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 twodimensional convolution of the “ideal” image with
the PSF:
g2=g1h NOTE: Both * and ** are used to represent convolution
where h is the impulse response, or PSF.
• In some medical imaging systems (e.g. planar Xray) 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 ‚1MTF
MTF and OTF
• For spatially invariant imaging systems, the Optical
Transfer Function (OTF) is defined as the
normalized frequency response i.e.,
OTF = H([1,[2)/H(0,0)
• The Modulation Transfer Function (MTF) is defined
as the magnitude of the OTF, i.e.,
MTF = |OTF| = | H([1,[2)|/| 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:
xn, m
f
f
¦ ¦sn a, m bh a,b
a f b f
Where x is the blurry image, s is the ideal image, and
h is the point spread function.
MTF
• The DFT of this PSF is given by:
H u, v
f
f
§ un um ·
2Si ¨ ¸
©N M¹
¦ ¦hn, me
n 0 m 0
N
N
1,,
2
2
M
M
1,,
v
2
2
u
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:
ª1 2 1º
1«
»
2
3
2
»
15 «
«¬1 2 1»¼
A Numerical Example
• The transform is given by:
H u, v
1ª
§ 2Sv ·
§ 2Su · § 2Sv ·º
§ 2Su ·
3
4
cos
4
cos
4
cos
¸
¨
¸
¨
¸ cos¨
¸»
¨
15 «¬
N
M
N
M
¹
©
¹
©
¹ ©
¹¼
©
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-1) .
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:
CAB = |SA-SB|
where CAB is the contrast between tissues A and B
and SA and SB 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:
CNRAB = CAB/VN = |SA-SB|/VN = |SNRA-SNRB|
where VN 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