To Bin or Not to Bin?Using Mathematics to Improve CAT scans

First X-ray Sketch of first CT scanner Modern Chest CAT scan

Angel R. Pineda

First X-ray Sketch of first CT scanner Modern Chest CAT scan

Angel R. PinedaCalifornia State University at Fullerton

Mathematics Department

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Fullerton CollegeNovember 17, 2009

Mathematics of Medical Imaging (MoMI)

C t G M bCurrent Group Members: Emily Bice (graduate student working on chemical species separation in MRI)

Kevin Park (undergraduate working on mathematical methods for accelerating MRI)

Selected Former Group Member: Joaquin Alvarado* (undergraduate working on Cramer-Rao Bounds in MRI)

2* Fullerton College graduate!

Acknowledgements

• Norbert Pelc and Rebecca Fahrig at Stanford• Norbert Pelc and Rebecca Fahrig at Stanford University

• Harry Barrett at University of Arizona

• Jeff Siewerdsen and Daniel Tward at JohnsJeff Siewerdsen and Daniel Tward at Johns Hopkins University

Avinash Kak and Malcolm Slaney• Avinash Kak and Malcolm Slaney

• Wikipedia

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Outline

• How do CAT scans work?

• What is the noise in CT images?

• How to find a tumor?

• To Bin or Not to Bin?

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Nobel History

• In 1901 Roentgen gets first Nobel Prize g gin Physics for the discovery of X-rays. An immediate application is medical projection imaging.

• In 1979 Hounsfield and Cormack• In 1979 Hounsfield and Cormack (mathematician) get Nobel Prize in medicine for X-ray Computed ed c e o ay Co pu edTomography which allows for cross-sectional imaging.

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C-Arm Flat Panel CT Scanner

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X-ray Projections (2-D Radon Transform)

From Kak and Slaney Principles of

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From Kak and Slaney, Principles of Computerized Tomographic Imaging

Inverting the Radon Transform

• Take the Fourier Transform of Projections

• Multiply by Judicious Weighting Function (Filter)

• Smear Back Filtered Projections (Backproject)

• Modern reconstruction in CT is based on Filtered Backprojection (Note: Hounsfield iteratively solved the linear system.)

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Show MATLAB Simulation of CT

Image with 60 projections Image with 120 projections Image with 180 projectionsg p j g p j g p j

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Warhol Cat Scan of the Cheshire Cat

Image with 60 projections Image with 120 projections Image with 180 projectionsg p j g p j g p j

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Noise in X-Ray Projections

• X-ray source produces x-rays with a Poisson y p ydistribution

• The body attenuates (absorbs) x rays by• The body attenuates (absorbs) x-rays by binomial sampling, such that x-rays exiting the body are still Poisson distributedthe body are still Poisson distributed

• Noise is important because the more X-ray you use the less noise you get but the moreyou use, the less noise you get but the more likely you are to cause cancer in the patient.

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Cartoon of arriving X-rays

Body

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Noise in Reconstructed Images

For a 2D CT system with continuous ydata and stationary noise:

2 2 2constant | | | ( ) |recon f h f dfσ+∞

∞= ∫−∞∫

where f is spatial frequency and p q yh(f) is the filter function.

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Binning

x-rays

pixels

binned pixels

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Same Results for Noise-Free Data

ReconstructionWith Binned Data

Reconstruction With High Resolution Data

Slices of Both Reconstructions

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Different Results for Noisy Data!

Reconstruction Reconstruction With

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With Binned Data High Resolution Data

Different Results for Noisy Data!

ReconstructionWith Binned Data

Reconstruction With High Resolution Data

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With Binned Data High Resolution Data

Two paths to a reconstructed Images edFilter (binned)Bin*

ectio

ns

stru

cteFilter (binned)Bin*

Proj

e

Rec

ons

Imag

e

Filter (not binned) R I

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*Note that binning is not invertible

Variance Results

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Importance of Noise Correlation• Detectability changes as correlation changes

Objectj

Task 1: Tumor Detection (2AFC)

Left RightLeft Right

In which of these two images is there a tumor?

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Task 2: Tumor Detection (2AFC)

Left RightLeft Right

In which of these two images is there a tumor?

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Task Performance (Fraction Correct)

Issues:

•Depends on Object.•Is a Random Measurement.•Depends on Contrast.•Will be done by a humanWill be done by a human.

Solution:

Mathematical Modeling of Human Observers

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Contrast Sensitivity

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Eye Filters and Detection Theory

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Linear Template

• Filter images and nodules based on model for

( ) DE D Dα β−

frequency response of human eye

( ) DEye D D eα β−=h β t d D i thwhere α, β are parameters and D is the

magnitude of the Fourier coefficients in l dcycles per degree.

†w E Es=26

w E Es=

Test Statistic

††t w g= gt i th t t t ti ti• t is the test statistic

• w is the mathematical templatei th i d t• g is the image data

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Distribution of the Test-Statistic

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Results Large Object

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Results Small Object

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Conclusions

CAT scans have lots of fun mathCAT scans have lots of fun math, modeling, statistics and computing

&

Just Say No to Binning!

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Thank You For Listening

• Any Questions?Any Questions?

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