Lecture 8: Propagation of Waves II in Subsurface Sensing and Imaging Systems, Study notes of Algorithms and Programming

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ECSE-4962 Introduction to Subsurface Sensing and Imaging Systems Lecture 8: Propagation of Waves II Kai Thomenius1 & Badri Roysam2 1Chief Technologist, Imaging Technologies, General Electric Global Research Center 2Professor, Rensselaer Polytechnic Institute Center for Sub-Surface Imaging & Sensing Outline of Course Topics • THE BIG PICTURE – What is subsurface sensing & imaging? – Why a course on this topic? • EXAMPLES: THROUGH TRANSMISSION SENSING – X-Ray Imaging – Computer Tomography – Intro into Optical Imaging • COMMON FUNDAMENTALS – propagation of waves – interaction of waves with targets of interest • PULSE ECHO METHODS – Examples • MRI – A different sensing modality from the others – Basics of MRI • MOLECULAR IMAGING – What is it? – PET & Radionuclide Imaging • IMAGE PROCESSING & CAD Acoustic vs. EM Waves 3.0x108 m/s in vacuum1,500 m/s in waterVelocity of propagation Characteristic impedance Velocity of propagation no medium necessary (ether) elastic mediumTransmission requirements electromagnetic waveslongitudinal or shear mechanical waves Wave types EM wavesAcoustic waves c = 1 ρoK c = 1 με Z = ρc Z = μ ε Amazingly similar (for the first-order wave equation) Analytic Solutions to Wave Equations In some simple cases, people have found closed-form solutions, for example: ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − −+−= s t tt rc aR θ ρρ ρρ κ κκπ cos 2 33 3 2 :sourcepoint a todue field Scattered 3 Κ is the compressibility or source, ρ is the density of the source. In most other cases, we need to resort to numerical solution Approximate Analytic Solutions • In a general situation, it can be very hard to solve the wave equation analytically • For some values of wavelengths and distances, it is possible to obtain good approximations to the full wave equation • Examples: – Near-field zone (Fresnel approximation) – Far-field zone (Fraunhofer approximation) – We’ll get to them in a bit Simple Wave-Matter Interactions • Attenuation / absorption – Change in amplitude/energy of the wave • Reflection – Waves bouncing off surfaces (echoes) • Refraction – Waves changing direction • Diffraction – Spreading of waves creating deviation from geometric paths • Change in propagation speed – Waves change phase • Scattering – Waves are redirected in many directions • Dispersion – Different frequency waves traveling at different speeds through the medium • Doppler – Change in frequency caused by interaction with a moving object Contrast Generation • Each of these types of interactions is potentially a source of “imaging contrast” – Changes in properties of the medium as we go from one point to another can be “revealed” by detectable differences in wave-medium interactions • Contrast agents – These are artificial substances that we can often inject – They produce and/or enhance contrast max min max min Weber's Contrast Formula Michelson's Formula = background background I I I I I I I − = − + Attenuation • Usually measured in units of “decibels (dB)” 2 1 2 2 2 1 1 Watt/cm 10 Watts/cm Attenuation = 10 log 10 log10 10 I I I dB I = = × = × = • Attenuation often changes by frequency of the wave – The attenuation spectrum is characteristic for a given medium, i.e., a spectral “signature” Attenuation frequency Differential attenuation = A source of contrast Absorption based Contrast Agents Normal X-ray With contrast agent injected into blood vessels Difference http://imaging.eng.uci.edu Sample Numbers for Diagnostic Ultrasound Waves 47.515.0 6 8 12 17 30 Imaging Depth (cm) 5.010.0 3.87.5 2.55.0 1.83.5 1.02.0 Attenuation coefficient for soft tissue (dB/cm) Frequency (MHz) Higher-frequency waves can resolve smaller objects, but Penetrate less than lower-frequency waves Attenuation is not all bad! • Often we want to attenuate waves deliberately • Such highly attenuating materials are called “dampers” –We’ll discuss them further when we talk about ultrasound imaging systems Transmission at Interfaces • Some points on transmission: – If Z2 = Z1, there is100% transmission. • Hence, an impedance mismatch is needed to get a reflection. – If Z2 > Z1, there is no polarity change, however, if Z2 < Z1, the echo will be inverted. • This is necessary for there to be continuity in values at the boundary. – Upper video clip: Z1 / Z2 = 0.5 – Lower video clip: Z1 / Z2 = 2.0 • For a nice discussion of this, check: http://physics.usask.ca/~hirose/ep225 /animation/reflection/anim- reflection.htm T = 2Z1 Z2 + Z1 1T R+ = Refraction at Interfaces • Refraction – From Latin “to turn aside” – At interfaces of media with differing propagation speeds. – Only occurs for oblique incidence – Changes the direction of the wave Reflection & Refraction Diffraction http://lectureonline.cl.msu.edu/~mmp/kap13/cd372.htm 1 2 1 2 1 2 1 2 v v i i v v i i < ⇒ < > ⇒ > Diffraction • Diffraction – Sommerfeld’s (1894) definition: • “… any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction.” • Closed-form solutions are available for simple cases • In practical complex situations, we have to resort to solving the wave equation Diffraction http://lectureonline.cl.msu.edu/~mmp/kap13/cd372.htm Scattering • Waves redirected in many directions • Usually – Scattering intensity is weaker than reflection – Increases with frequency – Specific in terms of angular distribution – Dependent upon the size and shape of scatterers relative to λ • In ultrasound, scattering permits imaging of tissue boundaries that are not perpendicular to the incident wave Surface Backscatter Forward Scatter Dispersion • Refractive index / speed of the wave is a function of wave frequency – Basis for prisms – Key to spectral imaging ( ) ( ) ( ) 0 cv n dn d λ λ λ λ = < Doppler Example: STATIONARY SOURCE (transducer) MOVING LISTENER (red blood cell) vcosθ v θ Particle passes through approaching wavefronts Stationary particle sees cT/λ wavefronts in time T Moving particle sees (c+vcosθ)T/λ wavefronts in time T The frequency our particle sees is given by λ θ+= cosvcfs fs = 1+ vcosθ c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ f 0 where λ = cf 0 Original transmit frequency Physical Interactions of EM waves with Matter Mostly passes throughChange of nuclear configuration100pm and smaller (gamma rays) At these shorter wavelengths, photons can actually disrupt the absorbing molecule by photodissociation or even produce photoionization of individual atoms. (1000-angstrom photons will photoionize electrons in the outer shells, whereas 100-angstrom or shorter photons will photoionize electrons in the inner shells.) Change of electron distribution10nm – 100pm (X-ray) Changes in electronic states of atoms in the molecule produce changes in electric dipoles of the atoms, that interact with the applied wave Change of electron distribution (electric component of EM wave more important) 1μm – 10nm (visible - ultraviolet) Mostly vibrations, rotations, and bending of molecules while it still remains in its electronic ground state. The molecule must be asymmetric. Vibrations need more energy than rotations (20 μm or shorter). Change of configuration (electric component of EM wave more important) 100μm – 1μm (Infrared) Mostly rotational effectsChange of orientation (rotation) (electric component of EM wave more important) 1cm – 100μm (Microwaves) Electron Spin Resonance: Electrons absorb/emit based in their spin property Change of electron spin (magnetic component of EM wave more important) 1m – 1 cm (Radio Frequency) Nuclear Magnetic Resonance: Nucleons absorb/emit based on their spin property Change of nuclear spin (magnetic component of EM wave more important) 10m – 1meter (Radio Frequency). CommentType of interactionWavelength Range Recap of the Lecture • Wave propagation at the heart of sensing and imaging systems –Differential wave-matter interactions are a primary source of imaging contrast • We have noted some parallels between acoustic waves and electromagnetic waves –Similar wave equations –We’ll discuss differences, in greater depth later –They are the basis for substance-specific imaging Homework for Lecture 8 • Using the data in slides 13 and 14, find the optimal x-ray energies (voltages) that maximize the contrast between each pair of the materials (bone-muscle, air-muscle, water-air,…etc) • Repeat the above exercise keeping in mind that higher energy x-rays are more damaging. – Assume that damage to tissue is proportional to energy, and determine the optimal energy levels for each of the above cases