BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods X-ray Imaging Part 1

Today X-ray imaging equation Spatial resolution –Sampling –Limitations Contrast in projection radiography

X-ray Imaging We start with a simple system –Area, parallel-beam X-ray source –Monochromatic X-rays –Area detector –Uniformly-attenuating subject Subject X-rays Detector

X-ray Imaging The imaging equation is: Subject X-rays Detector I0I0  g

X-ray Imaging If the subject is not uniform X-rays Detector I0I0 g(x,y) Subject  x,y,z 

X-ray Imaging If the source is not uniform X-rays Detector I 0 (x,y) g(x,y) Subject  x,y,z 

X-ray Imaging If the source is polychromatic X-rays Detector I 0 (x,y,h ) g(x,y) Subject  x,y,z,h 

X-ray Imaging If the detector has nonuniform sensitivity X-rays Detector I 0 (x,y,h ) g(x,y) Subject  x,y,z,h 

X-ray Imaging This is complicated – How do we simplify it? X-rays Detector I 0 (x,y,h ) g(x,y) Subject  x,y,z,h 

X-ray Imaging For our purposes, we will use a simplified version: BUT we need to be aware of our assumptions 1. 2.

Example Problem A nonuniform object as shown is imaged. What is its contrast in the detected image? X-rays Detector 10 cm 3 cm  =.01 cm -1  =.1 cm -1

A Bit on Sampling (see P&L 2.8) A pixelized image is sampled A fundamental result from Nyquist tells us the highest spatial frequency that can be represented in a sampled image: where p is the pixel spacing (in spatial units)

Nyquist In the frequency domain, a sampled signal’s power spectrum is reproduced at intervals of  x and  y (the pixel spacing in each dimension) Assume our original signal is bandlimited i.e., its frequency content is zero for

Nyquist Then, the spectrum of the sampled signal looks like this (board illustration):

Nyquist If the sampled signal is not bandlimited, what happens? This is aliasing – it is bad. Deal with it by using a low-pass anti-aliasing filter before sampling

Key Result Sampling limits spatial resolution. The highest spatial frequency you can represent is 1/2p, where p is pixel size. The FWHM is therefore around 2p.

Resolution What factors affect the PSF of an X-ray system? First, what do we mean by the PSF of this system? Detector Subject  x,y,z,h  Source

Resolution Factors affecting the PSF

Resolution Detector design affects spatial resolution –Screen-film Phosphor thickness and properties Film grain size –Digital detectors Scintillator thickness and properties Photodiode size Be careful when a manufacturer quotes resolution in terms of pixel size.

Resolution Focal spot size affects spatial resolution (penumbra)

Resolution Scatter affects the PSF, usually with low-level tails. Detector Source This location of detector receives primary (unscattered) photons from the source.

Resolution Scatter affects the PSF, usually with low-level tails. Detector Source This location also receives scattered photons from the entire medium (and even within the detector).

Resolution Scatter effect on PSF depends on many factors –Probability of scatter in the medium –Energy distribution of scatter These PSF tails affect contrast

Resolution Magnification –Magnification is depth-dependent.

Resolution Magnification –Let d be the distance from source to image plane –Let z be the distance from source to an object of width w d z w wdwd

Resolution Magnification –Then w d = –Magnification M = w d /w = d z w wdwd

Resolution Magnification is a function of depth –Good: The FWHM in the detector is projected back onto the 3D image space as the inverse of magnification. –Bad: Features at different depths have different magnifications and FWHMs. d z w wdwd

Resolution Four primary effects on PSF –Detector sampling –Focal spot size –Compton scatter –Magnification The PSF is depth-dependent – therefore, spatially-varying. Which of these affect the FWHM?

Contrast What is the source of contrast in a projection X-ray system? What causes two pixels to be of different intensities?

The Earlier Example The intensity detected at a pixel is related to the sum of attenuation along the path. X-rays Detector 10 cm 3 cm  =.01 cm -1  =.1 cm -1

The Earlier Example What if we increase the size of the object, but not its  ? X-rays Detector 10 cm 5 cm  =.01 cm -1  =.1 cm -1 contrast increases to.362

Contrast So, contrast depends on two primary factors –The difference in linear attenuation coefficient between feature and background –The thickness (size) of the feature All is a function of energy.

Contrast Properties of materials