Department of Computer Science Medical Imaging Mohammad Dawood Department of Computer Science University of Münster Germany

Recap

α decay β- decay β+ decay Gamma scintigraphy PET Sinogram

Ultrasound / Sonography

The frequency range of sound above 20kHz is known as ultrasound Sound spectrum The frequency range of sound above 20kHz is known as ultrasound

Sound wave propagation

Ultrasound

Ultrasound

Ultrasound - is produced through the conversion of electrical energy into mechanical energy - is detected by the reverse process, by converting mechanical energy into electrical energy. - The transducer is a device that is both a transmitter and receiver of the ultrasound signal and it serves a dual role in pulse echo imaging.

Inge Edler und Helmut Hertz (1954)

Ultrasound

Ultrasound Reflexion Scatter Transmission

Ultrasound Resolution Depth Frequency

Ultrasound

Ultrasound

Medium Speed of sound m/s Air 331 Water 1483 Tissue 1540-1595 Liver 1549 Blood 1570 Glycerin 1923 Bones (Compact) 3600

Image Reconstruction

Reconstruction Tomography

Reconstruction Law of Attenuation

Reconstruction Parallel projections of a plane

y Reconstruction Radon Transformation s f r n θ x

Reconstruction Radon Transformation (Line Integrals at different angles)

Reconstruction Radon Transformation Original Sinogram (Radon Transform)

Reconstruction Inverse Radon Transformation H: Hilbert transform

Reconstruction Inverse Radon Transformation Problems with Missing data and Noise!

Reconstruction Filtered Back Projection

Reconstruction Filtered Back Projection

Reconstruction FBP: Commonly used filters 1=Ram-Lak (ramp), 2=Shepp-Logan, 3=Cosine, and 4=Hamming

Reconstruction Filtered Back Projection 2D/3D filtering is costly Projections Backproject Filter 1D Filter 2D Image

Reconstruction Fourier slice theorem Take a two-dimensional function f(r), project it onto a line, and do a Fourier transform of that projection Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin parallel to the projection line

Reconstruction Fourier Slice Theorem

Reconstruction Iterative Reconstruction b: measured values x: unknown attenuation coefficients aij: weights f1 f2 … fn LOR1 LOR2 LORn

Reconstruction Iterative Reconstruction Kaczmarz Method (=ART: Algebraic Reconstruction Technique)

Reconstruction Iterative Reconstruction Kaczmarz Method (=ART: Algebraic Reconstruction Technique) 1. Start by setting x(0) = 0 2. Compute the forward projection from the n-th estimate, i.e. b(n) = A x(n) 3. Choose i and correct the current estimate x(n) 4. Iterate steps 2,3 until the difference between new forward projection b(n), computed in 2, and the old one is below tolerance

Reconstruction Iterative Reconstruction EM (Expectation Maximization)