An Introduction: Radon Transform, X-ray Transform, Inverse Problems

Transcription

1 Other applications: SPECT and Attenuated An Introduction:, X-ray Transform, TING ZHOU Northeastern University January 9, 2018

2 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

3 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

4 Other applications: SPECT and Attenuated References The Mathematics of by Frank Natterer

5 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

6 Other applications: SPECT and Attenuated Tomography and Imaging Derived from Greek words "τ oµoσ" (tomo)= slice and graphein = to write. Diagnostic radiology: the imaging of cross sections of the human body. Applied in medicine (diagnostics and treatment), biology, geophysics, archeology, astronomy, material science, industry, oceanography, atmospheric sciences, homeland security... (Non-intrusive, non-destructive)

7 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated Medical Imaging (CT scan; MRI; Ultrasound; Photoacoustics) Why do we need so many imaging methods? T ING Z HOU Northeastern University

8 Other applications: SPECT and Attenuated Synthetic-Aperture Radar (SAR) The surface of Venus, as imaged by the Magellan probe using SAR.

9 Other applications: SPECT and Attenuated Geophysics and Seismology "The new studies rely on a computer-intensive technique called whole waveform tomography, which yields the highest resolution images ever of Earth s interior. Whereas previous studies only used the initial bursts of energy from an earthquake, the new ones now incorporate information from all the squiggles in a seismogram." - Science Journal

10 Other applications: SPECT and Attenuated Computerized (Assisted) Tomography CAT Scan inventors Cormack and Hounsfield built the first CT scanners in 1960s, won 1979 Nobel Prize in medicine.

11 Other applications: SPECT and Attenuated CT scanners Cormack s scanner Modern scanners Cost: $300 Cost: $1,500,000

12 Other applications: SPECT and Attenuated CT = X-ray Tomography A cross-section (2D) of the human body is scanned by X-ray beams. The X-ray beam is attenuated when passing through the body.

13 Other applications: SPECT and Attenuated Beer s law The relative intensity loss of X-ray is proportional to the distance it travels. I(x) intensity of X-ray along the beam line L at point x. L Beer s law: f (x) dx for "all" L di I = f (x) }{{} attenuation coefficient / density at x f (x) dx = ln I(x) receiver:x 1 = ln(i source:x 1 /I 0 ) 0 L }{{} known! Processor = f (x) Plot = tomogram dx

14 Other applications: SPECT and Attenuated Beer s law The relative intensity loss of X-ray is proportional to the distance it travels. I(x) intensity of X-ray along the beam line L at point x. L Beer s law: f (x) dx for "all" L di I = f (x) }{{} attenuation coefficient / density at x f (x) dx = ln I(x) receiver:x 1 = ln(i source:x 1 /I 0 ) 0 L }{{} known! Processor = f (x) Plot = tomogram dx

15 Other applications: SPECT and Attenuated Transport of photons/particles (A general model) u(x, ω) density of particles at x moving in direction ω S 1. ω x u(x, ω) + f (x)u(x, ω) = σ(s, ω, ω)u(x, ω ) dω + s(x). S 1 f (x) - attenuation coefficient (density of the tissue); σ - scattering coefficient; s(x) - source density. In absence of scattering and source, one get s Beer s law. Optical tomography: fluorescence tomography etc. Highly scattering medium: diffusive optical tomography.

16 Other applications: SPECT and Attenuated CT (square body) Line integrals of f reconstruction of f

17 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

18 Other applications: SPECT and Attenuated in Imaging (inverse v.s. direct) Direct problem: Input signal = A = Output signal =? Inverse problem: Input signal = A =? = Output signal e.g. X-ray beam L = f =? = L f (x) dx

19 Other applications: SPECT and Attenuated Interesting questions for Uniqueness Inversion/Reconstruction (Optimization v.s. direct inversion) Stability Range of the direct operator Incomplete data Contrast* Resolution*

20 Other applications: SPECT and Attenuated Examples of Other Electrical Impedance Tomography (EIT) Inverse Scattering

21 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

22 Other applications: SPECT and Attenuated The R of f (x) in R 2 : Rf (L) := f (x) dx for all lines L R 2 L Can we reconstruct f (x) from Rf?

23 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

24 Other applications: SPECT and Attenuated 3D object and X-ray transform To image a medium in R 3 using CT scan, Layer by layer of 2D CT scan; X-ray transform X of f (x) along line L in R 3 : Xf (L) := f (x) dx. ( and X-ray transform differ in dimension n 3.) L

25 Other applications: SPECT and Attenuated Outline 1 Course Information (syllabus, course webpage) Other applications: SPECT and Attenuated

26 Other applications: SPECT and Attenuated SPECT: Single Photon Emission Computed Tomography A nuclear medical tomographic imaging technique using gamma rays, with a true 3D reconstruction (v.s. cross-sectional slices 2D reconstruction). The γ-ray density solves the equation ω x u(x, ω) + a(x)u(x, ω) = f (x). Here a(x) is the known attenuation coefficient and f (x) is the unknown source. The inverse source problem consists of reconstructing f (x) from u out (x, ω).

27 Other applications: SPECT and Attenuated Attenuated Knowing u out (x, ω), one can compute the AtRT R a of the source f (x) given by R a f (L) := f (x)e L(x) a(y) dy dx where L(x) is the line section from x to the receiver on L. L

28 Other applications: SPECT and Attenuated of Attenuated The inverse problem consists then in answering the following questions: 1 Knowing the AtRT R a f (L) for all L and a(x), what can we reconstruct in f (x)? What if f also depends on ω? 2 Can we reconstruct f (x) from half of the measurements or do we at least have uniqueness of the reconstruction? 3 Do we have a reliable numerical technique to obtain fast reconstructions?