NERS/BIOE 481 Lecture 05 Radiographic Image Formation Michael Flynn, Adjunct Prof HenryFord Nuclear Engr & Rad. Science Health System mikef@umich.edu mikef@rad.hfh.edu RADIOLOGY RESEARCH
IV - General Model – xray imaging Xrays are used to examine the interior content of objects by recording and displaying transmitted radiation from a point source. DETECTION DISPLAY (A) Subject contrast from radiation transmission is (B) recorded by the detector and (C) transformed to display values that are (D) sent to a display device for (E) presentation to the human visual system. 2 NERS/BIOE 481 - 2019
IV.A – Geometric projection (9 charts) A) Geometric Projection 1) Transmission geometry 2) Magnification 3) Focal spot blur 4) Object resolution 3 NERS/BIOE 481 - 2019
IV.A.1 – Transmission geometry The radiographic image formation process projects the properties of the object along straight lines from a point- like source to various positions on a detector surface. • The recorded signal Flat detector reflects material properties encounted along each ray path. • Distortion of the object Object can occur if the detector surface is oblique Point-like source The radiographic projection is a ‘perspective transmissive projection’ from the point of view of the source. Object features close to the source are magnified as are visual objects close to the viewer eye. 4 NERS/BIOE 481 - 2019
IV.A.2 – Magnification, M The diverging path of source the x-rays caused the recorded signal, S, in relation to detector position, x d , to be d so magnified relative to the object size, x i . d sd object x i d d od sd M d so d d d so od so detector d S 1 od d x d so 5 NERS/BIOE 481 - 2019
IV.A.3 – Focal spot blur Penumbral blur: The size of the focal Focal spot f emission spot emission area causes points and edges to be blurred. d so Blur for detector dimensions: d od B f f M 1 d od fd d so Blur scaled to object dimensions: M 1 1 B fd B fo f f 1 M M 6 NERS/BIOE 481 - 2019
IV.A.4 – Object resolution In general, the detector will further blur the position of incident radiation. Blur scaled to object dimensions: do B B M d If the focal spot and the detector blur have Gaussian distributions, they convolve to a Gaussian system resolution, scaled to the object, B d with width, B o . 2 2 2 B B B o do fo 2 2 B 1 2 d f 1 M M 7 NERS/BIOE 481 - 2019
IV.A.4 – Object resolution Large focal spot size f = 1.0 mm B d = 0.5 mm 1.4 B o (M) 1.2 B fo (M) B do (M) Blur, B mm 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 M 8 NERS/BIOE 481 - 2019
IV.A.4 – Object resolution Small focal spot size f = 0.2 mm B d = 0.5 mm 1.4 B o (M) 1.2 B fo (M) Blur, B mm 1 B do (M) 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 M 9 NERS/BIOE 481 - 2019
IV.A.4 – Object resolution In general, for a given focal spot size and detector blur, there is a magnification that produces minimal resolution in object dimensions. This can be found by setting the derivative of B O with respect to M equal to 0. dB dB dB 2 2 2 fo B B B o do 2 B 2 B 2 B 0 o do fo o do fo dM dM dM B dB B d B do d do M 2 dM M dB 1 f fo B f 1 fo 2 M dM M 10 NERS/BIOE 481 - 2019
IV.A.4 – Object resolution B B 1 f d d f 1 0 2 2 M M M M Substituting the 2 B 1 expressions from the prior 2 d f 1 0 page and rearranging M M yields the simple solution 2 2 B f M 1 0 d that the best object 2 resolution is obtained when B d M 1 the magnification is equal f to 1.0 plus the square of 2 B the ratio of the detector d M 1 f blur to the focal spot size. Bd = 0.5 mm, f = 0.2 mm M = 7.25 Bd = 0.5 mm, f = 1.0 mm M = 1.25 11 NERS/BIOE 481 - 2019
IV.A.4 – Example, magnification image Agricultural Imaging http://www.faxitron.com/life-sciences-ndt/agricultural.html Faxitron X-ray Corp • MX-20 Digital • 20 m m focal spot • 5X magnification “Ultra-high resolution x-ray imaging is an important tool in seed and plant inspection and analysis. “ Sunflower Seed 12 NERS/BIOE 481 - 2019
IV.B – Primary Signal (10 charts) B) Primary Signal - Radiography 1) Attenuation 2) The projection integral 3) Ideal detector a) photon counting b) energy integrating 13 NERS/BIOE 481 - 2019
IV.B.1 – projection nomenclature To mathematically describe signal and noise, we will consider the signal associated with projection vectors, p , whose directions are defined by detector coordinates, (u,v) , or source angles (q,f) . u ^ p v (f,q ) 14 NERS/BIOE 481 - 2019
IV.B.2 – projection path variable Radiation traveling along the projection p enters the ^ object and will travel a distance T before exiting the object and striking the detector. We will consider a pathlength variable, t , which is 0 at the object entrance and T at exit. T 0 t dt ^ p 15 NERS/BIOE 481 - 2019
IV.B.2 – projection integral Radiation traveling through an object along the projection vector p will be subject to attenuation by various material encountered along the path t . 1 2 3 4 5 6 7 n = f o f t e 3 T ( t ) dt t n t t t e 1 e 2 e 3 e n e 0 o 16 NERS/BIOE 481 - 2019
IV.B.2 – the Radon transform T P r ( , ) ln ( ) t dt • The argument of the 0 o exponential factor describing the attenuation through an y object path is known as the Radon transform. t • It’s form is that of a r q generalized pathlength x integral of a density function. m (x,y) • The inverse solution to the Radon transform, i.e. m (x,y) as a function of P(r, q ) , is used in computed tomography. In the Radon transform equation above, the attenuation shown as a function of the projection path variable, m (t) , is more formally written as m (r ,q) or m (x,y) The line integral of m (t) , P(r, q ) , is referred to a a ‘Projection Value’. The set of all values obtained in one exposure is called a ‘Projection View. 17 NERS/BIOE 481 - 2019
IV.B.3 – Energy dependant incident fluence • The x-ray fluence on the detector will vary as a function of x-ray energy. T ( , ) t E dt P P e 0 ( E ) ( E ) o • The energy dependence of the linear attenuation coefficient effects the shape of the differential energy spectrum presented to the detector. 18 NERS/BIOE 481 - 2019
IV.B.3 – Spectrum incident on object 100 kV, tungsten target, NO tissue 19 NERS/BIOE 481 - 2019
IV.B.3 – Spectrum incident on detector 100 kV, tungsten target, 20 cm tissue 20 NERS/BIOE 481 - 2019
IV.B.3 – ‘ideal’ versus actual detectors • The signals recorded by actual detectors are determined in a complex manner by the energy dependant absorption in the target and background materials and the energy dependant absorption in the detector. • We consider now signals recorded by ‘ideal’ detectors. Later we will examine actual detectors 21 NERS/BIOE 481 - 2019
IV.B.3 – Ideal image detector – counting type • An ideal photon counting detector will accumulate a record of the number of photons incident on the detector surface. • The detected signal for an ideal photon counting detector, Sc, can be written as: E max S A t dE ( E ) c d 0 Where is the exposure time, sec • t • f is the photon fluence rate, photons/mm 2 /sec, • A d is the effective area of a detector element. 22 NERS/BIOE 481 - 2019
IV.B.3 – Ideal image detector – energy integrating type • An ideal energy integrating detector will record a signal equal to the total energy of all photons incident on the detector surface. • The detected signal for an ideal energy integrating detector, Se , can be written as: E max S A t E dE ( E ) e d 0 Where is the exposure time, sec • t • f is the photon fluence rate, photons/mm 2 /sec, • A d is the effective area of a detector element. The majority of actual radiographic detectors are energy integrating; however, they are not ‘ideal’. 23 NERS/BIOE 481 - 2019
IV.C – Radiation Noise & Stats (14 charts) C) Radiation Detection Noise - Statistical principles 1) Radiation counting & noise 2) Poisson/Gaussian distributions 3) Propagation of error 24 NERS/BIOE 481 - 2019
IV.C.1 - Radiation Counting A simple radiation detector may be used to make repeated measurements of the number of radiation quanta striking the detector in a specified time period. Radiation Radiation Detector source 25 NERS/BIOE 481 - 2019
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