Lecture 14. Point Spread Function (PSF)

Lecture 14

Point Spread Function (PSF), Modulation Transfer Function (MTF), Signal-to-noise Ratio (SNR), Contrast-to-noise Ratio (CNR),

and Receiver Operating Curves (ROC)

Point Spread Function (PSF)

• Recollect that Image restoration refers to the

removal or minimization of known degradations in an image

• This includes deblurring of images degraded by the limitations of the sensor or its environment, noise filtering, and correction of geometric

distortions or non-linearities due to sensors.

Point Spread Function (PSF)

• The figure below shows a typical situation in an imaging system:

Point Spread Function (PSF)

• The image of a point source is blurred and degraded due to noise by an imaging system.

• If the imaging system is linear, the image of an object can be expressed as:

where K(x,y) is the additive noise function, f(DE) is the

object, g(x,y) is the image, and h(x,y;DE) is the Point

Spread Function (PSF). The “;” is used to distinguish the

input and output pairs of coordinates in this case.

Point Spread Function (PSF)

• A typical image restoration problem, give the formulation in the previous slide is of the form:

Find an estimate of f(a,b) given

ĺThe Point Spread Function, ĺThe blurred image, and,

ĺThe statistical properties of the noise and/or the factors affecting the noise contribution.

PSF

• The point spread function (PSF) describes the imaging system response to a point input, and is analogous to the impulse response.

• A point input, represented as a single pixel in the

“ideal” image, will be reproduced as something

other than a single pixel in the “real” image.

PSF

A “Point Source”

PSF

• The PSF need not be isotropic (radially symmetric).

• In ultrasound, X-ray CT, and radionuclide

tomography it is typical to have a nonisotropic PSF for various physical reasons.

• In MRI it is possible to have either isotropic or nonisotropic PSF depending on the type of

acquisition since spatial frequency coverage can be

different for the two in-plane directions.

PSF

• The output image may then be regarded as a two- dimensional convolution of the “ideal” image with the PSF:

g

=g

h

where h is the impulse response, or PSF.

• In some medical imaging systems (e.g. planar X- ray) the PSF can vary gradually over the field of view.

• In this case it is convenient to define a zone of

constant PSF (isoplanatic region) to allow use of

convolutional forms and the transform domain.

Isotropic PSFs

Anisotropic PSFs

PSF

• Optical (e.g. microscopy) also manifest

asymmetric point spread functions due to lens imperfections (both material and geometry).

• It is typical for the PSF to degrade as distance

from the center of the FOV is increased.

PSF: Examples

From Fundamentals of Digital Image Processing by A.K. Jain

Modulation Transfer Function (MTF)

• Another measure of system performance is the modulation transfer function (MTF).

• This is analogous to the frequency response typically used for one-dimensional applications.

• In the case of a complex transfer function, MTF is usually expressed as the magnitude portion of the function.

• MTF allows for simplified descriptions of an

imaging system’s spatial resolution capabilities.

MTF

• PSF expresses system performance in the spatial domain, while MTF expresses system

performance in the frequency domain.

• The two parameters are related by the Fourier transform:

 MTF 

PSF

PSF MTF

1

‚

‚

MTF and OTF

• For spatially invariant imaging systems, the Optical Transfer Function (OTF) is defined as the

normalized frequency response i.e., OTF = H([

,[

)/H(0,0)

• The Modulation Transfer Function (MTF) is defined as the magnitude of the OTF, i.e.,

MTF = |OTF| = | H([

,[

)|/| H(0,0) |

MTF: Examples

• MTFs of the PSFs displayed earlier

From Fundamentals of Digital Image Processing by A.K. Jain

MTF

• Consider the discrete convolutional representation of a blurring function:

Where x is the blurry image, s is the ideal image, and h is the point spread function.

  ¦ ¦

f

f

f









b a

b a h b m a n s m

n

x , , ,

MTF

• The DFT of this PSF is given by:

, 2 , 2 1

, 2 , 2 1

, ,

0

2

0

M v M

N u N

e m n h v

u H

m

M um N i un

n





 

 

¦

¦

MTF and Frequency Response

• The coefficients of H(u,v) are those for plane waves of various frequencies and orientations.

• These are the spatial frequency components that exactly represent the PSF (blurring function).

• H(u,v) is the transfer function, also referred to as the frequency response.

• Examination of the magnitude H(u,v) allows for

determination of limiting spatial resolution.

A Numerical Example

• Consider this example: a 3x3 blurring kernel typical of what has been seen before:

» »

»

¼ º

« «

«

¬ ª

1 2

1

2 3

2

1 2

1 15

1

A Numerical Example

• The transform is given by:

¼

« º

¬

ª ¸

¹

¨ ·

©

¸ §

¹

¨ ·

©

 §

¸¹

¨ ·

©

 §

¸¹

¨ ·

©

 §

M v N

u M

v N

v u u

H

S S S

S

2 cos cos 2 4

cos 2 4

cos 4 15 3

, 1

N=M=33

2D Plots of the example

Obtaining the MTF from a kernel PSF

• To obtain the MTF of a PSF represented as a kernel:

– Apply a scaling factor (= 6(array elements)) if you want a normalized (range 0-1) MTF plot.

– Embed the kernel in the center of an array of zeros equal to the size of the image to which it is to be applied.

– fftshift(abs(fft2(…)))

– Result will be 2D array representing the spatial

frequency version of the blurring/filter function.

MTF

• As with systems seen in one-dimensional applications, the MTF of imaging systems

typically demonstrate a roll-off (like a low-pass filter) with higher spatial frequencies.

• An MTF curve is typically used to express the spatial frequency response of a system, and expresses normalized contrast as a function of spatial frequency (expressed in units of inverse length, e.g. mm

) .

Signal-to-noise Ratio (SNR)

• Signal-to-noise ratio is an engineering term for the power ratio between a signal (meaningful information) and the background noise:

• Because many signals have a very wide dynamic

range, SNRs are usually expressed in terms of the

logarithmic decibel scale.

SNR

• In decibels, the SNR is 20 times the base-10 logarithm of the amplitude ratio, or 10 times the logarithm of the power ratio:

where P is average power and A is RMS amplitude.

• Both signal and noise power are measured within the system bandwidth.

SNR

• SNR is usually taken to indicate an average signal to noise ratio, as it is possible that (near) instantaneous signal to noise ratios will be considerably different.

• In general, higher signal to noise is better. (i.e.

cleaner.)

• In image processing, the SNR of an image is usually defined as the ratio of the mean pixel value to the standard deviation of the pixel values.

• Related measures are the "contrast ratio" and the

"contrast to noise ratio".

SNR

• In the case of MRI, the noise is distributed uniformly throughout the image.

• The SNR can be measured by computing the mean signal intensity over a certain region of interest (ROI) and dividing this by the standard deviation of the signal from a region outside the image.

• In other modalities, this is not true; the noise is not uniformly distributed over the image. As a

consequence, other methods must be used to estimate the SNR.

Contrast-to-noise Ratio (CNR)

• CNR is a measure for assessing the ability of an imaging procedure to generate clinically useful image contrast.

• The image contrast itself is not precise enough to qualify an image, because in a noisy image it is unclear where the contrast originates.

• It may be due to true tissue contrast, or it may be due to noise fluctuations.

• The human ability to distinguish between objects is proportional to contrast, and it decreases

linearly with noise.

CNR

• These laws of perception are taken into account by the definition of the contrast to noise ratio CNR = contrast/noise.

• Therefore the CNR gives an objective measure of useful contrast.

• For instance, if an acquisition technique generates images with twice the contrast of those produced by another technique, the noise must increase less than twice in order to provide clinically better images.

CNR

• A practical problem of CNR definition is that it relies on the measurement of the photon flux; this depends upon the display system and is difficult to perform.

• An equivalent yet much more feasible approach is to

use the signal difference in the original data instead of

assessing the contrast of the displayed image.

CNR

• Even if the image has a high signal-to-noise ratio, it is not useful unless there is a high enough CNR to be able to distinguish among different tissues and tissue types, and in particular between healthy and pathological tissue.

• Various definitions of image contrast exist, but the most common is:

C

= |S

-S

|

where C

is the contrast between tissues A and B and S

and S

are the signals from tissues A and B.

CNR

• The CNR between two tissues is defined in terms of their respective signal noise-to-ratios of the two tissues:

CNR

= C

/V

= |S

-S

|/V

= |SNR

-SNR

| where V

is the standard deviation of the noise.

* Note that this is analogous to the Signal

difference to noise ratio

Receiver Operating Curve (ROC) in imaging-based diagnoses

• There are four possibilities for a practitioner making a diagnosis:

– a true positive (where true refers to a correct diagnosis and positive refers to say a tumor being present),

– a true negative,

– a false positive, and, – a false negative.

ROC

ROC plot for tumor diagnosis

Figure – (Left) A table showing the four possible outcomes of a tumor diagnosis. (Right) The ROC represented by the dashed line represents a random diagnosis. The upper curve represents an improved diagnosis. The better the diagnosis, the larger is the integrated area under the ROC.

ROC

• The ROC plots the fraction of true positives versus the fraction of false positives for a series of images acquired under different conditions, or with a

different value of some parameter, or with different SNRs, or with different practitioners, for example.

• The area under the ROC is a measure of the effectiveness of the imaging system and/or the practitioner.

• The greater the area under the curve the more effective is the diagnosis.

ROC

• There are three measures commonly used in ROC analysis:

1. The accuracy is the correct number of diagnoses divided by the total number of diagnoses

2. The sensitivity is the number of true positives divided by the sum of the true positives and false negatives 3. The specificity is the number of true negatives divided

by the number of true negatives and false positives