kb= 2.6!
Figure 2: Compressed Image

6. Conclusion:
In summary,we talkedabouthowthe DCT isusedin digital image processingaswell as the implications
of image compression. We brieflytalkedaboutsome of the theory behind the DCT and found that a lot
of itis basedonhighlyadvancedmathematical andscientificconcepts. Lastly, we provided an example
of the DCT image compression techniques sing MatLab. The results were displayed and we found for
our particularexample thatthe outputimage was compressed by a factor of 2.6 without any major loss
of information as to what the big picture actual represents.