m. wu: enee631 digital image processing (spring'09) medical imaging topic: radon transform...

M. 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]

ENEE631 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.

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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 …


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 detection

Illustration of Illustration of Hough TransformHough Transform

Figures from http://en.wikipedia.org/wiki/Hough_transform

(, ) space: x cos + y sin =


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 course

Figure from Gonzalez’s 2/e book resource


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)


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 peaks

Figures from Lowe’s IJCV 2004


Imaging [13]

Examples of Difference of GaussianExamples of Difference of Gaussian

Examples from SIFT tutorial notes by Estrada/Jepson/Fleet (2004)


Imaging [14]

Examples from SIFT tutorial notes by Estrada/Jepson/Fleet (2004)


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 blocks

SIFT: Rotational Invariant Keypoint Descriptor SIFT: Rotational Invariant Keypoint Descriptor

Figure from Lowe’s IJCV 2004


Imaging [16]

Boundary Boundary RepresentationRepresentation

Chain code4-way or 8-way connectivity

Features/measurement obtained from object’s chain code– Perimeter; Area; Centroid

Figure from Gonzalez’s book resource


Imaging [17]

Boundary DescriptorsBoundary Descriptors

Simple descriptors– boundary length, diameter– curvature (rate of change of slope/tangent); B-splines

Shape number– The 1st difference of smallest magnitude of chain code

to normalize chain code’s start point and orientation

Fourier descriptor– Record point coordinates w/ a sequence of complex #– DFT coeff. of the complex sequence as Fourier descriptors– First several coeff. represent coarse info of the shape

Statistical moments– mean, variance, and higher-order moments of boundary

segments and/or 1-D representation of boundary e.g. r()


Imaging [18]

Radon Transform and its Applications in Radon Transform and its Applications in

Medical Image ProcessingMedical Image Processing


Imaging [19]

Non-Intrusive Medical DiagnosisNon-Intrusive Medical Diagnosis

(From Jain’s Fig.10.1)


Imaging [20]

Non-Intrusive Medical Diagnosis (cont’d)Non-Intrusive Medical Diagnosis (cont’d) Observe a set of projections (integrations) along different angles of a

cross-section– Each projection itself loses the resolution of inner structure– Types of measurements

transmission (X-ray), emission, magnetic resonance (MRI)

Want to recover inner structure from the projections– “Computerized Tomography” (CT)

(From Bovik’s Handbook Fig.10.2.1)

Emission tomography: measure emitted gamma rays by the decay of isotopes from radioactive nuclei of certain chemical compounds affixed to body parts.

MRI: based on that protons possess a magnetic moment and spin. In magnetic field => align to parallel or antiparallel. Apply RF => align to antiparallel. Remove RF => absorbed energy is remitted and detected by RF detector.


Imaging [21]

Radon TransformRadon Transform

A linear transform f(x,y) g(s,)– Line integral or “ray-sum”– Along a line inclined at angle

from y-axis and s away from origin

Fix to get a 1-D signal p(s)= g(s,)

dxdysyxyxfsg )sincos(),(),(

rotation) e(coordinat cossin

sincos where

)cossin,sincos(

y

x

u

s

duususf



Imaging [22]

Example of Image Radon TransformExample of Image Radon Transform

(From Matlab Image Processing Toolbox Documentation)

[Y-axis] distance, [X-axis] angle


Imaging [23]

Inverting A Radon TransformInverting A Radon Transform

To recover inner structure from projections

Need many projections to better recover the inner structure

Reconstruction from 18, 36, and 90 projections (~ every 10,5,2 degrees)

(From Matlab Image Processing Toolbox Documentation)


Imaging [25]

Rotational Invariance Property of Fourier TransformRotational Invariance Property of Fourier Transform

Consider a 2-D continuous signal & its 2-D Fourier transform

2-D FT is rotationally invariantFT of a rotated version of function f(t1, t2)is just to apply same rotation to F(1, 2) ,

cossin

sincos where

)cossin ,sincos(

) (),g(

2

1

2121

21

t

t

ttttf

ftt

tR

tR

)cossin ,sincosF()F(

])( exp[)(

1, note .] exp[)(

. ],[let . ] exp[) (

)](exp[),(),G(

2121

111

21

2122112121

RΩ

ttRΩt

RRRtRtRΩt

tRtttΩtR

djf

djf

ttdjf

dtdtttjttg

T

TT

T


Imaging [26]

Connection Between Radon & Fourier TransformConnection Between Radon & Fourier Transform

Observations– Look at 2-D FT coeff. along horizontal frequency axis

FT of 1-D signal 1-D signal is vertical summation (projection) of original 2-D

signal

– Look at FT coeff. along = 0 ray passing origin

FT of projection of the signal perpendicular to = 0

Projection TheoremProof using FT definition

& coordinate transform

(Figure from Bovik’s Handbook Fig.10.2.7)


Imaging [27]

Projection Theorem Projection Theorem (a.k.a. Projection-Slice Theorem)(a.k.a. Projection-Slice Theorem)

Let F(1, 2) be the FT of a cont’s function f(t1, t2)

Let Radon transform of f be given as p(s) with 1-D FT P()

Projection Theorem: P() = F( cos , sin ) for all i.e. FT of a slice is the corresponding slice of the 2-D FT– We can obtain the full 2-D FT in terms of 1-D freq. components of

the Radon transform at all angles.

duususfs

t

t

t

t

)cossin ,sincos()(p

u

s

cossin

sincos

u

s

u

s

2

1

2

1

RR

)sin ,cosF(

),(u)(s, note .)]sincos( exp[),(

]exp[ )cossin ,sincos(s

]exp[)()(P

21212121

ttdtdtttjttf

dudssjusuf

dssjsp


Imaging [28]

Inverting Radon by Projection TheoremInverting Radon by Projection Theorem

(Step-1) Filling 2-D FT data matrix with 1-D FT of Radon along different angles

(Step-2) 2-D IFT

Need Polar-to-Cartesian grid conversion for discrete scenarios– May lead to artifacts esp. with less densely sampled slices



Imaging [29]

Back-ProjectionBack-Projection

Sum up Radon projection along all angles passing the same pixel location (x, y)

dyxgyxf ),sincos(),(~

0


dxdysyxyxfsg )sincos(),(),(


Imaging [30]

Back Projection: ExampleBack Projection: Example


Imaging [31]

Back-projection = Inverse Radon ?Back-projection = Inverse Radon ?

Not exactly ~ Back-projection gives a blurred recovery– Intuition: most contribution is from the pixel (x,y), but still

has some tiny contribution from other pixels– B ( R f ) = conv( f, h1 )– Bluring function h1 = (x2 + y2)-1/2, FT( h1 ) ~ 1 / ||

where radial frequency 2 = x2 + y

2 (reflect how far from DC)

Need to apply inverse filtering to fully recover the original– Inverse filtering for “sharpening”: multiplied by || in FT

domain


Imaging [32]

Inverting Radon via Filtered Back ProjectionInverting Radon via Filtered Back Projection

yxyxyx ddyxjFyxf )](2exp[),(),(

f(x,y) = B H g

0

2

0 0

)]sincos(2exp[),(||

)]sincos(2exp[),(

ddyxjF

ddyxjF

polar

polar

Change coordinate(Cartesian => polar)

deGsgdyxg

ddyxjG

sj2

0

0

),(||),(ˆ where),sincos(ˆ

)]sincos(2exp[),(||

Projection Theorem( F_polar => G)

Back Projection (FT domain) filtering



Imaging [34]

Other Scenarios of Computerized TomographyOther Scenarios of Computerized Tomography

Parallel beams vs.Fan beams– Faster collection of

projections via fanbeams

involve rotationsonly

Recover from projections contaminated with noise– MMSE criterion to minimize reconstruction errors

See Gonzalez 3/e book and Bovik’s Handbook for details




Imaging [35]

Explore Imaging Model and SparsityExplore Imaging Model and Sparsity

Recall the talk on Model-based Imaging by Dr. Bouman

Exploit imaging model to improve SNR– Exploit sparsity in image data

– E.g. use L1 norm instead of L2 as in compressed sensing

=> be aware sparsity with artificial model like head phantom may not completely reflect the real data


Imaging [36]

Summary of Today’s LectureSummary of Today’s Lecture

Gradient based image features

Medical imaging topic: computerized tomography– Radon transform and Inverse Radon transform

by Projection Theorem by Filtered Back Projection

Next week: 2-D sampling theory

Readings– Stable Gradient Features ~ SIFT

David G. Lowe (2004). "Distinctive Image Features from Scale-Invariant Keypoints". International Journal of Computer Vision 60 (2): 91–110.

– Boundary representation: Gonzalez’s 3/e book 11.1 – 11.2

– Radon transform: Gonzalez 3/e book 5.11[More readings] Bovik’s Handbook Sec.10.2; Jain’s Sec.10.1-10.6, 10.9

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m. wu: enee631 digital image processing (spring'09) medical imaging topic: radon transform...

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