Medical Imaging Topic: Radon Transform & Inverse Radon Transf.Overview and LogisticsENEE631 End of Semester Plan & UpdateIllustration of Hough TransformPattern MatchingMore on Gradient-based Salient Feature PointsSIFT: Employ “Scale Space Extrema”Examples of Difference of GaussianSlide 14SIFT: Rotational Invariant Keypoint DescriptorBoundary RepresentationBoundary DescriptorsRadon Transform and its Applications in Medical Image ProcessingNon-Intrusive Medical DiagnosisNon-Intrusive Medical Diagnosis (cont’d)Radon TransformExample of Image Radon TransformInverting A Radon TransformRotational Invariance Property of Fourier TransformConnection Between Radon & Fourier TransformProjection Theorem (a.k.a. Projection-Slice Theorem)Inverting Radon by Projection TheoremBack-ProjectionBack Projection: ExampleBack-projection = Inverse Radon ?Inverting Radon via Filtered Back ProjectionOther Scenarios of Computerized TomographyExplore Imaging Model and SparsitySummary of Today’s LectureM. Wu: ENEE631 Digital Image Processing (Spring'09)Medical Imaging Topic: Medical Imaging Topic: Radon Transform & Inverse Radon Transf.Radon Transform & Inverse Radon Transf.Spring ’09 Instructor: Min Wu Electrical and Computer Engineering Department, University of Maryland, College Park bb.eng.umd.edu (select ENEE631 S’09) [email protected] Spring’09ENEE631 Spring’09Lecture 24 (4/29/2009)Lecture 24 (4/29/2009)M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [2]Overview and LogisticsOverview and LogisticsLast Time:–Non-intrusive image forensics–Useful image features and feature extraction techniquesProjection based: convert 2-D data to 1-D profileParameter space: detect line & other structure with low computation complexityGradient based: corner detection via eigen properties of derivative matrixToday:–More on gradient based image features –Radon transform and applications in medical image processingCourse LogisticsProject progress report: Due next Monday 5pm May 4 by emailProject Topic 2: start with 1-2 small datasets and 2-3 types of features; make critical evaluation/comparison; do your own feature extractions in final submission; can use and acknowledge open-source classifier.UMCP ENEE631 Slides (created by M.Wu © 2004)M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [3]ENEE631 End of Semester Plan & UpdateENEE631 End of Semester Plan & Update–Assignments 20% => 25%4 Homework, 5% eachNo final => 5th assignment (no programming) => 5%–Projects 45% => 50%Project 1 on wavelet image coding ~20%Project 2 ~25% => 30%– Exams 35% => 20%In-class midterm exam ~20%Final take-home exam ~15% => no final–Base points (and extra for active class participations) => 5%Project presentation & demo: Thurs. 5/21 …M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [4]–For each point (x0, y0), lines of all angles passing it form a sinusoid curve in (, ) space –(, ) curves corresponding to colinear points intersect at a point (0, 0)=> useful for line detectionIllustration of Illustration of Hough TransformHough TransformFigures from http://en.wikipedia.org/wiki/Hough_transform(, ) space: x cos + y sin = M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [6]Pattern MatchingPattern MatchingMinimum distance classifier: compare with mean feature vector of each class (if known or can be learned)Matching by correlation–Reduce sensitivity via normalized correlation (correlation coeff.)–Exhaustive search for size & orientations ~ computationally expensive–Learn more on pattern classificationOverview: Chapter 12 of Gonzalez’s bookENEE731 Statistical Pattern Recognition; CS Machine Learning courseFigure from Gonzalez’s 2/e book resourceM. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [9]More on Gradient-based Salient Feature PointsMore on Gradient-based Salient Feature PointsRecall operations for robust edge detection ~ e.g. Canny–Laplacian of Gaussian (LoG) [Gonzalez 3/e 10.2.6]effectively bandpass filtering to suppress noise when taking derivatives–Ridge of gradient magnitude and edge linkingLocal extrema of gradient; seek stable edge infoRecall multiresolution analysisAchieve invariance to scaling and rotation–Take account of multiple resolution scale levels–Represent orientation w.r.t. dominant or canonical direction=> Scale Invariant Feature Transform (SIFT) by D. LoweGive about 2000 stable “keypoints” for a typical 500 x 500 imageEach keypoint is described by a vector of 4 x 4 x 8 = 128 elements(over 4x4 array of 8-bin gradient histograms in keypoint neighborhood)M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [12]SIFT: Employ “Scale Space Extrema”SIFT: Employ “Scale Space Extrema”Examine Differences of Gaussian filtered images at nearby scale  and kAvoid low contrast & poorly defined DoG peaksFigures from Lowe’s IJCV 2004M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [13]Examples of Difference of GaussianExamples of Difference of GaussianExamples from SIFT tutorial notes by Estrada/Jepson/Fleet (2004)M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [14]Examples from SIFT tutorial notes by Estrada/Jepson/Fleet (2004)M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [15](weighted) Gradient orientation histogram near a stable DoG extrema–Peaks in histogram correspond to dominant orientation–Also include other directions close to the peak as keypoints for higher stability–Measure properties of a keypoint relative to its assigned orientation to gain rotational invariance Keypoint Descriptor records gradient pattern of each keypoint–As a set of 8-bin orientation histograms over 4x4 nearby blocksSIFT: Rotational Invariant Keypoint Descriptor SIFT: Rotational Invariant Keypoint Descriptor Figure from Lowe’s IJCV 2004M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec 24 – Radon Transf / Medical Imaging [16]Boundary Boundary RepresentationRepresentationChain code4-way or