BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Introductory Topics Part 2

Today Fourier Transform Properties A Generalized Imaging System Contrast Spatial Resolution Noise

Fourier Transform Remember: F is a complex quantity!! Note the change to be consistent with the book Units of u and v are cycles per distance (cm, pixel, etc)

A Generalized Imaging System Real world Raw data Raw image Displayed image Diagnosis, treatment AcquisitionProcessingDisplayObservation This may be a human. This may be a CAD system.

What is an image? A signal with at least two independent variables –Independent variables are spatial –Dependent variable is intensity Intensity of what?

An Imaging System To be useful, an imaging system must have: –Spatial sensitivity (The independent axes) –Intensity sensitivity (The dependent axis or axes) It must communicate –That a feature exists (It stands out from the background) –Where the feature is located

Contrast and Resolution Contrast –The degree to which a feature differs in intensity from surrounding features –The dependent or intensity axis Spatial Resolution –The degree to which an imaging system can precisely represent the position of a feature –The independent or spatial axis

Contrast The degree to which a feature differs in intensity from surrounding features There are many different mathematical definitions. For our purposes:

Contrast Example

Contrast Example Sometimes it is not easy to define the boundaries of the object to compute its contrast

Contrast Why do these look different when the background is.5 in each case?

Contrast Contrast is invariant to scaling, BUT display contrast can be manipulated. These are both renditions of the same raw image.

Contrast Raw image Displayed image Display A look-up-table (LUT) can be used to map raw image intensities to display intensities Input intensity Output intensity Input intensity Output intensity

Contrast Input intensity Output intensity Input intensity Output intensity Linear map Piecewise linear map

Contrast So, contrast can be improved easily in the display subsystem. –At what price? –Keywords: saturation, dynamic range Is the piecewise linear mapping an LSI system?

Contrast Contrast resolution: The ability of an imaging system to distinguish between two intensities –The smallest difference in intensity between objects that a system detects as being different –Humans can resolve about gray levels. –Computer displays are limited by the number of bits available for each pixel.

Spatial Resolution The degree to which an imaging system can precisely represent the position of a feature In an LSI system, this is a property of the PSF and MTF

Compare two PSFs Which system has better spatial resolution? System 1 System 2

Compare Two MTFs Zero frequency Is here

Spatial Resolution Most real-world imaging systems have low- pass characteristics –Although we can design digital filters that do not.

Spatial Resolution Note that in an LSI system, the PSF or MTF completely characterizes the spatial resolution everywhere. LSV system: Need a PSF for each position (Pretend it is locally shift-invariant) Nonlinear system: ? Pretend it is linear?

Spatial Resolution An abbreviated way of characterizing spatial resolution: –The full-width at half-maximum (FWHM) of the PSF Find the peak of PSF Take ½ of the peak Measure the width What are the units of FWHM?

Spatial Resolution FWHM does not tell you everything! What if the PSF is not isotropic? What if the PSF has tails?

Spatial Resolution Since it is hard to create a point source in some modalities, a line source is often used This gives a line-spread function (LSF). How does this relate to the PSF?

Spatial Resolution An alternative to FWHM: The bar phantom This system gets between 2.2 and 2.5 line pairs/mm.

Spatial Resolution Comparing MTFs Estimated MTFs of SPECT collimators

Resolution Affects Contrast Poorer resolution results in poorer contrast

Noise Noise is random variation in an image. –It is unpredictable. A typical linear model for additive noise would look like this: System f(x,y)g(x,y) n(x,y) +

Noise The additive noise is an image. It is a sample from a two-dimensional random process. The probability distribution associated with n is important. –You should know the normal distribution and the Poisson distribution.

Power Spectra Any image has a power spectrum. This represents the power in different frequencies of the image.

Power Spectra A random process also has a power spectrum. E[] is the expected value of its argument This represents the power in different frequencies of a sample from the random process, on average.

Noise Raw noise is usually well modeled as a white noise random process. White noise has equal power content at all frequencies. Frequency Noise power

Effects of Systems on Power Spectra An LSI system changes the power spectrum of an image in a predictable way. The same is true for a random process. System f(x,y) g(x,y) h(x,y)

Noise So, what happens to white noise when I run it through a low-pass filter? Frequency Noise power Frequency Noise power Filter

Noise The most complete way to characterize a random process is to specify its probability distribution on all of the pixels and their interrelationships (i.e., covariances). –The power spectrum tells us some of this, but not all. If a random process is stationary, its statistical properties do not change with position. (This is required for the power spectrum to exist.) If the random process is white, its pixels vary independently of one another. –We only need to specify a probability distribution for each pixel. If stationary and white, we only need a single probability distribution to represent all pixels. –The variance is a good single statistic.   –The area under the power spectrum is an estimate of the variance.

Noise Imaging System f(x,y) g(x,y) n(x,y) + Filtering System g(x,y) h(x,y) q(x,y)

Noise The signal is affected by both H and Q. The noise is affected only by Q.

Noise What happens to resolution and noise as we vary the cutoff of a low-pass filter?

Noise We can also design a restoration filter to improve resolution. How does this change the noise variance? The resolution? Frequency MTF

Key Point There is an essential and inescapable tradeoff between noise and resolution in every imaging system. Noise variance Resolution (FWHM)

Noise Measures related to noise –The signal-to-noise ratio (SNR) –These depend on what you think “signal” is.

Noise Measures related to noise –The contrast-to-noise ratio (CNR) –This relates closely to many applications.

Noise Measures related to noise –The noise-equivalent quanta (NEQ) –This is a plot of signal-to-noise ratio at each spatial frequency.

Other Dimensions Temporal Resolution –The ability to represent two events separated in time as distinct events Energy Resolution –The ability to determine two photons of different energies as distinct energies