U.S. patent application number 13/122379 was filed with the patent office on 2011-10-27 for system and method for processing images.
Invention is credited to Ting Yim Lee.
Application Number | 20110262022 13/122379 |
Document ID | / |
Family ID | 42073000 |
Filed Date | 2011-10-27 |
United States Patent
Application |
20110262022 |
Kind Code |
A1 |
Lee; Ting Yim |
October 27, 2011 |
SYSTEM AND METHOD FOR PROCESSING IMAGES
Abstract
A method of processing a plurality of time separated images
comprises selecting a plurality of imaging units in each image;
measuring a temporal difference in each imaging unit; and selecting
temporal differences above a threshold limit.
Inventors: |
Lee; Ting Yim; (London,
CA) |
Family ID: |
42073000 |
Appl. No.: |
13/122379 |
Filed: |
October 2, 2009 |
PCT Filed: |
October 2, 2009 |
PCT NO: |
PCT/CA2009/001397 |
371 Date: |
July 19, 2011 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61102206 |
Oct 2, 2008 |
|
|
|
Current U.S.
Class: |
382/131 |
Current CPC
Class: |
A61B 6/032 20130101;
A61B 6/504 20130101; A61B 5/055 20130101; A61B 6/507 20130101; G06T
2207/10088 20130101; G06K 9/52 20130101; G06T 7/0016 20130101; G06T
2207/10081 20130101; G06T 2207/10104 20130101; A61B 6/5258
20130101; A61B 6/037 20130101; A61B 6/486 20130101; A61B 5/0033
20130101; G06T 2200/24 20130101; A61B 6/481 20130101; G06K 9/6215
20130101 |
Class at
Publication: |
382/131 |
International
Class: |
G06K 9/00 20060101
G06K009/00 |
Claims
1. A method of processing a plurality of time separated images
comprising: selecting a plurality of imaging units in each image;
determining a temporal difference for each imaging unit; and
selecting temporal differences above a threshold limit.
2. The method of claim 1, further comprising ranking each measured
temporal difference prior to said selecting.
3. The method of claim 1, further comprising constructing a
processed image using the selected temporal differences.
4. The method of claim 1, wherein each imaging unit is one of a
pixel and a voxel.
5. The method of claim 1, wherein each imaging unit is one of a
plurality of pixels and a plurality of voxels.
6. The method of claim 1, wherein the temporal difference is based
on contrast enhancement.
7. The method of claim 6, wherein the temporal difference is a
difference of a contrast agent.
8. The method of claim 7, wherein contrast enhancement is based on
the concentration of the contrast agent.
9. The method of claim 7, wherein contrast enhancement is based on
the activation of the contrast agent.
10. The method of claim 1, further comprising characterizing the
measured temporal differences with a blind signal separation
statistical technique.
11. The method of claim 1, wherein determining the temporal
difference comprises calculating a covariance matrix of temporal
differences for the imaging units.
12. The method of claim 11, further comprising analyzing the
covariance matrix with a blind signal separation statistical
technique to determine common temporal patterns among the temporal
differences.
13. The method of claim 12, further comprising analyzing the
eigenvectors of the covariance matrix.
14. The method of claim 13, wherein eigenvalues of the eigenvector
represent the degree of temporal difference.
15. The method of claim 14, wherein the selecting of temporal
differences comprises selecting eigenvalues above the threshold
limit.
16. The method of claim 1, wherein the threshold limit is
adjustable.
17. The method of claim 12, wherein the blind signal separation
statistical technique is selected from the group consisting of
principal component analysis, independent component analysis, blind
source deconvolution, Karhunen-Loeve transform and Note!ling
transform.
18. The method of claim 1, wherein the plurality of time separated
images are obtained using a dynamic medical imaging technique.
19. The method of claim 18, wherein the dynamic medical imaging
technique comprises use of a contrast agent.
20. The method of claim 18, wherein the time separated images are
obtained using a protocol having a reduced radiation dose.
21. The method of claim 18, wherein the time separated images are
obtained using a protocol having a reduced amount of contrast
agent.
22. The method of claim 18, wherein the dynamic medical imaging
technique is magnetic resonance imaging (MRI), functional MRI,
positron emission tomograghy, nuclear medecine or computed
tomography.
23. The method of claim 18, wherein the dynamic medical imaging
technique is computed tomography (CT).
24. The method of claim 23, wherein the time separated images are
obtained using a CT Perfusion protocol having a reduced radiation
dose.
25. The method of claim 23, wherein the time separated images are
obtained using a CT Perfusion protocol having a reduced amount of
contrast agent.
26. A system for processing a plurality of time separated images
comprising: an interface for receiving a plurality of time
separated images; and a processing structure for selecting a
plurality of imaging units in each image, determining a temporal
difference for each imaging unit, and selecting temporal
differences above a threshold limit.
27. A computer readable medium embodying a computer program for
processing a plurality of time separated images, the computer
program comprising: computer program code for selecting a plurality
of imaging units in each image; computer program code for
determining a temporal difference for each imaging unit; and
computer program code for selecting temporal differences above a
threshold limit.
28. A system for processing a plurality of time separated images
according to the method of claim 1.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of U.S. Provisional
Application No. 61/102,206 to Lee filed on Oct. 2, 2008 and
entitled "SYSTEM AND METHOD FOR PROCESSING IMAGES", the content of
which is incorporated herein by reference.
FIELD OF THE INVENTION
[0002] The present invention relates generally to image processing
and, more particularly, to a system and method for processing image
data obtained by using a dynamic imaging technique
BACKGROUND OF THE INVENTION
[0003] Medical imaging encompasses techniques and processes used to
create images of the human body for clinical purposes, including
medical procedures for diagnosing or monitoring disease. Medical
imaging technology has grown to encompass many image recording
techniques including electron microscopy, fluoroscopy, magnetic
resonance imaging (MRI), nuclear medicine, photoacoustic imaging,
positron emission tomography (PET), projection radiography,
thermography, computed tomography (CT), and ultrasound.
[0004] Medical imaging can incorporate the use of compounds
referred to as contrast agents or contrast materials to improve the
visibility of internal bodily structures in an image.
[0005] Medical imaging was originally restricted to acquisition of
singular static images to capture the anatomy of an organ/region of
the body. Currently, the use of more sophisticated imaging
techniques allows dynamic studies to be made that provide a
temporal sequence of images which can characterize physiological or
pathophysiological information.
[0006] Dynamic medical imaging involves an acquisition process that
takes many "snapshots" of the organ/region/body of interest over
time in order to capture a time-varying behaviour, for example,
distribution of a contrast agent and hence capture of a specific
biological state (disease, condition, physiological phenomenon,
etc.). As the speed and digital nature of medical imaging evolves,
this acquisition data can have tremendous temporal resolution and
can result in large quantities of data.
[0007] Medical imaging technologies have been used widely to
improve diagnosis and care for such conditions as cancer, heart
disease, brain disorders, and cardiovascular conditions. Most
estimates conclude that millions of lives have been saved or
dramatically improved as a result of these medical imaging
technologies. However, the risk of radiation exposure from such
medical imaging technologies for patients must be considered.
[0008] In this regard, the increasing use of CT in medical
diagnosis has highlighted concern about the increased cancer risk
to exposed patients because of the larger radiation doses delivered
in CT than the more common, conventional x-ray imaging procedures.
This is particularly true with the two-phase CT Perfusion protocol.
For illustration purposes, using the dose-length product for a
protocol provided by a commercially available CT scanner (GE
Healthcare), the effective dose of a CT Stroke series consisting
of: 1) a non-enhanced CT scan to rule out hemorrhage; 2) a CT
angiography to localize the occlusion causing the stroke; and 3) a
two-phase CT Perfusion protocol to define the ischemic region with
blood flow and blood volume and predict hemorrhagic transformation
(HT) with blood-brain barrier permeability surface area product
(BBB-PS); is 10 mSv, of which 4.9 mSv is contributed by the
two-phase CT Perfusion protocol. The CT Stroke series is predicted
to induce 80 and 131 additional cancers for every exposed 100,000
male and female acute ischemic stroke (AIS) patients, respectively,
with the two-phase CT Perfusion protocol alone predicted to cause
half of the additional cancers (Health Risks from Exposure to Low
Levels of Ionizing Radiation: BEIR VII. The National Academies
Press, Washington D.C., 2006).
[0009] Unless the effective dose of the two-phase CT Perfusion
protocol is reduced, the benefits of medical imaging will be
undermined by the concern over cancer induction. Furthermore, the
concern over the risks of radiation exposure extends to other
medical imaging techniques, particularly for pediatric patients or
those patients undergoing repeated exposure.
[0010] The use of statistical filtering techniques to increase the
signal to noise ratio in low dose radiation imaging has been
described, for example in U.S. Pat. No. 7,187,794 issued Mar. 6,
2007. However, at present the application of statistical filtering
is limited to projection data prior to the construction of images.
Several disadvantages are associated with statistical filtering of
projection data. For example, the computational burden when working
with projection data is high because each image from a dynamic
sequence is typically reconstructed from .about.1,000 projections.
In addition, the filtering process has to be `hardwired` into the
image reconstruction pipeline of the scanner. As such, statistical
filters of projection data lack flexibility as they are typically
specific to a scanner platform.
[0011] Whether perceived or proven, concerns over the risk of
radiation exposure will have to be addressed to assure patients of
the long term safety of medical imaging techniques. Accordingly,
there is a need for medical imaging techniques that allow for a
reduction in radiation dose.
[0012] It is therefore an object of the present invention to
provide a novel system and method for processing images.
SUMMARY OF THE INVENTION
[0013] In an aspect, there is provided a method of processing a
plurality of time separated images comprising: selecting a
plurality of imaging units in each image; determining a temporal
difference for each imaging unit; and selecting temporal
differences above a threshold limit.
[0014] In another aspect, there is provided a system for processing
a plurality of time separated images comprising: an interface for
receiving a plurality of time separated images; and a processor for
selecting a plurality of imaging units in each image, determining a
temporal difference for each imaging unit, and selecting temporal
differences above a threshold limit.
[0015] In yet another aspect, there is provided a computer readable
medium embodying a computer program for processing a plurality of
time separated images, the computer program comprising: computer
program code for selecting a plurality of imaging units in each
image; computer program code for determining a temporal difference
for each imaging unit; and computer program code for selecting
temporal differences above a threshold limit.
BRIEF DESCRIPTION OF THE DRAWINGS
[0016] Embodiments will now be described, by way of example only,
with reference to accompanying drawings in which:
[0017] FIG. 1 is a flow chart of an image processing method;
[0018] FIG. 2 schematically illustrates the positioning of imaging
units in a plurality of time separated images;
[0019] FIGS. 3A to 3E show brain blood flow maps from CT images
processed with a statistical filtering technique;
[0020] FIG. 4 graphically shows blood flow Figure of Merit
calculated for the images shown in FIGS. 3A to 3D using the regions
outlined in FIG. 3E; and
[0021] FIG. 5 is a diagram of a system for implementing the image
processing method shown in FIG. 1.
DETAILED DESCRIPTION
[0022] Dynamic medical imaging involves an acquisition process that
takes many "snapshots" of an organ/region/body of interest (i.e. a
target region) over time in order to capture a time-varying
behaviour, for example uptake and/or wash out of a contrast agent.
This acquisition process results in the production of a plurality
of time separated images. The method and system described herein
involves processing of such time separated images.
[0023] Turning now to FIG. 1, the steps performed during processing
of time separated images according to the method are illustrated.
The plurality of time separated images may be obtained from dynamic
medical imaging of a patient with or without the use of an injected
contrast agent. A contrast agent can selectively increase the
contrast of the target region in an image, for example on the basis
of the target region's structure or physiological state. For
example, one may inject into a patient a compound which has a
biophysical, molecular, genetic or cellular affinity for a
particular organ, disease, state or physiological process. Such
contrast agents are selected to have a property that provides
enhanced information to a given imaging technique by altering
imaging conditions, for example by altering image contrast, to
reflect the behaviour of the compound in the body. This may be
achieved via increased X-ray attenuation at a localized site (e.g.
for CT/X-ray), altered paramagnetic properties (e.g. for MRI) or
the use of a radioisotope for nuclear medicine/PET. Contrast agents
for many imaging techniques are well-known. In some cases contrast
enhancement does not rely on an injected contrast agent, for
example the use of "black blood" or "white blood" in magnetic
resonance imaging (MRI) where specific pulse sequences are used to
change the magnetic saturation of the blood and thus its appearance
in the image, or tagged MRI sequences which alter the magnetic
behaviour of a particular tissue or fluid. An injected contrast
agent, or a tissue or fluid with altered behaviour, may all be
regarded as "imaging agents".
[0024] The plurality of time separated images are not limited to
any particular imaging technique and include, for example, dynamic
medical imaging using magnetic resonance imaging (MRI), computed
tomography (CT), nuclear medicine (NM) or positron emission
tomography (PET). The plurality of time separated images are also
not limited to particular image types. For example, grayscale or
colour images may be processed.
[0025] During the method, within each time separated image a
plurality of imaging units are selected (step 110). An "imaging
unit" may be any desired unit for separating an image into
equivalent portions including, for example, a pixel, a plurality of
pixels, a fraction of a pixel, a voxel, a plurality of voxels, a
fraction of a voxel etc.
[0026] The imaging unit data is represented by a value, for example
a digital value, and can thus be quantified and measured.
Typically, image intensity is measured for corresponding imaging
units in the time separated images. For dynamic imaging using
contrast agents a temporal difference can be determined with
respect to contrast concentration. The temporal difference of
contrast concentration can be represented as a plot of contrast
enhancement as a function of time.
[0027] Two alternatives for determining temporal differences of
contrast concentration and selecting temporal differences are shown
in FIG. 1. In one alternative shown in FIG. 1, once the plurality
of imaging units has been selected from each time separated image
at step 110, a temporal difference is determined for each imaging
unit (step 115). The determined temporal difference or
representations thereof are then ranked according to degree of
observed temporal difference (step 120). Temporal differences above
a threshold limit are then selected (step 125). A processed image
is then constructed on the basis of the selected temporal
differences (step 150).
[0028] In the other alternative shown in FIG. 1, the temporal
differences are not ranked. Rather, the temporal data for imaging
units are analyzed to select temporal differences above a threshold
limit using statistical techniques, and more particularly blind
signal separation statistical techniques. For example, with respect
to dynamic CT imaging using a contrast agent, once the plurality of
imaging units have been selected from each time separated image at
step 110, a covariance matrix of the co-variations of all pairs of
contrast enhancement plots in relation to a mean curve is
established (step 130) and analyzed with a blind signal separation
statistical technique (step 135), such as with a principal
components analysis being applied to the covariance matrix.
Eigenvector/eigenvalue pairs are calculated on the basis of the
statistical analysis (step 140). Resulting eigenvector/eigenvalue
pairs above a threshold limit are then selected on the basis of
their eigenvalues (step 145). A processed image is then constructed
using the selected eigenvectors, with different weightings of the
selected eigenvectors being used to construct individual imaging
units (step 160).
[0029] The plurality of time separated images may optionally be
registered to correlate corresponding locations in the time
separated images to each other. Registration of the plurality of
time separated images is only needed to correct for gross
misregistration. Otherwise, slight movement between the time
separated images may be tolerated and may even be removed by the
processing method described herein.
[0030] As another optional step, the plurality of time separated
images may be preprocessed so that the time separated images are
represented with quantifiable values or codes. For example, the
time separated images may be represented by digital values using
known digitization techniques. If the imaging data of the plurality
of time separated images is provided in a digital form, then a
digitization step is not necessary.
[0031] The determination of temporal differences of contrast
concentration will now be further described with reference to FIG.
2. FIG. 2 shows a series of six chronologically ordered time
separated images, t.sub.1 to t.sub.6. A 4.times.6 array of imaging
units is marked on each time separated image. In this example, five
(5) imaging units, IU1 to IU5, have been selected. As is clear from
comparing the entire series of time separated images, t.sub.1 to
t.sub.6, imaging unit IU1 is selected at the same position of the
array (i.e. column 1, row 2) for each of the time separated images.
Similarly, each of imaging units IU2 to IU5 is selected in
corresponding positions of the imaging unit array throughout the
series of time separated images. Accordingly, values for each
imaging unit at its corresponding position throughout the time
separated images can be plotted as a function of time to obtain a
temporal curve. The plotting step is illustrated by the arrows
linking the corresponding positions of imaging units IU1, IU2 and
IU5. The temporal curve for each imaging unit can then be analyzed
to determine a temporal difference. For example, a mean curve for
all of the temporal curves obtained can be calculated. The mean
curve is then subtracted from each of the temporal curves to obtain
a temporal difference curve for each selected imaging unit. It will
be understood, that the marked array and the selection of imaging
units IU1 to IU5, as well as the arrows representing plotting of
values for imaging units IU1, IU2, and IU5 as a function of time,
is for illustration purposes only, and that any number of imaging
units may be selected and analyzed as desired. Furthermore, the
determination of temporal differences by subtraction of a mean
curve from the temporal curves for imaging units is for
illustration only, and other methods of analyzing the temporal
curves, for example using derivatives and/or statistical techniques
are contemplated.
[0032] A mathematical basis for blind signal separation statistical
analysis of a plurality of time separated images is now provided
using time separated images from a CT Perfusion study as an
example.
[0033] Given a series of n dynamic (i.e. time separated) images
from a CT Perfusion study, the n dynamic images can be represented
compactly as a matrix:
{tilde over (X)}=[{tilde over (x)}.sub.1.sup.T {tilde over
(x)}.sub.2.sup.T . . . {tilde over (x)}.sub.p-1.sup.T {tilde over
(x)}.sub.p.sup.T].sup.T
where p is the number of pixels in a CT image (i.e.
p=512.times.512, 256.times.256, etc) and {tilde over (x)}.sub.i,
i=1, . . . , p is a n.times.1 vector whose elements are the pixel
values in the n images or the time vs. enhancement curve from the
i.sup.th pixel. Note that the (i,j).sup.th element of {tilde over
(X)} is the j.sup.th element x.sub.ij of {tilde over (x)}.sub.i.
Each {tilde over (x)}.sub.i, i=1, . . . , p can also be interpreted
as repeated observations of a single time vs. enhancement curve
{tilde over (.chi.)} (a random process consisting of n random
variables, .chi..sub.1, .chi..sub.2, . . . , .chi..sub.n-1,
.chi..sub.n).
[0034] Notation is interpreted as follows: [0035] (i) a vector is
represented by a tilde above a Greek or Roman alphabet character
with elements arranged in a column; [0036] (ii) a matrix is
represented by tilde over a capitalized Greek or Roman alphabet
character; [0037] (iii) the transpose of a vector or a matrix is
indicated by a superscript T on the symbol; and [0038] (iv) a
random variable is represented by a Greek alphabet character while
an observation (samples) of a random variable is represented by the
corresponding Roman alphabet character; the correspondence between
Greek and Roman alphabet character will be used as shown below in
the following discussion:
[0038] .chi.x
.psi.y
.zeta.z [0039] (v) a random process consisting of n random
variables is represented by a n.times.1 vector.
[0040] One approach to removing noise from the n dynamic images is
to find an `expansion` of the p.times.n matrix, {tilde over (X)},
and then truncate the expansion by certain pre-determined criteria.
A common expansion is the singular value decomposition of {tilde
over (X)}:
{tilde over (X)}= {tilde over (L)}{tilde over (V)}.sup.T (1)
where is a p.times.p orthogonal matrix, {tilde over (V)} is a
n.times.n orthogonal matrix, and {tilde over (L)} is a p.times.n
diagonal matrix with singular values of {tilde over (X)} as its
diagonal elements. Let l.sub.1, . . . ,l.sub.n be the singular
values of {tilde over (X)} (it is assumed without loss of
generality that {tilde over (X)} is of full column rank, n and
p>n). Then:
{tilde over (L)}=.left brkt-bot.{tilde over (l)}.sub.1 . . . {tilde
over (l)}{tilde over (l.sub.n)}.right brkt-bot.
where {tilde over (l)}{tilde over (l.sub.i)} i=1, . . . ,n is a
p.times.1 vector with zero elements except for the i.sup.th element
which is l.sub.i. Usually the singular values l.sub.1, . . .
,l.sub.n are arranged in descending order and small singular values
can be discarded as arising from noise. A simple way to reduce
noise in {tilde over (X)} is to keep only r<n singular values.
Then:
{tilde over (L)}=.left brkt-bot.{tilde over (l)}{tilde over
(l.sub.1)} . . . {tilde over (l)}{tilde over (l.sub.r)} {tilde over
(0)}.sub.r+1 . . . {tilde over (0)}.sub.n.right brkt-bot.
If the truncated series of singular values is used to reconstitute
{tilde over (X)} according to Equation (1), then:
X ~ .apprxeq. [ u ~ 1 u ~ 2 u ~ p ] [ 1 ~ 1 1 ~ r 0 ~ r + 1 0 ~ n ]
[ v ~ 1 T v ~ r T v ~ r + 1 T v ~ n T ] ##EQU00001##
[0041] where .sub.i i=1, . . . ,p is the i.sup.th column of and
{tilde over (v)}.sub.i i=1, . . . ,n is the i.sup.th column of
{tilde over (V)}. Evaluating the expansion, yields:
X ~ .apprxeq. [ u ~ 1 u ~ 2 u ~ p ] [ l ~ 1 1 ~ r 0 ~ r + 1 0 ~ n ]
[ v ~ 1 T v ~ r T v ~ r + 1 T v ~ n T ] .apprxeq. [ u ~ 1 u ~ 2 u ~
p ] [ l ~ 1 v ~ 1 T + + 1 ~ r v ~ r T ] .apprxeq. [ u ~ 1 u ~ 2 u ~
p ] ( [ l 1 0 r 0 r + 1 0 p ] v ~ 1 T + + [ 0 l r 0 r + 1 0 p ] v ~
r T ) .apprxeq. [ u ~ 1 u ~ 2 u ~ p ] [ l 1 v ~ 1 T l r v ~ r T 0 r
+ 1 0 p ] = i = 1 r l i u ~ i v ~ i T ( 2 ) ##EQU00002##
Thus, in the expansion of {tilde over (X)}, keeping r singular
values means that only the first r columns of and {tilde over (V)}
is needed.
[0042] The singular value decomposition (SVD) (expansion) of {tilde
over (X)} as formulated here, however, has two main disadvantages
in de-noising dynamic images from CT Perfusion, namely: [0043] 1.
The size of the matrix {tilde over (X)} can be as large as
(256).sup.2.times.40-60, making it computationally prohibitive to
find the SVD of {tilde over (X)}; and [0044] 2. Although the
criterion of discarding small singular values makes general sense,
it is difficult to justify the threshold used to keep or discard
singular values.
[0045] An alternative method to achieve the SVD of {tilde over (X)}
will now be described. As stated above, the time vs. enhancement
curves from each pixel (or pixel block) {tilde over (x)}.sub.i,
i=1, . . . ,p can be regarded as repeated samples (observations) of
the underlying random process (time vs. enhancement curve) {tilde
over (.chi.)}. Then the sample mean of the random process, x, can
be calculated as:
x _ = 1 p i = 1 p x ~ i ( 3 ) ##EQU00003##
x by definition is a n.times.1 vector (the sample mean time vs.
enhancement curve) and the j.sup.th component of x, according to
Equation (3), is:
x _ j = 1 p i = 1 p x ~ ij ##EQU00004##
[0046] where {tilde over (x)}.sub.ij is the j.sup.th component of
{tilde over (x)}.sub.i
The zero mean random process from {tilde over (.chi.)} is .psi.=
.chi.- .chi., where .chi. is the population mean of {tilde over
(.chi.)}. For each observation (sample) {tilde over (x)}.sub.i of
{tilde over (.chi.)}, the corresponding observation (sample) of is
given by:
{tilde over (y)}.sub.i={tilde over (x)}.sub.i- x.
The sample mean of {tilde over (.psi.)}, y, a n.times.1 vector is
given by:
y _ = 1 p i = 1 p y ~ i = 1 p i = 1 p ( x ~ i - x _ ) = ( 1 p i = 1
p x ~ i ) - x _ = 0 ~ ##EQU00005##
where {tilde over (0)} is a n.times.1 zero vector as expected since
{tilde over (.psi.)} is a zero mean random process.
[0047] Let {tilde over (Y)}=[{tilde over (y)}.sub.1.sup.T {tilde
over (y)}.sub.2.sup.T . . . {tilde over (y)}.sub.p-1.sup.T {tilde
over (y)}.sub.p.sup.T].sup.T be a p.times.n matrix whose rows are
{tilde over (y)}.sub.i.sup.T i=1, . . . ,p. Then:
Y ~ T Y ~ = [ y ~ 1 y ~ 2 y ~ p - 1 y ~ p ] [ y ~ 1 T y ~ 2 T y ~ p
- 1 T y ~ p T ] = i = 1 p y ~ 1 y ~ 1 T = i = 1 p [ y ~ i 1 2 y ~ i
1 y ~ i 2 y ~ i 1 y ~ in - 1 y ~ i 1 y ~ in y ~ i 2 y ~ i 1 y ~ i 2
2 y ~ i 2 y ~ in - 1 y ~ i 2 y ~ in y ~ i n - 1 y ~ i 1 y ~ i n - 1
y ~ i 2 y ~ in - 1 2 y ~ in - 1 y ~ i n y ~ in y ~ i 1 y ~ in y ~ i
2 y ~ in y ~ i n - 1 y ~ in 2 ] = [ .sigma. 1 2 .sigma. 12 2
.sigma. 1 n - 1 2 .sigma. 1 n 2 .sigma. 21 2 .sigma. 2 2 .sigma. 2
p - 1 2 .sigma. 2 n 2 .sigma. n - 11 2 .sigma. n - 12 2 .sigma. n -
1 2 .sigma. n - 1 n 2 .sigma. n 1 2 .sigma. n 2 2 .sigma. nn - 1 2
.sigma. n 2 ] ( 4 ) ##EQU00006##
where .sigma..sub.i.sup.2 is the sample variance of .psi..sub.i
i=1, . . . ,n and .sigma..sub.ij.sup.2 is the sample covariance of
.psi..sub.i and .psi..sub.j i,j=1, . . . ,n.
[0048] Or,
S ~ = 1 p - 1 Y ~ T Y ~ ##EQU00007##
is the sample covariance matrix of {tilde over (.psi.)} and is of
dimension n.times.n. In CT Perfusion n is typically 40-60. A linear
combination of .psi..sub.i, i=1, . . . ,n, .zeta., can be defined
as:
.zeta.=a.sup.T{tilde over (.psi.)}
where a is a given n.times.1 vector of coefficients. .zeta. is a
random variable (not process) with observations z.sub.i i=1, . . .
,p given by:
z.sub.i=a.sup.T{tilde over (.psi.)}
by definition of .zeta.. The sample mean of .zeta., z, is given
by:
z _ = 1 p i = 1 p z ~ i = 1 p i = 1 p a ~ T y ~ i = a ~ T 1 p i = 1
p y ~ i = 0 ##EQU00008##
Thus, .zeta. is also a zero mean random variable. The sample
variance of .zeta. is given by:
sample var ( .zeta. ) = 1 p - 1 i = 1 p ( z i ) 2 = 1 p - 1 i = 1 p
( a ~ T y ~ i ) ( a ~ T y ~ i ) T = 1 p - 1 i = 1 p a ~ T y ~ i y ~
i T a ~ = 1 p - 1 a ~ T ( i - 1 p y ~ i y ~ i T ) a ~ = a ~ T S ~ a
~ ##EQU00009##
[0049] Determination of the vector of coefficients, a, to maximize
the sample variance of .zeta., a.sup.T{tilde over (S)}a will now be
investigated. Since a.sup.T{tilde over (S)}a can be made as large
as required by scaling a with a constant, the optimization can only
be done by imposing a normalization condition, such as a.sup.T=a=1.
To maximize a.sup.T{tilde over (S)}a subject to a.sup.Ta=1, the
standard approach is to use the technique of Lagrange multipliers.
That is it is sought to maximize:
a.sup.T{tilde over (S)}a-.lamda.(a.sup.Ta-1)
where .lamda. is the Lagrange multiplier. Differentiation with
respect to a gives:
{tilde over (S)}a-.lamda.a={tilde over (0)}
Thus, .lamda. is the eigenvalue of {tilde over (S)} and a is the
corresponding eigenvector and
sample var(.zeta.)=a.sup.T{tilde over
(S)}a=a.sup.T.lamda.a=.lamda., the eigenvalue of {tilde over
(S)}
That is to maximize a.sup.T{tilde over (S)}a subject to a.sup.Ta=1,
a is the eigenvector of {tilde over (S)} that has the highest
eigenvalue.
[0050] Let .lamda..sub.1 be the largest eigenvalue with the
corresponding eigenvector a.sub.1, then
.zeta..sub.1=a.sub.1.sup.T{tilde over (.psi.)} is called the first
sample principal component of {tilde over (.psi.)}. It is a random
variable with observations defined by z.sub.i1=a.sub.1.sup.T{tilde
over (y)}.sub.iz.sub.i1 is also called the score of the i.sup.th
observation on the first principal component. The second principal
component, .zeta..sub.2=a.sub.2.sup.T{tilde over (.psi.)},
similarly, maximizes the sample variance of .zeta..sub.2,
a.sub.2.sup.T{tilde over (S)}a.sub.2, subject to being uncorrelated
to .zeta..sub.1 and the normalization condition
a.sub.2.sup.Ta.sub.2=1. The observations corresponding to
.zeta..sub.2 is z.sub.i2=a.sub.2.sup.T{tilde over (y)}.sub.i. The
sample covariance of .zeta..sub.1 and .zeta..sub.2 is:
sample cov ( .zeta. 1 , .zeta. 2 ) = 1 p - 1 i = 1 p z i 1 z i 2 =
1 p - 1 i = 1 p ( a ~ 1 T y ~ i ) ( a ~ 2 T y ~ i ) T = 1 p - 1 i =
1 p a ~ 1 T y ~ i y ~ i T a ~ 2 = 1 p - 1 a ~ 1 T ( i - 1 p y ~ i y
~ i T ) a ~ 2 = a ~ 1 T S ~ a ~ 2 = 1 p - 1 i = 1 p ( a ~ 2 T y ~ i
) ( a ~ 1 T y ~ i ) T = a ~ 2 T S ~ a ~ 1 ##EQU00010##
The requirement that .zeta..sub.2 is uncorrelated to .zeta..sub.1
is the same as requiring:
a.sub.2.sup.T{tilde over
(S)}a.sub.1=a.sub.2.sup.T.lamda..sub.1a.sub.1=.lamda..sub.1[a.sub.2.sup.T-
a.sub.2]=0 a.sub.2.sup.T{tilde over (S)}a.sub.1=0 and
a.sub.2.sup.Ta.sub.1=0
a.sub.1.sup.T{tilde over (S)}a.sub.2=({tilde over
(S)}a.sub.1).sup.Ta.sub.2=.lamda..sub.1[a.sub.1.sup.Ta.sub.2]=0a.sub.1.su-
p.T{tilde over (S)}a.sub.2=0 and a.sub.1.sup.Ta.sub.2=0
Thus, any of the conditions:
a.sub.2.sup.T{tilde over (S)}a.sub.1=0 a.sub.2.sup.Ta.sub.1=0
a.sub.1.sup.T{tilde over (S)}a.sub.2=0 a.sub.1.sup.Ta.sub.2=0
can be used to specify that .zeta..sub.2 is uncorrelated to
.zeta..sub.1.
[0051] To maximize the sample variance of .zeta..sub.2,
a.sub.2.sup.T{tilde over (S)}a.sub.2 , subject to the conditions
that .zeta..sub.2 is uncorrelated to .zeta..sub.1 and
a.sub.2.sup.Ta.sub.2=1 is the same as to maximize:
a.sub.2.sup.T{tilde over
(S)}a.sub.2-.lamda..sub.2(a.sub.2.sup.Ta.sub.2-1)-.phi.a.sub.2.sup.Ta.sub-
.1
where the condition a.sub.2.sup.Ta.sub.1=0 is used to specify that
.zeta..sub.2 is uncorrelated to .zeta..sub.1 and .lamda..sub.2 and
.phi. are Lagrange multipliers. Differentiation with respect to
a.sub.2 gives:
{tilde over (S)}a.sub.2-.lamda..sub.2a.sub.2-.phi.a.sub.1=0
Multiplying through by a.sub.1.sup.T gives:
a.sub.1.sup.T{tilde over
(S)}a.sub.2-.lamda..sub.2a.sub.1.sup.Ta.sub.2-.phi.a.sub.1.sup.Ta.sub.1=0
which, since the first two terms are zero and
a.sub.1.sup.Ta.sub.1=1, reduces to .phi.=0. Therefore,
{tilde over (S)}a.sub.2-.lamda..sub.2a.sub.2=0
so .lamda..sub.2 is once more an eigenvalue of {tilde over (S)},
and a.sub.2 the corresponding eigenvector.
[0052] Again, a.sub.2.sup.T{tilde over (S)}a.sub.2=.lamda..sub.2 or
the the sample variance of .zeta..sub.2 is .lamda..sub.2. Assuming
that {tilde over (S)} does not have repeat eigenvalues,
.lamda..sub.2 cannot be equal to .lamda..sub.1. If it did, it
follows that a.sub.2=a.sub.1, violating the constraint that
a.sub.2.sup.Ta.sub.1=0. Hence .lamda..sub.2 is the second largest
eigenvalue and a.sub.2 the corresponding eigenvector.
[0053] It can be shown using the same technique that for the third,
fourth, . . . , n.sup.th sample principal components, the vectors
of coefficients a.sub.3, a.sub.4, . . . , a.sub.n are the
eigenvectors of {tilde over (S)} corresponding to
.lamda..sub.3,.lamda..sub.4, . . . ,.lamda..sub.n, the third,
fourth largest, . . . , and the smallest eigenvalue. Furthermore,
the sample variances of the principal components are also given by
the eigenvalues of {tilde over (S)}.
[0054] Define {tilde over (Z)} to be the p.times.n matrix of the
scores of observations on principal components. In this case, the
i.sup.th row consists of the scores of the i.sup.th observation of
{tilde over (.psi.)} (i.e. {tilde over (y)}.sub.i) on all principal
components .zeta..sub.1, .zeta..sub.2, . . . ,.zeta..sub.n-1,
.zeta..sub.n, that is:
Z ~ = [ a ~ 1 T y ~ 1 a ~ j T y ~ 1 a ~ n T y ~ 1 a ~ 1 T y ~ i a ~
j T y ~ i a ~ n T y ~ i a ~ 1 T y ~ p a ~ j T y ~ p a ~ n T y ~ p ]
= [ a ~ 1 T y ~ 1 a ~ 1 T y ~ i a ~ 1 T y ~ p a ~ j T y ~ 1 a ~ j T
y ~ i a ~ j T y ~ p a ~ n T y ~ 1 a ~ n T y ~ i a ~ n T y ~ p ] T =
( [ a ~ 1 T a ~ j T a ~ n T ] [ y ~ 1 y ~ i y ~ p ] ) T = [ y ~ 1 T
y ~ i T y ~ p p T ] T [ a ~ 1 a ~ j a ~ n ] = Y ~ A ~ ( 4 a )
##EQU00011##
where is a n.times.n matrix whose columns are the eigenvectors of
{tilde over (S)} and is orthogonal.
[0055] The first principal component, .zeta..sub.1, maximized is
expressed by:
a ~ 1 T S ~ a ~ 1 = 1 p - 1 a ~ 1 T ( i = 1 p y ~ i y ~ i T ) a ~ 1
= 1 p - 1 a ~ 1 T ( i = 1 p ( x ~ i - x _ ) ( x ~ i - x _ ) T ) a ~
1 ##EQU00012##
({tilde over (x)}.sub.1- x) is the deviation of the time vs.
enhancement curve (TEC) from the i.sup.th pixel (pixel block) from
the mean of TECs from all pixels (pixel blocks). Thus, the
eigenvectors: a.sub.1, a.sub.2, a.sub.3, a.sub.4, . . . a.sub.n
represent the first, second, third, fourth, . . . , and least
dominant time vs. enhancement behaviour in the set of TECs from all
pixels. The loadings of the eigenvectors a.sub.1, a.sub.2, a.sub.3,
a.sub.4, . . . , a.sub.n in the TEC from the i.sup.th pixel (pixel
block) are:
a.sub.1.sup.T{tilde over (y)}.sub.i, a.sub.2.sup.T, {tilde over
(y)}.sub.i, a.sub.3.sup.T{tilde over (y)}.sub.i, a.sub.4.sup.T,
{tilde over (y)}.sub.i, . . . , a.sub.n.sup.T{tilde over
(y)}.sub.i
and is the i.sup.th row of the matrix {tilde over (Z)} while the
loadings of the eigenvector a.sub.i in the TECs from all pixels
(pixel blocks) are:
a.sub.i.sup.T{tilde over (y)}.sub.1, a.sub.i.sup.T{tilde over
(y)}.sub.2, a.sub.i.sup.T{tilde over (y)}.sub.3,
a.sub.i.sup.T{tilde over (y)}.sub.4, . . . , a.sub.i.sup.T{tilde
over (y)}.sub.p
and is the i.sup.th column of the matrix {tilde over (Z)}. The
i.sup.th column of {tilde over (Z)}, therefore, gives the loading
map of the eigenvector a.sub.i which corresponds to the i.sup.th
largest eigenvalue .lamda..sub.i.
[0056] To de-noise the dynamic images from a CT Perfusion study,
the reverse of the calculation of the loading maps of eigenvectors
is required. Reconstituting a smooth version of the original
dynamic images, {tilde over (Y)}({tilde over (X)}), from a
truncated series of eigenvectors a.sub.1, a.sub.2, a.sub.3,
a.sub.4, . . . , a.sub.r is necessary.
[0057] As in Equation (1), the p.times.n matrix {tilde over (Y)}
can be expanded by singular value decomposition as:
{tilde over (Y)}= {tilde over (L)}{tilde over (V)}.sup.T
where is a p.times.p orthogonal matrix, {tilde over (V)} is a
n.times.n orthogonal matrix, and {tilde over (L)} is a p.times.n
diagonal matrix with singular values of {tilde over (Y)} as its
diagonal elements. Let be l.sub.1, . . . ,l.sub.n be the singular
values of {tilde over (Y)} (it is assumed without loss of
generality that {tilde over (Y)} is of full column rank, n and
p>n). Using the singular value decomposition:
S ~ = 1 p - 1 Y ~ T Y ~ = 1 p - 1 ( U ~ L ~ V ~ T ) T U ~ L ~ V ~ T
= 1 p - 1 V ~ L ~ T U ~ T U ~ L ~ V ~ T = 1 p - 1 V ~ L ~ T L ~ V ~
T ##EQU00013##
Multiplying both sides of the equation by {tilde over (V)}
yields:
S ~ V ~ = 1 p - 1 V ~ L ~ T L ~ ( 5 ) ##EQU00014##
{tilde over (L)}=.left brkt-bot.{tilde over (l)}.sub.1 . . . {tilde
over (l)}.sub.i . . {tilde over (l)}.sub.n.right brkt-bot. where
{tilde over (l)}.sub.i i==1, . . . ,n is a p.times.1 vector whose
elements are all zero except the i.sup.th element which is equal to
that is {tilde over (l)}.sub.i.sup.T=[0 . . . l.sub.i . . . 0]
and
1 ~ i T 1 ~ j = { 1 i 2 i = j 0 i .noteq. j Then , L ~ T L ~ = [ 1
~ 1 T 1 ~ i T 1 ~ n T ] [ 1 ~ 1 1 ~ i 1 ~ n ] = [ m ~ 1 T m ~ i T m
~ n T ] ##EQU00015##
where {tilde over (m)}.sub.i.sup.T=[0 . . . l.sub.i.sup.2 . . .
0].
[0058] Let {tilde over (V)}=[{tilde over (v)}.sub.1 . . . {tilde
over (v)}.sub.i . . . {tilde over (v)}.sub.n]where {tilde over
(v)}.sub.i i=1, . . . ,n is a n.times.1 vector.
[0059] Then,
V ~ L ~ T L ~ = [ v ~ 1 v ~ i v ~ n ] [ m ~ 1 T m ~ i T m ~ n T ] =
i = 1 n v ~ i m ~ i T = i = 1 n v ~ i [ 0 1 i 2 0 ] = i = 1 n [ 0 1
i 2 v ~ i 0 ] = [ 1 1 2 v ~ 1 1 i 2 v ~ i 1 n 2 v ~ n ]
##EQU00016##
Substituting in Equation (5) yields:
S ~ V ~ = 1 p - 1 V ~ L ~ T L ~ = [ 1 1 2 v ~ 1 p - 1 1 i 2 v ~ i p
- 1 1 n 2 v ~ n p - 1 ] [ S ~ v ~ 1 S ~ v ~ i S ~ v ~ n ] = [ 1 1 2
v ~ 1 p - 1 1 i 2 v ~ i p - 1 1 n 2 v ~ n p - 1 ] S ~ v ~ i = 1 i 2
p - 1 v ~ i ##EQU00017## i = 1 , , n ##EQU00017.2##
From the discussion on eigenvectors and eigenvalues:
v ~ i = a ~ i V ~ = A ~ ##EQU00018## 1 i 2 p - 1 = .lamda. i
##EQU00018.2##
or the singular values of {tilde over (Y)}(l.sub.i) is equal to the
square root of (p-1) times the eigenvalue value of {tilde over
(S)}(.lamda..sub.i), where
S ~ = 1 p - 1 Y ~ T Y ~ , ##EQU00019##
i.e. l.sub.i= {square root over ((p-1).lamda..sub.i)}
Thus, {tilde over (Y)}.sup.T{tilde over (Y)}{tilde over
(v)}.sub.i=l.sub.i.sup.2{tilde over (v)}.sub.i or {tilde over
(Y)}.sup.Y{tilde over (Y)}a.sub.i i=1, . . . ,n (5a)
[0060] Next, consider
S ~ * = 1 p - 1 Y ~ Y ~ T = 1 p - 1 U ~ L ~ V ~ T ( U ~ L ~ V ~ T )
T = 1 p - 1 U ~ L ~ V ~ T V ~ L ~ T U ~ T = 1 p - 1 U ~ L L ~ T U ~
T ( 6 ) ##EQU00020##
(note that {tilde over (S)}* is not the transpose of {tilde over
(S)})
[0061] Multiplying both sides of the equation by yields:
S ~ U ~ = 1 p - 1 U ~ L ~ L ~ T ##EQU00021## L ~ L ~ T = [ 1 ~ 1 1
~ i 1 ~ n ] [ 1 ~ 1 T 1 ~ i T 1 ~ n T ] = i = 1 n 1 ~ i 1 ~ i T
##EQU00021.2## 1 ~ i 1 ~ i T = [ 0 1 1 i 0 p ] px 1 [ 0 1 1 i 0 p ]
1 xp = [ 0 0 i 0 1 n 0 1 p 0 i 1 1 i 2 0 i n 0 i p 0 n 1 0 n i 0 n
n 0 n p 0 p 1 0 p i 0 pn 0 pp ] ##EQU00021.3## 1 ~ i 1 ~ j T = [ 0
1 1 i 0 j 0 p ] px 1 [ 0 1 0 i 1 j 0 p ] 1 xp = 0 ~ pxp
##EQU00021.4## Thus , L ~ L ~ T = [ 1 1 2 0 i 0 1 n 0 1 p 0 i 1 1 i
2 0 i n 0 i p 0 n 1 0 n i 1 n 2 0 n p 0 p 1 0 p i 0 pn 0 pp ] = [ m
~ 1 * T m ~ i * T m ~ n * T 0 ] ##EQU00021.5##
Let {tilde over (m)}.sub.i*=[0 . . . l.sub.i.sup.2 . . .
0.sub.p].sup.T be a p.times.1 vector as defined, then
L ~ L ~ T = [ m ~ 1 * T m ~ i * T m ~ n * T 0 n + 1 0 p ]
##EQU00022##
Let =[ .sub.1 . . . .sub.i . . . .sub.p] where .sub.i i=1, . . . ,p
is a p.times.1 vector.
[0062] Then,
U ~ L ~ L ~ T = [ u ~ 1 u ~ i u ~ n u ~ n + 1 u ~ p ] [ m ~ 1 * T m
~ i * T m ~ n * T 0 n + 1 0 p ] = i = 1 n u ~ i m ~ i T 1 = i = 1 n
u ~ i [ 0 1 i 2 0 p ] + i = n + 1 p u ~ i [ 0 0 i 0 p ] = i = 1 n [
0 1 i 2 u ~ i 0 p ] = [ 1 1 2 u ~ 1 1 i 2 u ~ i 1 n 2 u ~ n 0 n + 1
0 p ] ##EQU00023##
Substituting in Equation (6) yields:
S * ~ U ~ = 1 p - 1 U ~ L ~ L ~ T = [ 1 1 2 u ~ 1 p - 1 1 i 2 u ~ i
p - 1 1 n 2 u ~ n p - 1 0 n + 1 0 p ] [ S * ~ u ~ 1 S * ~ u ~ i S *
~ u ~ n 0 n + 1 0 p ] = [ 1 1 2 u ~ 1 p - 1 1 i 2 u ~ i p - 1 1 n 2
u ~ n p - 1 0 n + 1 S ~ u ~ i { 1 i 2 p - 1 u ~ i i .ltoreq. n 0 u
~ i n + 1 .ltoreq. i .ltoreq. p ##EQU00024##
[0063] Thus, .sub.i i=1, . . . ,n is the eigenvector of {tilde over
(S)}* corresponding to the i.sup.th eigenvalue
1 i 2 p - 1 , ##EQU00025##
which happens also to be the i.sup.th eigenvalue of {tilde over
(S)}, while all .sub.i i=n+1, . . . ,p are eigenvectors of {tilde
over (S)}* with zero eigenvalue. Also
{tilde over (Y)}{tilde over (Y)}.sup.T .sub.i=l.sub.i.sup.2 .sub.i
i=1, . . . ,n (6a)
As stated above, Equation (5a) is:
{tilde over (Y)}.sup.T{tilde over (Y)}{tilde over
(v)}.sub.i=l.sub.i.sup.2{tilde over (v)}.sub.i
[0064] Multiplying both sides by {tilde over (Y)} yields:
{tilde over (Y)}{tilde over (Y)}.sup.T{tilde over (Y)}{tilde over
(v)}.sub.i=l.sub.i.sup.2{tilde over (Y)}{tilde over (v)}.sub.i
Comparing with Equation (6a):
{tilde over (Y)}{tilde over (v)}.sub.i=c .sub.i where c is a
constant (7a)
[0065] Similarly multiplying both sides of Equation (6a) by {tilde
over (Y)}.sup.T yields:
{tilde over (Y)}.sup.T{tilde over (Y)}{tilde over (Y)}.sup.T
.sub.i=l.sub.i.sup.2{tilde over (Y)}.sup.T .sub.i
Comparing with Equation (5a):
{tilde over (Y)}.sup.T .sub.i=c*{tilde over (v)}.sub.i where c is a
constant (7b)
From Equation (7b):
[0066] v ~ i = 1 c * Y ~ T u ~ i ##EQU00026##
Substituting into Equation (7a) yields:
{tilde over (Y)}{tilde over (Y)}.sup.T .sub.i=c*c .sub.i
Substituting into Equation (6a) yields:
l.sub.i.sup.2 .sub.i=c*c .sub.i
Therefore, c*c=l.sub.i.sup.2
[0067] Without loss of generality, c* is set to:
c*=c=l.sub.i= {square root over ((p-1).lamda..sub.i)}i=1, . . . ,n
and n<p
Thus, for the singular value decomposition of {tilde over (Y)}:
{tilde over (Y)}= {tilde over (L)}{tilde over (V)}.sup.T
where is a p.times.p orthogonal matrix:
=[ .sub.1 . . . .sub.i . . . .sub.p] and .sub.i i=1, . . . ,p is a
p.times.1 vector;
{tilde over (V)} is a n.times.n orthogonal matrix:
{tilde over (V)}=[{tilde over (v)}.sub.i . . . {tilde over
(v)}.sub.i . . . {tilde over (v)}.sub.n] and {tilde over (v)}.sub.i
i=1, . . . ,n is a n.times.1 vector;
and {tilde over (L)}is a p.times.n diagonal matrix with singular
values of {tilde over (Y)}, l.sub.1, . . . ,l.sub.n, as its
diagonal elements; it is also assumed without loss of generality
that {tilde over (Y)} is of full column rank, n and p>n.
[0068] It is shown that: [0069] {tilde over (v)}.sub.i=a.sub.i
where a.sub.i is the i.sup.th eigenvector of
[0069] S ~ = 1 p - 1 Y ~ T Y ~ ##EQU00027##
corresponding to the eigenvaluie
1 i 2 ( p - 1 ) ; ##EQU00028##
and from Equation (7c)
u ~ i = 1 c * Y ~ v ~ i = 1 ( p - 1 ) .lamda. i Y ~ a ~ i
##EQU00029##
As discussed above, to de-noise {tilde over (Y)} the SVD expansion
can be truncated after the r.sup.th singular value:
Y ~ .apprxeq. i = 1 r 1 i u ~ i v ~ i T = i = 1 r 1 i u ~ i a ~ i T
= i = 1 r 1 i 1 ( p - 1 ) .lamda. i Y ~ a ~ i a ~ i T = i = 1 r Y ~
a ~ i a ~ i T Y ~ .apprxeq. Y ~ i = 1 r a ~ i a ~ i T ( 8 )
##EQU00030##
[0070] Equation (8) also affords a very simple geometric
interpretation as explained in the following.
Y ~ .apprxeq. Y ~ i = 1 r a ~ i a ~ i T = i = 1 r Y ~ a ~ i a ~ i T
= ( Y ~ a ~ 1 Y ~ a ~ 2 Y ~ a ~ r - 1 Y ~ a ~ r ) ( a ~ 1 T a ~ 2 T
a ~ r - 1 T a ~ r T ) = Y ~ A ~ r A ~ r T ( 9 ) ##EQU00031##
where .sub.r is a n.times.r matrix whose i.sup.th column is a.sub.i
and from Equation (4a):
{tilde over (Z)}.sub.r={tilde over (Y)} .sub.r
and the rows and columns of {tilde over (Z)}.sub.r have the same
interpretation as those of {tilde over (Z)}. Rewrite Equation
(9):
Y ~ .apprxeq. Y ~ i = 1 r a ~ i a ~ i T = i = 1 r Y ~ a ~ i a ~ i T
= ( Y ~ a ~ 1 Y ~ a ~ 2 Y ~ a ~ r - 1 Y ~ a ~ r ) ( a ~ 1 T a ~ 2 T
a ~ r - 1 T a ~ r T ) = ( ( y ~ 1 T y ~ 2 T y ~ p - 1 T y ~ p T ) a
~ i ( y ~ 1 T y ~ 2 T y ~ p - 1 T y ~ p T ) a ~ 2 ( y ~ 1 T y ~ 2 T
a ~ p - 1 T a ~ p T ) a ~ r - 1 ( y ~ 1 T y ~ 2 T y ~ p - 1 T y ~ p
T ) a ~ r ) ( a ~ 1 T a ~ 2 T a ~ r - 1 T a ~ r T ) = ( y ~ 1 T a ~
1 y ~ 1 T a ~ 2 y ~ 1 T a ~ r - 1 y ~ 1 T a ~ r y ~ 2 T a ~ 1 y ~ 2
T a ~ 2 y ~ 2 T a ~ r - 1 y ~ 2 T a ~ r y ~ p - 1 T a ~ 1 y ~ p - 1
T a ~ 2 y ~ p - 1 T a ~ r - 1 y ~ p - 1 T a ~ r y ~ p T a ~ 1 y ~ p
T a ~ 2 y ~ p T a ~ r - 1 y ~ p T a ~ r ) ( a ~ 1 T a ~ 2 T a ~ r -
1 T a ~ r T ) = ( i = 1 r ( y ~ 1 T a ~ i ) a ~ i T i = 1 r ( y ~ 2
T a ~ i ) a ~ i T i = 1 r ( y ~ p - 1 T a ~ i ) a ~ i T i = 1 r ( y
~ p T a ~ i ) a ~ i T ) ( 10 ) ##EQU00032##
[0071] Equation (10) suggests that the TEC for the i.sup.th pixel
(pixel block) is a weighted summation of r eigenvectors with the
weights given by the loading of the i.sup.th pixel TEC on the
eigenvectors.
[0072] The eigenvectors a.sub.1,a.sub.2, . . . , a.sub.r-1, a.sub.r
can be regarded as forming an orthonormal basis in n dimensional
space. The projection of {tilde over (y)}.sub.i on a.sub.i is
{tilde over (y)}.sub.i.sup.Ta.sub.i and {tilde over (y)}.sub.i can
be reconstituted as
i = 1 r ( y ~ 1 T a ~ j ) a ~ j T . ##EQU00033##
[0073] The alogorithm to de-noise {tilde over (X)}=[{tilde over
(x)}.sub.i.sup.T . . . {tilde over (x)}.sub.i.sup.T . . . {tilde
over (x)}.sub.p.sup.T] is as follows: [0074] 1. Calculate the
p.times.n zero mean matrix, {tilde over (Y)}=[{tilde over
(y)}.sub.1.sup.T . . . {tilde over (y)}.sub.i.sup.T . . . {tilde
over (y)}.sub.p.sup.T].sup.T, corresponding to {tilde over (X)},
where
[0074] {tilde over (y)}.sub.i={tilde over (x)}.sub.i- x and
x _ = 1 p i = 1 p x ~ i ##EQU00034## [0075] 2. Calculate the
n.times.n sample covariance matrix, {tilde over (S)}:
[0075] S ~ = 1 p - 1 Y ~ T Y ~ ##EQU00035## [0076] 3. Calculate the
n pairs of eigenvalue and eigenvectors,
[0076] (.lamda..sub.i,a.sub.i)i=1, . . . ,n [0077] 4. Calculate the
n.times.n smoothing matrix by keeping only the largest r
eigenvalues:
[0077] i = 1 r a ~ i a ~ i T ##EQU00036## [0078] 5. Smooth {tilde
over (Y)}=[{tilde over (y)}.sub.1 . . . {tilde over (y)}.sub.i . .
. {tilde over (y)}.sub.p] by calculating the product:
[0078] Y ~ * = [ y ~ 1 * y ~ i * y ~ p * ] .apprxeq. Y i = 1 r a ~
i a ~ i T ##EQU00037## [0079] 6. Calculate a smoothed version of
{tilde over (X)}* as follows:
[0079] {tilde over (X)}*=[{tilde over (y)}.sub.1*+ x . . . {tilde
over (y)}.sub.i*+ x . . . {tilde over (y)}.sub.p*+ x].
[0080] An example of the method described herein will now be
demonstrated with regard to experimental results. The principal
component analysis (PCA) described herein is tested for the ability
to achieve dose reduction of CT scans.
[0081] The PCA method will now be summarized based on a CT
Perfusion study that generates a plurality of time separated images
that capture the passage of contrast through the brain.
Corresponding to each pixel in the series of images, a curve is
generated that represents the contrast concentration in that pixel
as a function of time. A difference curve, equal to the difference
of the curve from the mean of all pixel curves is also generated. A
covariance matrix of the co-variations of all pairs of difference
curves is calculated and represents all possible variations about
the mean curve. The principal component analysis method analyzes
the covariance matrix to find common temporal patterns among the
curves by finding linear combinations of the difference curves that
represent most of the variations about the mean curve. It can be
shown that such linear combinations, or principal components, are
the eigenvectors of the covariance matrix and that the eigenvector
with the largest eigenvalue is the most prominent temporal feature
among all the difference curves, the eigenvector with the second
largest eigenvalue is the second most prominent temporal feature
and so on, while eigenvectors with small eigenvalues are from noise
in the CT images. A corollary is that each difference curve can be
represented as a sum of all the principal components and by
discarding components with small eigenvalues, noise can be
effectively suppressed. To denoise the time series of images, the
mean curve is added to all the difference curves reconstituted by
using the first few principal components with large
eigenvalues.
[0082] While currently available statistical filters remove noise
in projection data before image reconstruction, the principal
component analysis method removes noise from the reconstructed
images. Experimental results described below show that performing
PCA leads to dose reduction in CT imaging.
[0083] The PCA method is tested versus the traditional filtered
back-projection (FBP) method in radiation dose reduction for CT
Perfusion applications. The dose saving of PCA versus FBP on
measuring brain perfusion using CT Perfusion was tested in a
healthy pig, which was scanned four times with the x-ray tube
current changed from 190 mA to 100 mA; 75 mA and 50 mA,
respectively, while keeping other scanning parameters the same.
[0084] FIG. 3 shows brain blood flow maps of the pig calculated
using the J-W model with CT Perfusion images obtained using (A) 190
mA and FBP reconstruction; (B) 100 mA, FBP reconstruction and PCA;
(C) 70 mA, FBP reconstruction and PCA; and, (D) 50 mA, FBP
reconstruction and PCA. FIG. 3(E) shows regions of interest (3
outlined circles) used for calculation of Figure of Merit of the
blood flow maps shown in FIG. 4. The regions are superimposed on a
CT image which shows a coronal section through the brain of the
pig. The dark areas are the skull and jaw bones.
[0085] FIG. 4 shows that in terms of standard deviation/mean of
blood flow (Figure of Merit) in the three regions shown in FIG.
3(E): 70 mA with PCA was better than 190 mA with FBP, while 50 mA
PCA was worse than 190 mA with FBP (p<0.05). The image quality
of the blood flow maps in FIG. 3 also reflects this progression.
Therefore, the PCA method is able to reduce radiation dose required
to obtain images in CT Perfusion studies and in maintaining the
quality of derived blood flow maps compared to the use of FBP
alone. More specifically, the PCA method achieved approximately
three fold reduction in radiation dose compared to the use of FBP
alone. The effect of the PCA method on dose reduction in production
of BBB-PS maps was not demonstrated because the blood-brain barrier
remained intact. However, since blood flow, blood volume and BBB-PS
are calculated together in the J-W model, the dose reduction shown
using blood flow maps is expected to apply to BBB-PS maps.
[0086] The principal component analysis methodology described
herein has been implemented in a computer executable software
application using C++. The time required to process ninety-five
(95) 512.times.512 CT brain images is less than two (2) minutes on
a general purpose computing device such as for example a personal
computer (PC).
[0087] The method described herein can be implemented on hardware
such as system 400 shown in FIG. 5. An input 510 receives a
plurality of time separated images, which can be previously stored,
received directly from an imaging device, or communicated over a
wired or wireless communication link from a remote location. A
general purpose computing device 505 receives the time separated
images from the input and executes a software application thereby
to process the time separated images as described above. The
general purpose computing device 505 interfaces with the user
through a display 520, a keyboard 515 and/or a mouse or other
pointing device 525. Results, for example the constructed processed
images, can be presented on display 520 and/or output to any
suitable output device 530, for example, a printer, a storage
device, or a communication device for communicating the results to
a remote location.
[0088] The software application described herein may run as a
stand-alone application or may be incorporated into other available
applications to provide enhanced functionality to those
applications. The software application may include program modules
including routines, programs, object components, data structures
etc. and may be embodied as computer readable program code stored
on a computer readable medium. The computer readable medium is any
data storage device that can store data, which can thereafter be
read by a computer system. Examples of computer readable media
include for example read-only memory, random-access memory,
CD-ROMs, magnetic tape and optical data storage devices. The
computer readable program code can also be distributed over a
network including coupled computer systems so that the computer
readable program code is stored and executed in a distributed
fashion.
[0089] The above-described embodiments are intended to be examples
and alterations and modifications may be effected thereto, by those
of skill in the art, without departing from the scope of the
invention which is defined by the claims appended hereto.
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