introduction to digital image processing

Ge#ngyourfeetwetwithMedicalImageProcessing

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Outline•  Digitalimageprocessing

– Basicsofedgedetec?onalgorithms

•  Medicalimagereconstruc?on–  Filteredbackprojec?onforCT

– SENSEMRIreconstruc?on

EDGE DETECTION IN MATLAB

Slide created by Ratnamanjuri Devi, Research Assistant, MIRC

INTRODUCTION �  The abrupt changes or discontinuities in the intensity values in an image represent the

edges present in it.

�  Edge Detection is an approach used in Image Segmentation for detecting these edges.

�  These discontinuities are detected using first and second order derivatives

�  The first order derivative of choice is called the gradient.

�  The gradient of a 2D function f(x, y) (image) is defined by:

∇f = [Gx, Gy] = [∂f/∂x, ∂f/∂y]

where, Gx and Gy are gradients in the x and y direction respectively.

�  The gradient vector points in the direction of maximum rate of change, which are the edges.

�  Second order derivatives are usually computed using the Laplacian method.

�  The Laplacian of a 2D function f(x,y) is defined by:

∇2f(x, y) = ∂2f(x y)/∂x2 +∂2f(x, y)/∂y2

�  Thus , edge detection can be carried out in one of the two ways:

→ Find places where the magnitude of the first derivative of the intensity is greater than a specified threshold.

→ Find places where the second derivative of the intensity has a zero crossing.

�  The function ‘edge’ is used in MATLAB for edge detection.


GRADIENT EDGE DETECTORS

These edge detectors use their respective masks shown on the right to digitally approximate the first derivatives Gx

and Gy, using which the gradient is calculated;

g = [Gx2 + Gy2]1/2

The pixel is then said to be an edge pixel if g ≥ T, where T is the specified threshold.

�  Sobel Edge Detector:

Syntax:

G = edge (f, ‘Sobel’, T, dir); where dir can be ‘Horizontal’, ’Vertical’ or ‘both’.

�  Prewitt Edge Detector:

Syntax:

G = edge (f,’Prewitt’, T, dir);

It is simpler to implement than Sobel but more susceptible to noise.

�  Roberts Edge Detector:

Syntax:

G = edge (f,’Roberts’), T, dir);

Simplest and oldest edge detector. It is used to detect edges at 45° angles.


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Sobel

Roberts

Prewitt

SECOND ORDER DERIVATIVE EDGE DETECTORS �  Laplacian of Gaussian:

§  The image is first smoothened using Gaussian filter. Hence, removing high frequency noises.

§  It then yields a Laplacian which yields a double edge image.

§  Finding the zero crossing between the two edges gives us the edge.

Syntax:

G = edge(f, ‘log’, T, sigma); where sigma is the standard deviation of the Gaussian filter, which determines the degree of image smoothening.

�  Canny Edge Detector:

§  Most powerful edge detector provided by function ‘edge’.

§  Image is smoothened using a Gaussian filter, after which its Gradient is calculated using either of the three detectors.

§  It uses two thresholds, one for weak edges and other for strong edges

Syntax:

G = edge(f, ‘canny’, T, sigma); where T = [T1 T2]


OUTPUT IMAGES

Original Image ‘rice.png’

Prewitt Sobel

Laplacian of Gaussian

Roberts

Canny


BIBLIOGRAPHY

�  Digital Image Processing using MATLAB, Rafael Gonzalez, Richard E. Woods and Steven L. Eddins.


FilteredBackProjectionAMATLABimplementation

SlidecreatedbyRatnamanjuriDevi,ResearchAssistant,MIRC

JosephHornack,TextonMRI

BasicOutlineoftheProgram

Algorithm:

•  Readtheimage.Here,theShepp-Loganphantomofsize256x256.

•  GeneratetheSinogramoftheimageassuminganincidentparallelbeamrotatedthrough180°oftheimage.

•  Reconstructimagefromthesinogramusingfilteredbackprojection.

•  DisplayOriginalandReconstructedImages.


START

GenerateSinogram

ReconstructImagefromSinogram

END

ReadImage:

I=phantom(256)

DisplayOriginalandReconstructedImages

SinogramGeneration

Algorithm:

•  Theimageisread.

•  Foreverydegreerotationoftheimagestartingfrom0°to179°(atotalof180°),thesumofpixelsfallingalongeachofthe256linesiscalculated.

•  Thesumofeachoftheserotationsresultinanarrayofsize1x256andoccupiesthepositionofbeingtheithcolumnofthematrix‘sinogram’,i-1beingtheiterativerotation.

•  Thesinogramgeneratedisthusamatrixofsize256x180.


START

ReadImage

Inputthetak=0°to179°

Rotateimageatthetakdegree

sinogram(:,i+1)=sumofrotatedimage

Displaysinogra

m

A

i=i+1

i<180

i=1

yes

no

Flowchart

ImageReconstructionusing

FilteredBackProjection

Algorithm:

•  Createarampfilterofsize256x1infrequencydomain

•  FFTofeachcolumnofthesinogramismultipliedwiththisfilterandfilteredsinogramisstoredinsinogram_filtafterIFFT.

•  Eachcolumnofthisfilteredsinogramisreplicatedtoforma256x256matrixeach.

•  Thesesinogramsarerotatedbackthrough0,-1…-179°andaddedtogether.

•  Theresultingmatrixisnormalizedtogivethereconstructedimage.

•  Theimagesaredisplayed.


A

CreatearampfilterM_filt

j=1

j<180

Temp1=FFTofsinogram(:,j)*M

_filt

Sinogram_filt(:,j)=

IFFT(temp1)

j=j+1

K<180

Userepmattoreplicate

sinogram_filt(:,k)toa256x256

matrix

Rotatematrixbykdegrees.

phat=phat+rotatedmatrixk=k+1

k=1,Initializeanemptymatrixphat(256,256)

Phat/max(phat(:)) Display

images END

Flowchart

OutputImages


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Sinogram FilteredSinogram

OriginalPhantom ReconstructedPhantom

SENSERECONSTRUCTION

SlidesPreparedBy:ShamshiaTabassum,RA,MIRC

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INTRODUCTION•  SENSi-vityEncoding(SENSE)-parallelimagingtechnique,whereanarrayof

mul?ple,simultaneouslyoperatedreceivercoilsisusedforsignalacquisi?on

•  SENSEisbasedonuseofmul?pleRFcoilsandreceivers.Thecoilelementsareusedtomeasuretheanatomicalregionofinterestsimultaneously

•  Thisallowsreduc?onofthetotalscan?mebyafactoruptothenumberofcoilelements

•  Reconstruc?onintheimagedomain

Reconstruc-onstrategies•  strongreconstruc-on-aimsatop?mizingvoxelshape

•  weakreconstruc-on-thevoxelshapecriterionisweakerinfavoroftheSNR

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METHODSenseReconstruc-on1.  Thecrea?onofanaliased,reducedFOVimageforeachcoilarray

element2.  AfullFOVimageiscreatedbyreversingthealiasingeffect(i.e.

unfolding)

S1AA+S1BB S2AA+S2BB

S3AA+S3BB S4AA+S4BB

[y1]=[S1AS1B][y2]=[S2AS2B][A][y3]=[S3AS3B][B][y4]=[S4AS4B]SolvingMatrix-----------à

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SENSEImaging-AnExample

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RESULTS

Figure3:Parallelacquisi?oninMRIusing8coilarray(a):cartoondepic?on(b):frommatlabcode(clockwise) %%PSUEDOCODE

load('brain_8ch.mat');[m,n,t]=size(im);fori=1:timagesc(abs(im(:,:,i)));end %%MULTICOILRECONSTRUCTION

mul?coil_recon=abs(sum(im,3));mul?coil_sq_recon=sqrt(abs(sum(abs(im.^2),3)));

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Figure5:Generatedundersampledimages(clockwise)

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%%SENSERECONSTRUCTIONimf=sense_sg(ima,map,Rx,Ry);figure;imagesc(abs(imf));

%%GENERATEUNDERSAMPLEDIMAGESRx=1;Ry=2;ima=undersample(im,Rx,Ry);

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REFERENCE[1]PruessmannKP,WeigerM,ScheideggerMB,BoesigerP,

“SENSE:Sensi?vityencodingforfastMRI”,Magne?cResonanceinMedicine1999;42(5):952-962

[2]DavidJLarkman,RitaGNunes,“Parallelmagne?c

resonanceimaging”,Phys.Med.Biol.52(2007)R15–R55

[3]“RoleofParallelImaginginMagne?cResonance

Imaging”,DouglasC.Noll&BradleyP.Suqon,DepartmentofBiomedicalEngineering,UniversityofMichiganFieldFunc?onalMRI

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Acknowledgements/Contactinfo

•  MIRCmembers,MedicalElectronicsmembers•  Management,DSI•  AcademicandIndustrialpartners

•  MIRC_HOME_OFFLINE(hqp://www.dayanandasagar.edu/mirc-home.html)•  MIRC_SHARING_OFFLINE•  MIRC_PEOPLE_OFFLINE•  [email protected]

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introduction to digital image processing

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