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Next-Generation Next-Generation Bioinformatics SystemsBioinformatics SystemsJelena KovačevićJelena Kovačević
Center for Bioimage InformaticsCenter for Bioimage InformaticsDepartment of Biomedical EngineeringDepartment of Biomedical EngineeringCarnegie Mellon UniversityCarnegie Mellon University
22S
AcknowledgmentsAcknowledgments
Current PhD students
AminaChebira
TadMerryman
GowriSrinivasa
PhD students
DoruCristianBalcan
ElviraGarciaOsuna
PabloHenningsYeomans
JasonThornton
Collaborators
VijaykumarBhagavatula
GeoffGordon
JoséMoura
MarkusPüschel
MariosSavvides
BobMurphy
Undergrads
Woon HoJung
Funding
LionelCoulot
HeatherKirshner
33S
GoalGoal
Imaging in systems biologyImaging in systems biology
Use informatics toUse informatics to acquire, store, manipulate acquire, store, manipulate
and share large bioimaging and share large bioimaging databasesdatabases
Leads toLeads to automated, efficient and automated, efficient and
robust processing robust processing
NeedNeed Host of sophisticated tools Host of sophisticated tools
from many areasfrom many areas
Computation
Knowledge Extraction
Acquisition
Application area
44S
Application AreasApplication Areas
BioimagingBioimaging Current focus in biology: mapping out the protein landscapeCurrent focus in biology: mapping out the protein landscape Fluorescence microscopy used to gather data on Fluorescence microscopy used to gather data on
subcellular eventssubcellular events ►►
Biometrics Biometrics Biosensing for providing securityBiosensing for providing security
to the financial industryto the financial industry at US bordersat US borders
Use person’s biometric characteristic to Use person’s biometric characteristic to identify/verifyidentify/verify ►►
55S
AcquisitionAcquisition
IssuesIssues z-stacks and time series resolutionz-stacks and time series resolution
Context-dependentContext-dependent Slow-changing process needs to be acquired with Slow-changing process needs to be acquired with
coarser resolutioncoarser resolution Changes need to be detected and reacted toChanges need to be detected and reacted to
Efficiency of acquisitionEfficiency of acquisition Acquire only Acquire only wherewhere and and whenwhen needed needed adaptivity adaptivity
Sample questionSample question How can we efficiently acquire How can we efficiently acquire
fluorescence microscopy images? fluorescence microscopy images? ►►
66S
Knowledge ExtractionKnowledge Extraction
Sample questionsSample questions How can we automatically and efficiently classify proteins How can we automatically and efficiently classify proteins
based on images of their subcellular locations? based on images of their subcellular locations? ►► How can we identify/verify person’s identity based on How can we identify/verify person’s identity based on
his/her biometric characteristic? his/her biometric characteristic? ►►
Toolbox needed to solve the problemToolbox needed to solve the problem Signal processing/data miningSignal processing/data mining Multiresolution tools allow for Multiresolution tools allow for
adaptiveadaptive and and efficientefficient processing processing ►►
77S
ComputationComputation
The problem: The problem: fast numerical softwarefast numerical software Hard to write fast codeHard to write fast code Best code platform-Best code platform-
dependentdependent Code becomes obsolete Code becomes obsolete
as fast as it is writtenas fast as it is writtenreasonableimplementation
vendor library
or SPIRAL generated
10x
88S
SPIRALSPIRALCode Generation for DSP AlgorithmsCode Generation for DSP Algorithms
The SolutionThe Solution Automatic generation Automatic generation
and optimization of and optimization of numerical softwarenumerical software
Tuning of Tuning of implementation and implementation and algorithmalgorithm
A new breed of A new breed of intelligent intelligent SW design toolsSW design tools
SPIRAL: a prototype for SPIRAL: a prototype for the domain of DSP the domain of DSP algorithms algorithms ►►
ww
w.s
pira
l.net
fast algorithm asSPL formula
C/Fortranprogram
DSP transform (user specified)
Platform adapted code
Formula translator controls
runtime on given platform
Formula generator controls
Sea
rch
engi
ne
99S
BioimagingBioimaging
AcquisitionAcquisition How can we efficiently How can we efficiently
acquire fluorescence acquire fluorescence microscopy images? microscopy images? ►►
Knowledge extractionKnowledge extraction How can we automatically How can we automatically
and efficiently classify and efficiently classify proteins based on images proteins based on images of their subcellular of their subcellular locations? locations? ►►
ComputationComputation Automatic code generation Automatic code generation
and optimization and optimization ►►
Computation
Knowledge Extraction
Acquisition
Bioimaging
1010S
MotivationMotivation
Current focus in biological sciencesCurrent focus in biological sciences System-wide research System-wide research “omics”“omics”
Human genome project Human genome project
Next frontierNext frontier ProteomicsProteomics Subcellular location one of major componentsSubcellular location one of major components
Grand challengeGrand challenge Develop an intelligent next-generation bioimaging system Develop an intelligent next-generation bioimaging system
capable of fast, robust and accurate classification of capable of fast, robust and accurate classification of proteins based on images of their subcellular locationsproteins based on images of their subcellular locations
1111S
MR Acquisition of MR Acquisition of Fluorescence Microscopy ImagesFluorescence Microscopy Images
ProblemProblem Why acquire in areas of Why acquire in areas of
low fluorescence?low fluorescence? Acquire only Acquire only whenwhen and and
wherewhere needed needed
Measure of successMeasure of success Problem dependentProblem dependent Here: Here:
Strive to maintain the Strive to maintain the achieved classification achieved classification accuracyaccuracy
Efficient acquisition leads toEfficient acquisition leads to Faster acquisitionFaster acquisition Possibility of increasing Possibility of increasing
acquisition resolutionacquisition resolution Possible increase in Possible increase in
classification accuracy due classification accuracy due to increased resolutionto increased resolution
ER
1212S
ApproachApproach Develop algorithm on an Develop algorithm on an
acquired data set at acquired data set at maximum resolutionmaximum resolution
Implement a microscope’s Implement a microscope’s scanning protocolscanning protocol
Algorithm:Algorithm:Mimic “Battleship” strategyMimic “Battleship” strategy Acquire around the hitsAcquire around the hits
MR Acquisition of MR Acquisition of Fluorescence Microscopy ImagesFluorescence Microscopy Images
2D2D3D3D
1313S
M
N
2l
Algorithm: DetailsAlgorithm: Details
Probe
Intensity > T?
Initialize probe locations
yes
Add probe
locations
yes
Probe locations left?
no
no
M
N
2l
1414S
Trade-OffsTrade-Offs
What will we lose?What will we lose? Scanning simplicityScanning simplicity
What will we gain?What will we gain? Faster acquisition processFaster acquisition process
Time is proportional to the Time is proportional to the savings in samplessavings in samples
Need to take into account Need to take into account the time to operate the time to operate scanning unitscanning unit
Higher resolution in 3DHigher resolution in 3D The laser intensity can be The laser intensity can be
reducedreduced Reduces photobleaching Reduces photobleaching
Some sources Some sources indicated linear indicated linear relationship, relationship, some othersome other
1515S
MR sampling algorithmTrivial approach
Percent of samples kept / 100
Mitochondrial compression versus distortion
MS
E
Results in 3DResults in 3D
MR
Alg
ori
thm
(9
.81:
1)
Tri
vial
Ap
pro
ach
(9
:1)
Approximation Difference Image
1616S
Results in 2DResults in 2D
Compression Ratio
Acc
ura
cy
[%]
1717S
Current and Future WorkCurrent and Future Work
Implementation issuesImplementation issues Can one operate galvo-Can one operate galvo-
mirrors fast enough to mirrors fast enough to capitalize on the gain?capitalize on the gain?
Algorithmic issuesAlgorithmic issues Add knowledge from Add knowledge from
classification (feedback)classification (feedback) Build modelsBuild models
http://www.olympusconfocal.com/theory/confocalintro.htmlhttp://www.olympusconfocal.com/theory/confocalintro.html
1818S
Funding and ReferencesFunding and References
FundingFunding NSF-0331657, “Next-Generation Bio-Molecular Imaging and Information NSF-0331657, “Next-Generation Bio-Molecular Imaging and Information
Discovery,” NSF, $2,500,000, 10/03-9/08. Co-PI.Discovery,” NSF, $2,500,000, 10/03-9/08. Co-PI.
Journal papersJournal papers T.E. Merryman and J. Kovačević, T.E. Merryman and J. Kovačević,
“An adaptive multirate algorithm for acquisition of fluorescence microscopy data s“An adaptive multirate algorithm for acquisition of fluorescence microscopy data sets,"ets," IEEE Trans. Image Proc., special issue on Molecular and Cellular Bioimaging, IEEE Trans. Image Proc., special issue on Molecular and Cellular Bioimaging, September 2005. September 2005.
Conference papersConference papers T.E. Merryman, J. Kovačević, E.G. Osuna and R.F. Murphy, T.E. Merryman, J. Kovačević, E.G. Osuna and R.F. Murphy,
"Adaptive multirate data acquisition of 3D cell images,""Adaptive multirate data acquisition of 3D cell images," Proc. IEEE Int. Conf. Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc., Philadelphia, PA, March 2005.Acoust., Speech, and Signal Proc., Philadelphia, PA, March 2005.
1919S
Knowledge Extraction
MR Classification of ProteinsMR Classification of Proteins
Why MR?Why MR? Introduction of simple MR Introduction of simple MR
features produced a features produced a statistically significant jump statistically significant jump in accuracyin accuracy
Introduce adaptivity with Introduce adaptivity with little computational costlittle computational cost
This is tubulin
Segmentation
Classification
2020S
Data SetsData Sets
3D HeLa 3D HeLa ►►
2D HeLa 2D HeLa ►►
3T3 3T3 ►►
Huang & Murphy, Journal of Biomedical Optics Huang & Murphy, Journal of Biomedical Optics 9(5), 893–912, 20049(5), 893–912, 2004
2121S
3D HeLa Data Set3D HeLa Data Set
Cells from Henrietta Lacks Cells from Henrietta Lacks (d. 1951, cervical cancer)(d. 1951, cervical cancer)
Confocal Scanning Laser Confocal Scanning Laser Microscope (100x)Microscope (100x)
DNA stain (PI), DNA stain (PI), all protein stain (Cy5 all protein stain (Cy5 reactive dye) and reactive dye) and fluorescent anti-body for a fluorescent anti-body for a specific proteinspecific protein
50-58 sets per class50-58 sets per class 14-24 2D slices per set14-24 2D slices per set Resolution Resolution
0.049 x 0.049 x 0.2 0.049 x 0.049 x 0.2 μμmm
Huang & Murphy, Journal of Biomedical Optics Huang & Murphy, Journal of Biomedical Optics 9(5), 893–912, 20049(5), 893–912, 2004
Covers all major Covers all major subcellular structures subcellular structures ►►
2222S
3D HeLa Data Set3D HeLa Data Set
Covers all major subcellular Covers all major subcellular structures structures ►► Golgi apparatus (giantin, Golgi apparatus (giantin,
gpp 130)gpp 130) Cytoskeleton (actin, Cytoskeleton (actin,
tubulin)tubulin) Endoplasmic reticulum Endoplasmic reticulum
membrane (ER)membrane (ER) Lysosomes (LAMP2)Lysosomes (LAMP2) Endosomes (transf. Endosomes (transf.
receptor)receptor) Nucleus (nucleolin)Nucleus (nucleolin) Mitochondria outer Mitochondria outer
membranemembrane
http://www.biologymad.com/http://www.biologymad.com/
2323S
2D HeLa Data Set2D HeLa Data Set
Cells from Henrietta Lacks Cells from Henrietta Lacks (d. 1951, cervical cancer)(d. 1951, cervical cancer)
Widefield w nearest Widefield w nearest neighbor deconvolution neighbor deconvolution (100x)(100x)
DNA stain and fluorescent DNA stain and fluorescent anti-body for a specific anti-body for a specific proteinprotein
78-98 sets per class78-98 sets per class Resolution 0.23 x 0.23 Resolution 0.23 x 0.23 μμmm
Boland & Murphy, Bioinformatics Boland & Murphy, Bioinformatics 17(12), 1213-1223, 200117(12), 1213-1223, 2001
Mitochondria
Tubulin
LAMP2
Giantin
Gpp130
Nucleolin
DNA
Actin
ER
Tfr
2424S
Classification: Previous systemClassification: Previous system
PreprocessingPreprocessing Manual shiftingManual shifting Manual rotationManual rotation
Feature computationFeature computation Subcellular Location Subcellular Location
Features (SLF)Features (SLF) Drawn from many different Drawn from many different
feature categoriesfeature categories Texture, morphological, Texture, morphological,
Gabor and waveletGabor and wavelet
Gabor and wavelet Gabor and wavelet features improved features improved accuracy significantlyaccuracy significantly(from 88% to 92%)(from 88% to 92%)
ClassificationClassification Combination of classifiersCombination of classifiers
Input image
Preprocessing
Feature extraction
Classification
Class
2525S
MR Classification of ProteinsMR Classification of Proteins
Points toPoints to Frames Frames ►► MD framesMD frames Wavelet/frame packets Wavelet/frame packets ►►
What do we need?What do we need? Want to keep MRWant to keep MR
(based on results with (based on results with Gabor and wavelet Gabor and wavelet features)features)
Avoid manual processingAvoid manual processing Rotation invarianceRotation invariance Shift invarianceShift invariance
AdaptivityAdaptivity
2626S
Does Adaptivity Help?Does Adaptivity Help?
Would like to use wavelet packets Would like to use wavelet packets ►► Do not have an obvious cost measureDo not have an obvious cost measure
Line of workLine of work
Find out if adaptivity helpsFind out if adaptivity helps
If it does, find a cost function to use with wavelet packetsIf it does, find a cost function to use with wavelet packets Frame packetsFrame packets
Challenge: Same class, different storyChallenge: Same class, different storyTubulin
2727S
Training PhaseTraining Phase
Number of classes CNumber of classes C Number of training images/class NNumber of training images/class N
Clustering images Full wavelet tree Feature extraction K-means clustering
Gaussian modelingWeight computationVotingWeights
Training image
Gaussianmodels
2828S
Full Wavelet Tree Full Wavelet Tree DecompositionDecomposition
Grow a full tree Grow a full tree ►► Depth L levelsDepth L levels Total number of subbands STotal number of subbands S
Clustering images
Full wavelet tree
2929S
Feature ExtractionFeature Extraction
Use Haralick texture features Use Haralick texture features ►► One feature vector per subband sOne feature vector per subband s Indexed by class c, training image n, subband sIndexed by class c, training image n, subband s
Clustering images
Full wavelet tree
Feature extraction
3030S
K-Means ClusteringK-Means Clustering
Clustering in a fixed subbandClustering in a fixed subband Max K clusters/classMax K clusters/class
Clusteringimages
of class c
Feature vector for image I from class c and subband s
Cluster mean
X
Clustering images
Full wavelet tree
Feature extraction
K-means clustering
3131S
Gaussian ModelingGaussian Modeling
Model each cluster with a Gaussian pdfModel each cluster with a Gaussian pdf Probability the training image belongs to class iProbability the training image belongs to class i
Output: single probability vectorOutput: single probability vector
Clustering images
Full wavelet tree
Feature extraction
K-means clustering
Gaussian modeling
Training image
3232S
Class CClass 1
From Feature Space to Probability SpaceFrom Feature Space to Probability Space
Subband S
Image 1 from
Class C
Image 1 from
Class 1
Subband 1
Image N from
Class 1
Image N from
Class C
3333S
Weight Computation: Weight Computation: InitializationInitialization
Decision for vector tDecision for vector tc,n,sc,n,s
Class CClass 1
Subband S
Image 1 from
Class C
Image 1 from
Class 1
Subband 1
Image N from Class 1
Image N from Class C
Clustering images
Full wavelet tree
Feature extraction
K-means clustering
Gaussian modeling
Weight computation
Training image
3434S
Weight Computation : Initialization Weight Computation : Initialization
Initial weight for subband s: probability of correct decisionInitial weight for subband s: probability of correct decision
Class CClass 1
Subband S
Image 1 from
Class C
Image 1 from
Class 1
Subband 1
Image N from Class 1
Image N from Class C
correct incorrect incorrect correct
incorrectcorrect correct correct
3535S
Weight ComputationWeight Computation
Compute probability vector for each imageCompute probability vector for each image
Class CClass 1
Subband S
Image 1 from
Class C
Image 1 from
Class 1
Subband 1
Image N from Class 1
Image N from Class C
Class 1
Subband S
Image 1 from
Class 1
Subband 1
3636S
Weight AdjustmentWeight AdjustmentVotingVoting
Make a decisionMake a decision Decision correctDecision correct
Do nothing, take next imageDo nothing, take next image
Decision incorrectDecision incorrect Adjust the weights, take next imageAdjust the weights, take next image
Make Make runs through all the imagesruns through all the images Does the algorithm converge?Does the algorithm converge?
Clustering images
Full wavelet tree
Feature extraction
K-means clustering
Gaussian modeling
Weight computation
VotingWeights
Training image
Gaussianmodels
3737S
Testing PhaseTesting Phase
Compute probabilities for each subbandCompute probabilities for each subband
Compute the overall probability vectorCompute the overall probability vector
Make the decisionMake the decision
Weights
Gaussianmodels
Full wavelet tree Feature extractionProbability space
Voting
Testing image
Class label
3838S
ResultsResults
C = 10 C = 10 classesclasses N = 45 N = 45 training imagestraining images T = 5 T = 5 testing images testing images 10-fold cross validation10-fold cross validation
Training phaseTraining phase 4444 clustering imagesclustering images 45-fold cross validation L = 2,3 levels of Haar wavelet decomposition K = 10 max number of clusters per class
3939S
ResultsResults
Images Images ►►
Output of the classifier [%], K=5Output of the classifier [%], K=5
TubTub GppGpp NucNuc GiaGia MitMit DNADNA ERER LMPLMP ActAct TfRTfR AvgAvg % %
Previous Previous systemsystem 6464 6464 6666 8686 6666 8686 7474 7272 100100 4040 71.871.8
MR MR systemsystem 7474 8484 9898 9090 6868 9494 8080 8686 100100 4848 82.282.2
4040S
Results: Accuracy vs Number of EpochsResults: Accuracy vs Number of Epochs
Variation of Accuracy with Number of Iterations
687072747678808284
0 10 20 30 40 50 60 70 80 90 100
Number of Iterations
Av
era
ge
Ac
cu
rac
y (
%)
K = 3 K = 5 K = 7 K = 10 K = 15
K K 33 55 77 1010 1515
No MR Acc (%)No MR Acc (%) 70.6 70.6 71.871.8 69.069.0 69.469.4 68.068.0
4141S
Classification EnhancementClassification Enhancement
4242S
Weight Adjustment: 2Weight Adjustment: 2ndnd Try Try
Keep the previous best weightKeep the previous best weight Can do no worse than previous systemCan do no worse than previous system
Images ►Images ►
Output of the classifier [%], K=10Output of the classifier [%], K=10
TubTub GppGpp NucNuc GiaGia MitMit DNADNA ERER LMPLMP ActAct TfRTfR Avg Avg %%
Previous Previous systemsystem 6464 7272 6464 8484 5656 8484 6060 7070 9696 4444 6969
MR MR systemsystem 7272 8484 9292 9090 5858 9494 8282 8686 100100 5656 8181
4343S
Principal Component AnalysisPrincipal Component Analysis
• Using eigenspace Using eigenspace representations for Haralick representations for Haralick texture featurestexture features
Texture classification (TC)Texture classification (TC)• Decomposition better than no Decomposition better than no
decompositiondecomposition(with or without PCA)(with or without PCA)
• There is information in the subbandsThere is information in the subbands
TC + PCATC + PCA • Improves accuracyImproves accuracy
(with or without decomposition)(with or without decomposition)
Dimensionality reduction (DR)Dimensionality reduction (DR)• Increases accuracy slightly without Increases accuracy slightly without
much complexitymuch complexity
Exp.Exp. No MRNo MR MRMR
TCTC 69.0%69.0% 81.0%81.0%
TC + TC + PCAPCA
81.8%81.8% 87.4%87.4%
TC + TC + PCA/DRPCA/DR
67.0%67.0% 82.6%82.6%
4444S
Effect of Translation VarianceEffect of Translation Variance
No translationNo translation accuracy(MR frames)accuracy(MR frames) >> accuracy(MR)accuracy(MR)
TranslationTranslation MR MR dropsdrops MR frames MR frames stablestable
No translationNo translation TranslationTranslation
MRMR 81.4%81.4% 80.8%80.8%
MR framesMR frames 83.2%83.2% 83.2%83.2%
4545S
Conclusions and Future DirectionsConclusions and Future Directions
Adaptivity definitely helps!Adaptivity definitely helps! Accuracy stable with the increased # of epochsAccuracy stable with the increased # of epochs
Investigate the algorithm for convergenceInvestigate the algorithm for convergence
K-means clustering introduces randomnessK-means clustering introduces randomness There is no notion of global, local minimaThere is no notion of global, local minima Reducing K reduces randomnessReducing K reduces randomness
WeightingWeighting Should be done for each class separately Should be done for each class separately Would lead to WP treesWould lead to WP trees
Find cost functionFind cost function Construct frame packetsConstruct frame packets
4646S
ReferencesReferences
Conference papersConference papers G. Srinivasa, A. Chebira, T. Merryman and J. Kovačević, “G. Srinivasa, A. Chebira, T. Merryman and J. Kovačević, “Adaptive Adaptive
multiresolution texture features for protein image classificationmultiresolution texture features for protein image classification”, ”, Proc. Proc. BMES Annual Fall MeetingBMES Annual Fall Meeting, Baltimore, MD, September 2005. , Baltimore, MD, September 2005.
K Williams, T. Merryman and J. Kovačević, “K Williams, T. Merryman and J. Kovačević, “A Wavelet Subband Enhancement A Wavelet Subband Enhancement to Classificationto Classification”, ”, Proc. Annual Biomed. Res. Conf. for Minority StudentsProc. Annual Biomed. Res. Conf. for Minority Students , , Atlanta, GA, November 2005. Submitted.Atlanta, GA, November 2005. Submitted.
A. Mintos, G. Srinivasa, A. Chebira and J. Kovačević, “A. Mintos, G. Srinivasa, A. Chebira and J. Kovačević, “Combining Wavelet Combining Wavelet Features with PCA for Classification of Protein ImagesFeatures with PCA for Classification of Protein Images”, ”, Proc. Annual Proc. Annual Biomed. Res. Conf. for Minority StudentsBiomed. Res. Conf. for Minority Students , Atlanta, GA, November 2005. , Atlanta, GA, November 2005. Submitted.Submitted.
T. Merryman, K. Williams and J. Kovačević, “T. Merryman, K. Williams and J. Kovačević, “A multiresolution enhancement to A multiresolution enhancement to generic classifiers of subcellular protein location imagesgeneric classifiers of subcellular protein location images”, ”, Proc. IEEE Intl. Proc. IEEE Intl. Symp. Biomed. Imaging, Symp. Biomed. Imaging, Arlington, VA, April 2006. In preparation.Arlington, VA, April 2006. In preparation.
G. Srinivasa, T. Merryman, A. Chebira, A. Mintos and J. Kovačević, “G. Srinivasa, T. Merryman, A. Chebira, A. Mintos and J. Kovačević, “Adaptive Adaptive multiresolution techniques for subcellular protein location image multiresolution techniques for subcellular protein location image classificationclassification”, ”, Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc., Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc., Toulouse, France, May 2006. Invited paper. In preparation.Toulouse, France, May 2006. Invited paper. In preparation.
4747S
Automatic Code GenerationAutomatic Code Generation
Work in progressWork in progress
4848S
BiometricsBiometrics
AcquisitionAcquisition NIST databaseNIST database
Knowledge extractionKnowledge extraction How can we identify/verify How can we identify/verify
person’s identity based on person’s identity based on his/her biometric his/her biometric characteristic? characteristic? ►►
ComputationComputation Automatic code generation Automatic code generation
and optimization and optimization ►►
Computation
Knowledge Extraction
Acquisition
Biometrics
4949S
MotivationMotivation
Security to the financial industry Security to the financial industry ►► 89,000 cases of identity theft in 200089,000 cases of identity theft in 2000 Losses incurred by Visa/MasterCard $68.2 million Losses incurred by Visa/MasterCard $68.2 million
Security at US bordersSecurity at US borders Multimodal biometric systemsMultimodal biometric systems
Grand challengeGrand challenge Develop an intelligent next-generation biometric system Develop an intelligent next-generation biometric system
capable of fast, robust and accurate identification and capable of fast, robust and accurate identification and verification of human biometric characteristics.verification of human biometric characteristics.
5050S
ChallengesChallenges
Variable conditionsVariable conditions Different lighting, indoors/outdoors, different poses, …Different lighting, indoors/outdoors, different poses, …
Small training setsSmall training sets Uncooperative biometricsUncooperative biometrics
(access to only one picture of a suspected criminal)(access to only one picture of a suspected criminal)
Huge databasesHuge databases Computation becomes an issueComputation becomes an issue Database sizes: up to hundreds of thousandsDatabase sizes: up to hundreds of thousands
5151S
State of Commercial ProductsState of Commercial Products
NIST (National Institute of Standards)NIST (National Institute of Standards) Mandated by the Government to measure accuracy of Mandated by the Government to measure accuracy of
biometric technologies (Patriot Act)biometric technologies (Patriot Act) In cooperation with FBI, State Department, DARPA, In cooperation with FBI, State Department, DARPA,
National Institute of Justice, Transportation Security National Institute of Justice, Transportation Security Administration , United States Customs, Service, Administration , United States Customs, Service, Department of Energy, Drug Enforcement Administration, Department of Energy, Drug Enforcement Administration, INS, etc.INS, etc.
5252S
Face Recognition Vendor Tests Face Recognition Vendor Tests FRVT 2002FRVT 2002 121,589 images of 37,437 individuals121,589 images of 37,437 individuals OutdoorsOutdoors
71.5% true accept rate @ 0.01% false accept rate 90.3% true accept rate @ 1.0% false accept rate
IndoorsIndoors 50% true accept rate @ 1.0% false accept rate
Size of the databaseSize of the database the recognition rate decreases linearly with
the logarithm of the database size(85% @ 800 people, 83% @ 1,600 people, 73% for 37,437 people)
Challenges Poor-quality images, small training sets, database size
5353S
Fingerprint Vendor Technology Evaluation Fingerprint Vendor Technology Evaluation FpVTE 2003FpVTE 2003
48,105 sets, 25,309 individuals, 393,370 distinct fingerprints
Verification results 99.4% true accept rate @ 0.01% false accept rate 99.9% true accept rate @ 1.0% false accept rate
Challenges Poor-quality images Database size
5454S
Correlation-Based Biometrics SystemCorrelation-Based Biometrics System
One of the standard One of the standard methodsmethods Based on correlation filtersBased on correlation filters Template matching Template matching
performed on performed on the entire imagethe entire image
Two systemsTwo systems IdentificationIdentification VerificationVerification
MR system ►MR system ►
Who am I?
Who is this?
This is Ben
I am Ben
Is this Ben?
Yes/No
Template matching
matchno
match
5555S
Correlation FiltersCorrelation Filters
Specific to one classSpecific to one class Produce correlation peaks when applied to their classesProduce correlation peaks when applied to their classes Output: correlation planeOutput: correlation plane
Match score: sharpness of peakMatch score: sharpness of peak
shift-invariantshift-invariant goodness of the match between input and stored imagegoodness of the match between input and stored image
5656S
Correlation Filter DesignCorrelation Filter Design
MACE (Minimum Average Correlation Energy) filterMACE (Minimum Average Correlation Energy) filter Origin of each correlation plane Origin of each correlation plane
constrained to 1 for in-class and 0 out-of-classconstrained to 1 for in-class and 0 out-of-class Minimizes ACE (Average Correlation Energy)Minimizes ACE (Average Correlation Energy)
SolutionSolution FilterFilter Minimum energy Minimum energy
Fitness metricFitness metric How well the correlation filter will performHow well the correlation filter will perform
X of size nxt, FT of training images as columnsX of size nxt, FT of training images as columns
u of size tx1, origin constraintsu of size tx1, origin constraints
D of size nxn, n total number of pixelsD of size nxn, n total number of pixels
h of size nx1, filter valuesh of size nx1, filter values
5757S
MR Approaches in BiometricsMR Approaches in Biometrics
MR systemMR system Introduces adaptivityIntroduces adaptivity Template matching Template matching
performed on different performed on different space-frequency regionsspace-frequency regions
Builds a different Builds a different decomposition for each decomposition for each classclass
5858S
Training Phase: Tree DeterminationTraining Phase: Tree Determination
Use wavelet packets to build adaptive space-Use wavelet packets to build adaptive space-frequency decomposition frequency decomposition ►►
Pruning criterion Pruning criterion
5959S
Training Phase: Filter DesignTraining Phase: Filter Design
Build a correlation filter for each subspaceBuild a correlation filter for each subspace Decompose all in-class training images with the Decompose all in-class training images with the
appropriate treeappropriate tree Compute the correlation filterCompute the correlation filter
Testing PhaseTesting Phase
Match metricMatch metric
6060S
Data SetsData Sets
NIST 24 NIST 24 fingerprint databasefingerprint database MPEG-2 videoMPEG-2 video 10 people 10 people
(5 male & 5 female)(5 male & 5 female) 2 fingers2 fingers 20 classes20 classes 100 images/class100 images/class Subjects instructed to roll fingers continuallySubjects instructed to roll fingers continually Used 10-15 images for training: 8 in-class and the rest out-of-classUsed 10-15 images for training: 8 in-class and the rest out-of-class
Easy class
Difficult class
6161S
Identification resultsIdentification results
0.00
5.00
10.00
15.00
20.00
25.00
30.00
SCF Average EER = 7.21% WDCF Average EER = 1.18%
EE
R (
%).
SCF 0.09 0.03 7.69 0.09 0.92 13.04 21.74 26.09 1.29 4.35 0 0.11 21.88 7.78 4.12 7.61 17.39 7.61 0.11 2.20
WDCF 0 0 0 0 0 4.35 0.83 0 0 0 0 0 8.70 0 0 9.78 0 0 0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Standard Correlation FiltersWavelet Correlation Filters
0
10
20
30
40
50
60
70
80
90
100
SCF Average IER = 18.41% WDCF Average IER = 1.68%
Iden
tifi
cati
on
Err
or
Rat
e (%
)
SCF 0 0 9.78 3.26 4.35 35.90 33.70 89.96 6.52 9.78 0 3.26 66.30 15.22 8.70 21.74 33.70 14.13 0 11.96
WDCF 0 0 0 0 0 5.43 0 0 0 0 0 0 15.22 0 0 13.04 0 0 0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Standard Correlation FiltersWavelet Correlation Filters
Verification results ►Verification results ►
ResultsResults
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Shift-InvarianceShift-Invariance
DWT is shift-varyingDWT is shift-varying Amount of shift variance Amount of shift variance
depends on level jdepends on level j
Evaluate the effectsEvaluate the effects Shift the input imageShift the input image Compute PCEsCompute PCEs
24
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Current and Future WorkCurrent and Future Work
Use frames instead of bases Use frames instead of bases Takes care of shift varianceTakes care of shift variance
Build rotation-invariant framesBuild rotation-invariant frames
Implies true 2D designImplies true 2D design
Build frame packetsBuild frame packets Issue of cost function in overlapping spacesIssue of cost function in overlapping spaces
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Automatic Code GenerationAutomatic Code Generation
FormulaFormula Uniquely represents our transformUniquely represents our transform
Code generationCode generation SPIRAL takes the formula and produces C codeSPIRAL takes the formula and produces C code
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ReferencesReferences
Journal papersJournal papers P. Hennings Yeomans, J. Thornton, J. Kovačević and B.V.K.V. Kumar, P. Hennings Yeomans, J. Thornton, J. Kovačević and B.V.K.V. Kumar,
"Wavelet packet correlation methods in biometrics,''"Wavelet packet correlation methods in biometrics,'' Applied Optics, special issue Applied Optics, special issue on Biometric Recognition Systems, vol. 44, no. 5, February 2005., pp. 637-646. on Biometric Recognition Systems, vol. 44, no. 5, February 2005., pp. 637-646.
Conference papersConference papers J.T. Thornton, P. Hennings Yeomans, J. Kovačević and B.V.K.V. Kumar, J.T. Thornton, P. Hennings Yeomans, J. Kovačević and B.V.K.V. Kumar,
``Wavelet packet correlation methods in biometrics,''``Wavelet packet correlation methods in biometrics,'' Proc. IEEE Int. Conf. Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc., Philadelphia, PA, March 2005. Acoust., Speech, and Signal Proc., Philadelphia, PA, March 2005.
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MR Signal Representation ToolsMR Signal Representation Tools
What?What? Analysis and processing at different resolutionsAnalysis and processing at different resolutions Resolution: amount of informationResolution: amount of information
Why?Why? LocalizationLocalization AdaptivityAdaptivity Computational efficiencyComputational efficiency
How?How? Decomposition into “time-frequency” atomsDecomposition into “time-frequency” atoms ““Divide and conquer”Divide and conquer”
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LocalizationLocalization
Zoom in on Zoom in on singularitiessingularities
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t
fDirac basisWPWT
ER
Actin
STFTFT
AdaptivityAdaptivity
““Holy Grail” of Signal Holy Grail” of Signal Analysis/Processing Analysis/Processing Understand the “blob”-like Understand the “blob”-like
structure of the energy structure of the energy distribution in the time-distribution in the time-frequency spacefrequency space
Design a representation Design a representation reflecting thatreflecting that
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How?How?
Divide and conquerDivide and conquer Represent a signal in Represent a signal in
terms of its building blocksterms of its building blocks
+ +
+ +
= *
*
*
*
=
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= x
How?How?
x =x = synthesizesynthesize (do something)(do something) analyzeanalyze xx
XX = analyzeanalyze xx
= x
x =x = synthesizesynthesize (do something)(do something) analyzeanalyze xx
xx = synthesizesynthesize XX
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MR Signal Representation ToolsMR Signal Representation Tools
We build tools responding to requirements from a We build tools responding to requirements from a specific applicationspecific application Shift invarianceShift invariance
Leads to redundant representations --- framesLeads to redundant representations --- frames
AdaptivityAdaptivity Leads to wavelet (frame) packetsLeads to wavelet (frame) packets
MD nature of the signalMD nature of the signal Leads to nonseparable MR decompositionsLeads to nonseparable MR decompositions
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FramesFrames
Nonredundant decompositionsNonredundant decompositions Robustness to noiseRobustness to noise Robustness to lossesRobustness to losses Freedom in designFreedom in design Shift-invarianceShift-invariance
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Bases versus Frames?Bases versus Frames?
Bases are nonredundantBases are nonredundant Loss of one transform coefficient is irreplaceableLoss of one transform coefficient is irreplaceable Sensitivity to noise is greatSensitivity to noise is great Space of possible solutions is restricted Space of possible solutions is restricted
Solution: framesSolution: frames
0
1
n-1
0
1
n-1
0
1
n-1
0
1
n-1
Processing
InverseTransformTransform
n x n n x n
m-1 m-1
m x n n x m
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Robustness to NoiseRobustness to Noise
Noise is spread over more components: easier to cleanNoise is spread over more components: easier to clean
0
1
n-1
Frame Fm x n
n-1
0
1
n-1
Reconstr. F*n x m
0
1
n-1
Transmission
0
1
m-1 m-1
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Robustness to LossesRobustness to Losses
LossesLosses Modeled as erasuresModeled as erasures To reconstruct, inverse transform must existTo reconstruct, inverse transform must exist Mathematically: any (n x n) submatrix of the frame matrix must Mathematically: any (n x n) submatrix of the frame matrix must
be full rankbe full rank maximally robust to erasures (MR)maximally robust to erasures (MR)
0
1
n-1
Frame Fm x n
n-1
0
1
n-1
Reconstr. F*n x m
0
1
n-1
Transmission
0
1
m-1 m-1
X
X
Losses
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What are Frames?What are Frames?
Generating system for RGenerating system for Rnn or C or Cnn
Usually represented by a matrix FUsually represented by a matrix F0
1
m-1
0
1
n-1
F xFrame coefficients y
=
=
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Frame PropertiesFrame PropertiesMaximally robustMaximally robust
(MR)(MR)
TightTight
(T)(T)
Columns areColumns are
orthonormalorthonormal
Equal normEqual norm
(EN)(EN)
All rows haveAll rows have
equal normequal norm
X
X
0
1
m-1
Any (n x n)Any (n x n)
submatrix is full ranksubmatrix is full rank
n
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0 1 m-1n-1
What Do We Want to Do?What Do We Want to Do?
We want to build frames We want to build frames with structure in stepswith structure in steps First impose maximum First impose maximum
robustnessrobustness MRMR
Then impose tightnessThen impose tightness tighttight MR MR
Finally, add equal normFinally, add equal norm tight tight ENENMRMR
Construction by seedingConstruction by seeding
0 1 n-1
Tools: Polynomial algebras and transformsTools: Polynomial algebras and transforms
m
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Invariance of Frame PropertiesInvariance of Frame Properties
FAB is F A, B invertible
0
0 MR
FA is MRF A, D invertible
0
0 A is UN TF D, U unitaryTF
U TFV is TF U, V unitary, nonzero aa
0
0 EN
FU is ENF D, U unitary, nonzero aa
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Building Frame FamiliesBuilding Frame Families
We impose these one by oneWe impose these one by one MRMR maximally robust to erasuresmaximally robust to erasures
use polynomial transformsuse polynomial transforms
then, then, F = PF = Pb,b,[1, …, N] [1, …, N] is an MR frameis an MR frame
TFTF tight framestight frames use orthogonal polynomials use orthogonal polynomials construct a polynomial transformconstruct a polynomial transform construct the closest orthogonal polynomial transformconstruct the closest orthogonal polynomial transform
ENEN equal normequal norm use DFT to get complex ENMR framesuse DFT to get complex ENMR frames use frame invariance properties to get real ENMR framesuse frame invariance properties to get real ENMR frames
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Funding and ReferencesFunding and References
FundingFunding NSF-0515152, “Frame Toolbox for Bioimaging, Biometrics and Robust NSF-0515152, “Frame Toolbox for Bioimaging, Biometrics and Robust
Transmission”, 09/05-08/08. PI.Transmission”, 09/05-08/08. PI.
Journal papersJournal papers V. K Goyal, J. Kovačević and J.A. Kelner, V. K Goyal, J. Kovačević and J.A. Kelner,
``Quantized frame expansions with erasures,''``Quantized frame expansions with erasures,'' Journal of Appl. and Comput. Journal of Appl. and Comput. Harmonic AnalysisHarmonic Analysis, vol. 10, no. 3, May 2001, pp. 203-233., vol. 10, no. 3, May 2001, pp. 203-233.
V. K Goyal and J. Kovačević, V. K Goyal and J. Kovačević, ``Generalized multiple description coding with correlated transforms,''``Generalized multiple description coding with correlated transforms,'' IEEE Trans. IEEE Trans. Inform. Th.Inform. Th., vol. 47, no. 6, September 2001, pp. 2199-2224., vol. 47, no. 6, September 2001, pp. 2199-2224.
V. K Goyal, J. A. Kelner and J. Kovačević, V. K Goyal, J. A. Kelner and J. Kovačević, ``Multiple description vector quantization with a coarse lattice,''``Multiple description vector quantization with a coarse lattice,'' IEEE Trans. IEEE Trans. Inform. Th.Inform. Th., vol. 48, no. 3, March 2002, pp. 781-788., vol. 48, no. 3, March 2002, pp. 781-788.
J. Kovačević, P.L. Dragotti and V. K Goyal, J. Kovačević, P.L. Dragotti and V. K Goyal, ``Filter bank frame expansions with erasures,''``Filter bank frame expansions with erasures,'' IEEE Trans. Inform. Th., special IEEE Trans. Inform. Th., special issue in Honor of Aaron D. Wynerissue in Honor of Aaron D. Wyner, vol. 48, no. 6, June 2002, pp. 1439-1450. , vol. 48, no. 6, June 2002, pp. 1439-1450. Invited paper.Invited paper.
P.G. Casazza and J. Kovačević, P.G. Casazza and J. Kovačević, ``Equal-norm tight frames with erasures,''``Equal-norm tight frames with erasures,'' Advances in Computational Mathematics, special issue on FramesAdvances in Computational Mathematics, special issue on Frames , pp. 387-430, , pp. 387-430, 2002. Invited paper.2002. Invited paper.
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References (cont’d)References (cont’d)
Conference papersConference papers V. K Goyal, J. Kovačević and M. Vetterli, V. K Goyal, J. Kovačević and M. Vetterli,
“Quantized frame expansions as source-channel codes for erasure channels,”“Quantized frame expansions as source-channel codes for erasure channels,” Proc. Wavelets Proc. Wavelets and Appl. Workshopand Appl. Workshop, Ticino, Switzerland, September 1998., Ticino, Switzerland, September 1998.
V. K Goyal, J. Kovačević and M. Vetterli, V. K Goyal, J. Kovačević and M. Vetterli, “Quantized frame expansions as source-channel codes for erasure channels,”“Quantized frame expansions as source-channel codes for erasure channels,” Proc. Data Proc. Data Compr. Conf.Compr. Conf., Snowbird, UT, March 1999., Snowbird, UT, March 1999.
P. L. Dragotti, J. Kovačević and V. K Goyal, P. L. Dragotti, J. Kovačević and V. K Goyal, “Quantized oversampled filter banks with erasures,”“Quantized oversampled filter banks with erasures,” Proc. Data Compr. Conf.Proc. Data Compr. Conf., Snowbird, UT, March 2001, pp. 173-182. , Snowbird, UT, March 2001, pp. 173-182.
A. C. Lozano, J. Kovačević and M Andrews, A. C. Lozano, J. Kovačević and M Andrews, “Quantized frame expansions in a wireless environment,”“Quantized frame expansions in a wireless environment,” Proc. Data Compr. Conf.Proc. Data Compr. Conf., Snowbird, , Snowbird, UT, March 2002, pp. 480-489. UT, March 2002, pp. 480-489.
A. C. Lozano, J. Kovačević and M Andrews, A. C. Lozano, J. Kovačević and M Andrews, “Quantized frame expansions in a wireless environment,”“Quantized frame expansions in a wireless environment,” Proc. DIMACS Workshop on Source Proc. DIMACS Workshop on Source Coding and Harmonic AnalysisCoding and Harmonic Analysis, Rutgers, NJ, May 2002. , Rutgers, NJ, May 2002.
M. Püschel and J. Kovačević, M. Püschel and J. Kovačević, “Real, Tight Frames with Maxi“Real, Tight Frames with Maximmal Robustness to Erasures”al Robustness to Erasures”, , Proc. Proc. Data Compr. Conf.Data Compr. Conf., Snowbird, UT, March 2005, pp. 63-72., Snowbird, UT, March 2005, pp. 63-72.
Book chaptersBook chapters P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and J. Tremain, P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and J. Tremain,
``A physical interpretation of finite tight frames.''``A physical interpretation of finite tight frames.'' Harmonic Analysis and ApplicationsHarmonic Analysis and Applications, C. Heil, , C. Heil, Ed., Birkhauser, Boston, MA, 2004.Ed., Birkhauser, Boston, MA, 2004.
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Wavelet PacketsWavelet Packets
First stage: full decompositionFirst stage: full decomposition
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Cost(parent) >< Cost(children)?Cost(parent) < Cost(children)
Wavelet PacketsWavelet Packets
Second stage: pruningSecond stage: pruning ExamplesExamples Bioimaging Bioimaging ►► Biometrics Biometrics ►►
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References on MRReferences on MR
Light readingLight reading ““Wavelets: Seeing the Forest -- and the TreesWavelets: Seeing the Forest -- and the Trees”, D. Mackenzie, Beyond Discovery, ”, D. Mackenzie, Beyond Discovery,
December 2001.December 2001.
BooksBooks ““A Wavelet Tour of Signal Processing”, S. Mallat, Academic Press, 1999.A Wavelet Tour of Signal Processing”, S. Mallat, Academic Press, 1999. ““Ten Lectures on Wavelets”, I. Daubechies, SIAM, 1992.Ten Lectures on Wavelets”, I. Daubechies, SIAM, 1992. ““Wavelets and Subband CodingWavelets and Subband Coding”, M. Vetterli and J. Kovačević, Prentice Hall, 1995.”, M. Vetterli and J. Kovačević, Prentice Hall, 1995. ““Wavelets and Filter Banks”, G. Strang and T. Nguyen, Wells. Cambr. Press, 1996.Wavelets and Filter Banks”, G. Strang and T. Nguyen, Wells. Cambr. Press, 1996.
BioimagingBioimaging ““A Review of Wavelets in Biomedical ApplicationsA Review of Wavelets in Biomedical Applications”, M. Unser and A. Aldroubi, ”, M. Unser and A. Aldroubi,
Proc. IEEE, April 1996.Proc. IEEE, April 1996. ““Wavelets in Temporal and Spatial Processing of Biomedical DataWavelets in Temporal and Spatial Processing of Biomedical Data”, A. Laine, ”, A. Laine,
Annu. Rev. Biomed. Eng., 2000.Annu. Rev. Biomed. Eng., 2000. ““Guest Editorial: Wavelets in Medical ImagingGuest Editorial: Wavelets in Medical Imaging”, M. Unser, A. Aldroubi and A. Laine, ”, M. Unser, A. Aldroubi and A. Laine,
IEEE Trans. On Medical Imaging, March 2003.IEEE Trans. On Medical Imaging, March 2003. ““
Wavelets in Bioinformatics and Computational Biology: State of the art and PerspeWavelets in Bioinformatics and Computational Biology: State of the art and Perspectivesctives”, P. Lio, Bioinformatics Review, 2003.”, P. Lio, Bioinformatics Review, 2003.
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ReferencesReferences
References on MR acquisition References on MR acquisition ►► References on MR protein classification References on MR protein classification ►► References on MR biometric recognition References on MR biometric recognition ►► References on MR References on MR ►► References on frames References on frames ►►
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Current ProjectsCurrent Projects
BioimagingBioimaging Efficient MR acquisition of fluorescence microscopy imagesEfficient MR acquisition of fluorescence microscopy images MR segmentation of multi-cell imagesMR segmentation of multi-cell images MR classification of proteins based on images of their MR classification of proteins based on images of their
subcellular locationssubcellular locations Automatic code generation for MR bioimaging algorithmsAutomatic code generation for MR bioimaging algorithms
BiometricsBiometrics MR identification/verification (fingerprints, faces, irises,…)MR identification/verification (fingerprints, faces, irises,…) Automatic code generation for MR biometric algorithmsAutomatic code generation for MR biometric algorithms
MR ToolsMR Tools FramesFrames Algebraic theory of signal processing (Algebraic theory of signal processing (SMARTSMART))
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ConclusionsConclusions
The “dream”:The “dream”:
automated, efficient and automated, efficient and reliable processingreliable processing
of large biosignal databasesof large biosignal databases
EmphasisEmphasis Introduction of MR toolboxIntroduction of MR toolbox Adaptivity and Adaptivity and
computational efficiency computational efficiency are keyare key
Computation
Knowledge Extraction
Acquisition
Systems Biology
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AcknowledgmentsAcknowledgments
Current PhD students
AminaChebira
TadMerryman
GowriSrinivasa
PhD students
DoruCristianBalcan
ElviraGarciaOsuna
PabloHenningsYeomans
JasonThornton
Collaborators
VijaykumarBhagavatula
GeoffGordon
JoséMoura
MarkusPüschel
MariosSavvides
BobMurphy
Undergrads
Woon HoJung
Funding
LionelCoulot
HeatherKirshner
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Supplementary MaterialSupplementary Material
Jelena KovačevićJelena Kovačević
Center for Bioimage InformaticsCenter for Bioimage InformaticsDepartment of Biomedical EngineeringDepartment of Biomedical EngineeringCarnegie Mellon UniversityCarnegie Mellon University
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ContentsContents
BioimagingBioimaging 3T3 data set3T3 data set SegmentationSegmentation Haralick texture featuresHaralick texture features
ComputationComputation Spiral detailsSpiral details
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3T3 Data Set3T3 Data Set
Cells from mouse embryoCells from mouse embryo Spinning Disk Confocal Spinning Disk Confocal
Microscope (60x)Microscope (60x) GFP for a specific proteinGFP for a specific protein
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SegmentationSegmentation
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Haralick Texture FeaturesHaralick Texture Features
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False accept rateFalse accept rate(FAR)(FAR)
False reject rateFalse reject rate(FRR)(FRR)
Equal error rateEqual error rate(EER)(EER)
Specificity/Sensitivity/Error RatesSpecificity/Sensitivity/Error Rates
Different jargon in different Different jargon in different communitiescommunities
SensitivitySensitivity
SpecificitySpecificity
DisorderDisorder
presentpresent absentabsent
Test Test resultresult
positivepositive aa bb
negativenegative cc dd
ClassClass
ownown otherother
Class. Class. resultresult
ownown aa bb
otherother cc dd
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SPIRALSPIRALCode Generation for DSP AlgorithmsCode Generation for DSP Algorithms
Transform: Transform: MatrixMatrix
Rules: Rules: Decompose transform into other onesDecompose transform into other ones
Formula: Formula: Uniquely represents the transformUniquely represents the transform
Code generation: Code generation: From formula produce C codeFrom formula produce C code