CS代考 BMEN90021, Lecture set 2

Set 2: Image Metrics
We have seen linear filters (2d image convolution) and some examples of other filtering operations such as unsharp masking and median filtering. 
What about edge detection filters?
• This is a toy example that would be far better in practice.

Original MRI image Filtered image
BMEN90021, Lecture set 2

Edge-preserving filter
A simple horizontal edge detector filter is 111
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Original MRI image
Filtered image
BMEN90021, Lecture set 2

Edge-preserving filter example
Apply the horizontal edge detector filter 111
000 -1 -1 -1
to the following 10×10 image, where black = 0, white = 1:
3 BMEN90021, Lecture set 2

3-d images
Just as a 2-d image is a 2-d matrix of grayscale
intensities, a 3-d image is a 3-d matrix.
It is only possible to visualise 2-d “slices” of a 3-d image at a time.
A 2-d image is written I(i,j) or I(x,y).
A 3-d image is written I(i,j,k) or I(x,y,z).
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Spatial resolution
The number of pixels in an image (“10 Megapixel
camera”) does not tell us the spatial resolution.
Point spread function (PSF)
Line spread function (LSF) Edge spread function (ESF)
Several common measures of spatial resolution:
PSF, LSF and ESF all related to a Modulation transfer function (MTF) in the spatial frequency domain.
Hurrah! Fourier transforms!
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Point spread function
Consider a very small “point source” positioned with the imaging field-of-view.
The relationship between the object, O, and the image, I, can be represented by:
I(x,y,z) = O(x,y,z) * h(x,y,z)
Here “*” is convolution.
h(x,y,z) is the three-dimensional PSF.
Equivalently, h(x,y,z) is a filter.
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Point spread function
In a perfect imaging system, the PSF is a 𝛿-function.
In reality, htotal(x, y, z) = hdetector(x, y, z) ⇤ hsampling(x, y, z) ⇤hreconstruction(x,y,z)⇤hfilters(x,y,z)
• The degree of effect of PSF depends on object being imaged:
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Resolution criteria
Definition of spatial resolution of an image:
“The smallest separation of two point sources necessary for the sources to be resolved.”
2 point sources can be resolved if separated by a
Full-width-at-half-maximum (FWHM)
distance greater than the FWHM.
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Gaussian PSF
• In1-d: h(x)= p 1 exp✓(xx0)2◆
For many imaging techniques, the PSF is well- approximated by a Gaussian function.
Mean (centre)
FWHM of a Gaussian function is given by
FWHM=2 2lln2⇡2⇡.362.36
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Line spread function
The LSF is defined in terms of the PSF:
LSF (y) = P SF (x, y)dx
Add up all values in x direction, for a particular y
A 1-dimensional quantity Useful in x-ray imaging
BMEN90021, Lecture set 2

Modulation transfer function Remember that
htotal(x, y, z) = hdetector(x, y, z) ⇤ hsampling(x, y, z) ⇤hreconstruction(x,y,z)⇤hfilters(x,y,z)
Convolution in the spatial domain is multiplication in the spatial frequency domain.
Therefore, often easier to talk about the Modulation
transfer function (MTF):
MTF(kx,ky,kz) = Z Z Z PSF(x,y,z)ej2(kxx+kyy+kzz) dxdydz
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Fourier transform review
Familiar expressions in the time / frequency domains:
S(f) = Z 1 s(t)ej2ftdt Z1
s(t)= 1 S(f)ej2ftdf 1
Equivalent expressions in the spatial / spatial frequency domains:
Spatial frequency function S(k) = Z 1 (x)ej2kxdx Z1
1-d spatial image (intensities on a line) (x) = 1 S(k)ej2kxdk 1
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Fourier transform in 2-d & 3-d Images require the multidimensional versions:
S(kx,ky) = Z 1 Z 1 (x,y)ej2(kxx+kyy)dxdy Z1 Z1
(x,y) = 1 1 S(kx,ky)ej2(kxx+kyy)dkx dky 1 1
S(kx,ky,kz) = Z 1 Z 1 (x,y,z)ej2(kxx+kyy+kzz)dxdydz Z1 Z1
(x,y,z) = 1 1 S(kx,ky,kz)ej2(kxx+kyy+kzz)dkx dky dkz 1 1
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Fourier transform review What does a Fourier transform do?
Tells us what combination of sinusoids will sum together to give our signal.
Gives us the frequency spectrum of the signal.
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Fourier Transforms – 1D
Which signals match which Fourier transforms? WHY?
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Fourier Transforms
Continuous Assume signal is Discrete periodic with period N
F (k) = Inverse: Z
f (x)e2⇡ixkdx F [k] =
f [x]e2⇡ikx N
1 F (k)e+2⇡ixkdx 1
f [x] = 1 X F [k]e+2⇡ikx N x=0
Frequency:
k = cycles per N samples
Frequency:
k = cycles per SI-unit
k = cycles per second
(seconds, metres)
BMEN90021, Lecture set 2

Fourier Transforms – 2D
Image Magnitude FT Zoomed y
What does the Fourier transform look like?
Spatial Frequency, k = cycles per N samples (i.e. width of image)
a) Howmanycyclesperimageinx-direction? b) Howmanycyclesperimageiny-direction?
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Back to the MTF…
Original objects
Imaging the LSFs
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Image quality: SNR
Signal to noise ratio (SNR) is defined in decibels.
In medical imaging, other formulae are used, however all
are a Ratio of Signal to Noise. One common definition is:
SNR = mean signal in a region-of-interest divided by standard deviation of noise in a background region.
S N R = Aσ S N R = Aσ
• Right image will have
higher SNR than left.
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Signal averaging
Signal averaging: adding copies of images of the same object.
Why does it improve SNR? At each pixel…
• Measuredimageintensity,z,asfunctionoftrueintensity,s,andnoise,nis:
z = s + n, n ∼ N(0,σn2)
• SNR of measured intensity: SNRz = |s|
z a v = s + K1 ∑K n k k=1
• Averagedintensity,zav,asfunctionofKcopies:
• SNRofaveraged SNR =s measurements: zav
|s|P = K|s|= KSNR
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⇢1 K n z varK nk

Contrast to noise ratio
Even if the SNR is high, it is important that there is enough CONTRAST to differentiate between tissue types.
Most common definition of image contrast:
CAB =|SASB|
where SA and SB are signals from two tissue types.
Most common definition of CNR:
CNRAB = CAB = |SA SB| =|SNRA SNRB| noise noise
Increasing noise
BMEN90021, Lecture set 2

Contrast to noise ratio Spatial resolution of image also affects CNR.
If FWHM of the PSF is on the order of the size of an image feature, then image blurring reduces contrast…
Decreasing spatial resolution
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Imaging “phantoms”
Phantoms are not ghosts… Phantoms are either
Physical objects of known geometry and composition that are scanned, or
newmaticsound.com
flickr.com
elimpex.com
Numerical objects of known geometry and composition that have image analysis methods applied to them. eg. Shepp- Logan phantom.
bigwww.epfl.ch
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eg. Lego phantom in use
Rosenzweig et al, ISMRM 2017
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