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Claims  |
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I claim:
1. A method for processing an image, said image comprising a plurality of
pixels representing a region of interest partitioned into a plurality of
voxels, said method comprising the steps of:
providing a plurality of multidimensional signals, each of said signals
being associated with each of said voxels, said plurality of voxels being
mapped into said plurality of signals in a signal space by a mapping M;
computing a density function representative of the frequency of occurrence
of each of said multidimensional signals in said signal space; and
assigning a tone value to each of said plurality of voxels in accordance
with said density function
2. A method for processing an image, said image comprising a plurality of
pixels representing a region of interest partitioned into a plurality of
voxels, comprising the steps of:
providing a plurality of multidimensional signals, each of said
multidimensional signals comprising a plurality of parameters containing
information about said region of interest, each of said signals being
further associated with each of said voxels, said plurality of voxels
being mapped into said plurality of signals in a signal space by a mapping
M, said mapping M being a multidimensional random variable having a
density function;
computing said density function representative of the frequency of
occurrence of each of said multidimensional signals in said signal space;
convolving said density function with a resolution function so as to
provide a real valued probability density function defined for each of
said plurality of voxels;
assigning a tone value to each of said plurality of voxels corresponding to
the value of said probability density function for each of said voxels;
and
displaying a tone level image wherein each voxel is represented by a pixel
with said assigned tone value.
3. A method for processing an image as defined in claim 2, wherein said
convolving step comprises the step of convolving a Gaussian function of
mean zero and variance .sigma. with said density function
4. A method for processing an image as defined in claim 3, wherein said
Gaussian has a variance .sigma. equal to or greater than the variance .nu.
of said density function of said signal mapping M.
5. A method for processing an image as defined in claim 4, wherein said
variance .sigma. of said Gaussian is a multiple of said variance .nu. of
said density function of said signal mapping M.
6. A method for processing an image as defined in claim 3, wherein said
variance .sigma. defines the resolution of said image in said signal
space, whereby high values of the variance .sigma. with respect to the
variance .nu. produce a low resolution image in said signal space and
whereby low values of the variance .sigma. with respect to the variance
.sigma. produce a high resolution image in said signal space.
7. A method for processing an image as defined in claim 3, wherein said
image has a contrast defined by said tone value and said signal space is
defined with a metric, said contrast between two voxels of said plurality
of voxels being independent on the metric in said signal space.
8. A method for processing an image as defined in claim 2, wherein said
convolving step includes the step of convolving a cosine function with
said density function of said mapping M.
9. A method for processing an image as defined in claim 2, wherein said
convolving step includes the step of convolving a pulse function having a
width .sigma. with said density function of said mapping M.
10. A method for processing an image as defined in claim 2, wherein said
convolving step includes the step of convolving a sinc function with said
density function of said mapping M.
11. A method for processing an image as defined in claim 2, wherein said
convolving step includes the step of convolving an exponential function
with said density function of said mapping M.
12. A method for processing an image as defined in claim 2, wherein said
convolving step includes the step of convolving a filter function with
said density function of said mapping M.
13. A method for processing an image as defined in claim 2, wherein said
density function is given by the following formula:
##EQU14##
wherein .delta. is the delta function, where w is the voxel in the set of
voxels comprising said region of interest, and s is any possible signal
value that a voxel could have as its measured value.
14. A method for processing an image as defined in claim 2, wherein said
convolution product is performed using Fourier transforms.
15. A method for processing an image as defined in claim 2, wherein said
convolution product is performed using Fast Fourier transforms.
16. A method for processing an image as defined in claim 2, wherein said
probability density function is given by the following equation:
H.sigma.(s)=G.sigma.(s)*H.sub.o (s),
wherein * denoted the convolution product, G.sigma.(s) represents said
distribution function and H.sub.o (s) represents the density function of
same mapping M, and where .sigma. is any real number, .sigma. being the
standard deviation of said distribution function G.sigma.(s).
17. A method for processing an image as defined in claim 2, wherein said
image is a medical image
18. A method for processing an image as defined in claim 2, wherein said
signals are NMR signals.
19. A method for processing an image as defined in claim 2, wherein said
signals are obtained by computer tomography.
20. A method for processing an image as defined in claim 2, wherein said
signals are obtained from X-ray projections of said region of interest.
21. A method for processing an image as defined in claim 2, wherein said
tone value is a grey value.
22. A method for processing an image as defined in claim 2, wherein said
tone value is a color.
23. A method for processing an image as defined in claim 2, wherein said
step of assigning a tone value is performed by segmentation.
24. A method for processing an image as defined in claim 2, wherein said
convolving step is performed by using a multidimensional resolution
function so as to obtain a single image incorporating said information
from all of said plurality of parameters.
25. A method for processing an image as defined in claim 2, wherein said
convolving step is performed by using a plurality of unidimensional
resolution functions so as to obtain a plurality of images corresponding
to each of said parameters of said multidimensional signals in said signal
space.
26. A method for processing an image, said method comprising a plurality of
pixels representing a region of interest partitioned into a plurality of
voxels, comprising the steps of:
providing a plurality of multidimensional signals, each of said signals
being associated with each of said voxels, said plurality of voxels being
mapped into said plurality of signals in a signal space by a mapping M,
said mapping M being a multidimensional random variable and defining a
probability measure in said signal space;
computing the Radon-Nikodym derivative of said probability measure of said
mapping M, said Radon-Nikodym derivative defining a probability density
function;
assigning a tone value to each of said plurality of voxels corresponding to
the value of said probability density function for each of said voxels;
and
displaying a tone level image wherein each voxel is represented by a pixel
with said assigned tone value.
27. An apparatus for processing an image comprising a plurality of pixels
representing a region of interest partitioned into a plurality of voxels,
said apparatus comprising:
an input imaging device for providing a plurality of multidimensional
signals, each of said signals being associated with each of said voxels,
said plurality of voxels being mapped into said plurality of signals in a
signal space by a mapping M, said mapping M being a multidimensional
random variable and defining a probability measure in said signal space;
means for computing a density function representative of the frequency of
occurrence of each of said multidimensional signals in said signal space;
means for convolving said density function with a resolution function so as
to provide a real valued probability density function defined for each of
said plurality of voxels;
means for assigning a tone value to each of said plurality of voxels
corresponding to the value of said probability density function for each
of said voxels; and
means for displaying a tone level image wherein each voxel is represented
by a pixel with said assigned tone value.
28. An apparatus for processing an image comprising a plurality of pixels
representing a region of interest partitioned into a plurality of voxels,
said apparatus comprising:
an input imaging device for providing a plurality of multidimensional
signals, each of said signals being associated with each of said voxels,
said plurality of voxels being mapped into said plurality of signals in a
signal space by a mapping M, said mapping M being a multidimensional
random variable and defining a probability measure in said signal space;
means for computing the Radon-Nikodym derivative of said probability
measure of said mapping M, said Radon-Nikodym derivative defining a
probability density function;
means for assigning a tone value to each of said plurality of voxels
corresponding to the value of said probability density function for each
of said voxels; and
means for displaying a tone level image wherein each voxel is represented
by a pixel with said assigned tone value.
29. An apparatus for processing an image comprising a plurality of pixels
representing a region of interest partitioned into a plurality of voxels,
said apparatus comprising:
an input imaging device for providing a plurality of multidimensional
signals, each of said signals being associated with each of said voxels,
said plurality of voxels being mapped into said plurality of signals in a
signal space by a mapping M, said mapping M being a multidimensional
random variable and defining a probability measure in said signal space;
means for computing a density function representative of the frequency of
occurrence of each of said multidimensional signals in said signal space;
means for assigning a tone value to each of said plurality of voxels
corresponding to the value of said probability density function for each
of said voxels; and
means for displaying a tone level image wherein each voxel is represented
by a pixel with said assigned tone value.
30. An apparatus for processing an image according to claim 29, wherein
said input imaging device is a NMR device.
31. An apparatus for processing an image according to claim 29, wherein
said input imaging device is an X-ray camera.
32. An apparatus for processing an image according to claim 29, wherein
said input imaging device is a CT device. |
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Claims |
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Description |
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A portion of the disclosure of this patent document contains material which
is subject to copyright protection. The copyright owner has no objection
to the facsimile reproduction by anyone of the patent document or the
patent disclosure in its entirety, as it appears in the Patent and
Trademark Office patent file or records, but otherwise reserves all
copyright rights, whatsoever.
FIELD OF THE INVENTION
The present invention relates to an image processor for displaying
gray-level images and particularly to image processors used for medical
applications. The present invention also relates to a method for analyzing
and extracting information from an analog or digital image signal so as to
produce a high contrast between two regions having similar signal
characteristics and physically proximate to one another.
BACKGROUND OF THE INVENTION
The process of detecting and characterizing a specific portion of a region
to be imaged has increasingly been used in the treatment of information on
images associated with various disciplines such as medicine, biomedicine,
nuclear physics, satellite imaging, non-destructive inspection (NDI) and
other imaging fields. Considerable effort has been expended to enhance the
contrast of an image with respect to two distinct domains of a region of
interest that are physically close and have similar signal
characteristics.
In non-destructive inspections, the aim of an efficient imaging technology
is to ascertain the presence and the location of small flaws in the
material to be inspected. Imaging technologies have also become extremely
valuable in the medical field and have proven to be quite effective in the
field of medical diagnosis. In particular, computed tomography (CT X-ray
scans) and Magnetic Resonance Imaging (MRI) are used extensively in
medical imaging for generating high quality images of the body. In medical
imaging, a frequent objective is to distinguish normal from abnormal
tissue, for example a tumor growth on an organ. This presents a very
difficult imaging problem in at three respects. First, the signals
representative of the abnormal tumor cells are often quite similar to the
signals representative of the normal cells of the organ. Second, since it
is desirable to detect such abnormal growths as early as possible, the
total volume or number of abnormal cells, is much smaller than the total
volume or number of normal cells. Third, the abnormal cells are
immediately adjacent to or in close proximity to the normal cells.
In most imaging techniques, and in particular medical imaging, the region
of interest, i.e., the region of the object being imaged, is typically
partitioned into a plurality of volume elements called voxels. A signal
derived from the imaging technology (e.g. an X-ray signal in CT or an NMR
signal in MRI) is averaged over each individual volume element or voxel.
The composite of these averaged signals for all of the voxels comprising
the region of interest form a representation of the object being imaged in
signal space. These signals indirectly correspond to physical and/or
chemical characteristics of the materials comprising the voxels being
imaged. The properties of the materials within the region of interest
directly affect the signals corresponding to their respective voxel
elements. The manner in which the signals are affected is, of course,
dependent upon the particular phenomenon, i.e., X-ray, NMR, etc., involved
in the imaging technique. The signals may comprise a single component or a
plurality of components. If the signals consist of a single component, as
in computed tomography, they form a scalar field. If the signals comprise
more than one component, as in MRI, the signal from each voxel can be
characterized as a signal vector in a signal space having a dimension
equal to the number of components comprising each signal vector. In
magnetic resonance imaging, three components of interest are typically
selected to describe the characteristics of the region of interest being
imaged. These components are the spin-lattice relaxation time T.sub.1, the
spin-spin relaxation time T.sub.2, and a spin density for hydrogen. The
components of the signal vectors are assumed to form a set of basis
vectors which span the signal space for all regions of interest within the
object being inspected. This assumption is not always true since the
individual components of the signals are often correlated, i.e., not
independent. For instance, in magnetic resonance imaging, the proton
density is related to the values of the times of relaxation T.sub.1 and
T.sub.2. An increase in water content not only increases the hydrogen
density and hence the proton density, but it also increases the times of
relaxation, T.sub.1 and T.sub.2. The proton density and relaxation times
T.sub.1 and T.sub.2 do, however, form a good approximation to a basis for
a three dimensional signal space. Thus, the signal from a voxel within the
region of interest is mapped into a 3-dimensional vector in the signal
space. The magnitude and direction of each vector is determined by the
intensity of the signals corresponding to each of the three components.
Other components may also be considered in MRI, such as microscopic
diffusion and microscopic rotational states of the resonant nuclei.
An image corresponding to the signal values of the voxels can be displayed,
for example, on a standard CRT in digital format. The image is partitioned
into a set of picture elements called pixels, there being a one-to-one
correspondence between the voxels of the region of interest to be imaged
and between the pixels of the displayed image. The displayed image is
obtained by assigning a value, for instance a grey tone or a color, to
each pixel. The grey tone or color of each pixel is determined by the
properties of the signal vector associated with its respective voxel in
the region of interest. In an ideal case, all voxels which represent the
same physical characteristic of the object, for example, a particular type
of tissue in medical imaging, should be associated with the same point in
signal space. However, imaging devices are not ideal and various sources
of noise result in the same type of tissue being associated with a
distribution of signal vectors in signal space. Additionally, two
different types of tissue, for example grey matter brain tissue and white
matter brain tissue, may have signal vectors which are very similar. Thus,
one of the key objectives in imaging technology is to characterize every
significant domain in the region of interest (also referred to as "ROI")
to be imaged so that distinct domains are distinguished in the displayed
image, even if their respective signal characteristics are similar.
This problem of characterization of a specific subregion is rendered even
more difficult by the presence of noise in the signal. Two types of noise
are present in most signals: electronic noise and biological noise. The
effects and properties of electronic noise are well understood and
standard techniques have been developed to deal with it in imaging. The
electronic noise is usually considered to be a random variable. As
subregions of the region of interest are represented by a
multi-dimensional distribution within the signal space, it may occur that
those distributions for various subregions may overlap. This creates a
problem in the identification of a grid pattern for a digitized subregion
insofar as two subregions having similar characteristics may not be
sufficiently contrasted in the final output image.
Various imaging techniques have been proposed to enhance contrast and
circumvent the problem set forth above. The methods known to date for
displaying image corresponding to certain signal characteristics fall into
two major categories. In a first approach, the image is partitioned into a
finite set of segments. This method, known as image segmentation, has been
extensively used. In a second method, known as continuous grey-scale
imaging, a grey value is assigned to each pixel in accordance with the
intensity of the signal vector. Both of these techniques have various
drawbacks that will now be briefly discussed.
The objective of the segmentation method is to group together voxels and
produce an image by assigning a color to each of these groups. In a
proposed segmentation method, the signal data are clustered into groups
with similar signal characteristics. A distinct uniform color is then
assigned to each of the clusters having the same signal characteristics.
Recently, there has been an increased interest in a data clustering
algorithm called fuzzy c-means clustering. By definition, a fuzzy set U is
a function from a subset of R.sup.p into the interval [0,1] which assigns
to each element of the subset X a grade of membership. The membership
function is a function valued between zero and one and is a key concept of
fuzzy sets. In this method, a signal vector having p components in the
signal space is assigned to each of the voxels. The values of the
components of the vector are the pixel intensities of the acquired images.
The calculation of the membership function determining to which cluster a
particular voxel belongs requires the definition of a distance metric.
Typically, this distance metric is chosen to be the simple Euclidean
distance in the signal space. The clustering begins by then randomly
choosing a set of c-cluster centers v.sub.i within the signal space and
defining a membership function u.sub.ik which would give the membership
grade for the pixel k being a member of the fuzzy cluster i. Cluster
centers are thereby redefined, and the procedure is iterated until the
movement of cluster centers is made as small as possible. Pixels are
assigned to clusters based on which of the c-membership functions is
largest. One drawback of this algorithm is that the number of c clusters
is arbitrarily chosen. Typically, the number of clusters is deliberately
chosen to be higher than the expected number of individual subregion
types.
Several methods are of interest to implement this algorithm. A first method
called hierarchical processing using pyramid algorithms has turned out to
be a valuable method because it allows the image to be examined at several
levels of resolution at one time. The person skilled in the art will be
able to find a thorough description of such a method in the following
article: "Tissue Type Identification by MRI Using Pyramidal Segmentation
and Intrinsic Parameters," Ortendhal Douglas, et al., Proceeding of the
Ninth International Conference on Information Processing in Medical
Imaging, Washington, D.C. (1985). In a second method, shape descriptors
are used for identifying regions of suspected partial volume averaging. In
a third method, histogram analysis can also be advantageously used as a
form of gray level statistics. In this histogram method, the signal values
are plotted in histograms according to their frequency of occurrence.
However, in this histogram equalization method, the voxels are first
assigned a gray level value which is then plotted in a histogram according
to the frequency of occurrence. Such a histogram is therefore referred to
as a gray level histogram. A summary of these segmentation methods can be
found in: Ortendhal D.A., et al, MRI Image Segmentation Using Fuzzy Set
Clustering and Spatial Correlations. "Book of Abstracts", Society of
Magnetic Resonance in Medicine, 6th Annual Meeting, Aug. 17-21, 1987; and
in Sklansky J., et al: Image Segmentation and Feature Extraction, IEEE
Transactions on Systems, Man and Cybernetics, Vol. SMC-8, No. 4, Apr.
1978. In another proposed segmentation method, a boundary is drawn in the
image whenever signal variation between two neighboring voxels is greater
than a preset threshold. The drawback of this segmentation method still
resides in the arbitrariness of the selection of the threshold. Numerous
other ways of partitioning the image have also been proposed without
resolving the inherent problems entailed by segmentation of an image.
The main disadvantage of the segmentation method remains in the
arbitrariness of the clustering of the image, irrespective of the
technique used to perform the segmentation itself (fuzzy set,
threshold,...). More specifically, in the clustering algorithm, there is
no way of determining what number of clusters should be used for
partitioning the region of interest to be imaged. This also applies to the
choice of the threshold. Since the threshold value or the number of
clusters is arbitrary and the color assignment discontinuous, errors in
the color segmentation are likely to occur. In particular, voxels of
distinct domains having similar signal values may be represented by the
same color. Several patents make use of the segmentation method or a
refinement thereof.
Reference is first made to the Watanabe patent, U.S. Pat. No. 3,805,239. In
the Watanabe patent, there is disclosed an apparatus comprising a memory
device for storing in matrix format electrical signals corresponding to
the gray levels of the respective picture elements of a pattern. In this
apparatus, the differences between gray levels of a central picture
element and of eight surrounding picture elements are determined in order
to obtain some of the differences between the central picture element and
the surrounding picture elements. These differential sums are then added
up for each matrix containing a given picture element. After completing
such additions with respect to numerous matrices in which different
picture elements constitute the central one, the gray level of the picture
element taken as the central one and which gives a maximum value from
among the totals of differential sums thus computed is then detected.
There is also provided a device for reading out of the memory device data
on a prescribed gray level higher or lower than the gray level of maximum
value which is used as a threshold value.
Another patent, U.S. Pat. No. 4,340,911 to Kato also uses the segmentation
method. In this patent, the densities of three different tissues are
plotted in a histogram. The three peaks corresponding to the frequency
distribution of those three tissues are used to determine the threshold
values necessary to produce the desired gradation of the image. Boundary
levels are then determined to perform the gradation processing. Contrast
between two tissues can be raised by lowering the level of the minimum
density of the image corresponding to one of the tissues, down to the
level of the fog density of the image.
The patent to Toraichi (U.S. Pat. No. 4,538,227) also uses the segmentation
technique. In the Toraichi patent, the image processor obtains information
about an organ to be imaged such as the boundary diagram, the volume, the
centroid movement view and a three-dimensional view. The gray level values
of given X-ray projections are plotted in a histogram. The extraction of
the image boundary from the provided processed image is thus accomplished
by segmenting a given image into a plurality of small regions, forming the
small regions into ternary coded signals in comparison with thresholds
K.sub.1 and K.sub.2 which are determined by the histogram of the gray
scale of the gray level values.
Another approach to image processing consists in assigning a gray scale
value to each of the pixels mapping into the voxels of the region of
interest, whereby the intensity of the gray value is a function of the
signals emitted by the voxels. This imaging technique, referred to as
continuous gray scale imaging, is widely used. The basics of this method
will now be briefly summarized.
A gray scale is by definition a one-dimensional real number line. To each
voxel in the image, there is assigned a real value corresponding to a gray
scale value. Actually, the grey scale is a discrete set of grey tones, but
the assumption that the grey scale is a continuous scale is legitimate to
a first approximation. As the signal space is typically a
multi-dimensional space and the gray scale value a one-dimensional space,
the mapping from the signal space into the gray scale real number line is
necessarily not homomorphic. Projections from the signal space into the
gray real number line are typically used to perform the grey value
assignment.
In the projection method, a gray scale value is assigned to each pixel
corresponding to the value of the signal vector projected onto a
preselected axis, usually a coordinate axis. When a coordinate axis is
chosen as a projection axis, the projection technique amounts to
projecting the signal vector onto one of the signal basis vectors, and
then assigning a gray scale value corresponding to the amplitude of this
projection on the projection axis. A voxel is thus associated with a
scalar which is associated to a particular gray value. In most current
projection techniques, this projection is performed without further
modulation by the mapping between the region of interest and the signal
space. The projection method yields satisfactory images of distinct
domains which are also spatially disjoint. However, such a method fails to
discriminate between neighboring regions having similar signal values and
spatially close to one another. The output image exhibits "poor contrast".
The concept of contrast is essential in image processing and is defined as
the difference in the gray scale values assigned to two distinct voxels.
In the projection method, the contrast is therefore dependent upon the
distance between the two points in the signal space to which the two
voxels are initially mapped. Thus, if the two voxels that are mapped have
similar characteristics, more specifically, if the values of the
projections of their corresponding signal vectors are nearly equal, the
distance on the projection axis in the signal space between the two mapped
voxels will be small. As the contrast depends on the difference between
the two gray scale values, two voxels having similar signal
characteristics are poorly contrasted in the projection method. To enhance
the contrast between two subregions which map to nearby points in the
signal space in a particular imaging technology, it becomes necessary to
resort to other imaging technologies (e.g. use of pharmacological agents,
etc).
The following example, taken from Nuclear Magnetic Resonance Imaging,
clearly illustrates the foregoing analysis. In NMR imaging, three
parameters are associated to each voxel. Three projections can thus be
envisioned, corresponding to each of the coordinate axes and leading to
three images. In each of these images, every voxel on each of the axes
T.sub.1, T.sub.2 (spin-lattice relaxation and spin-spin relaxation time,
respectively) and the proton density .rho. is assigned a gray scale value.
The first image is a representation of the region of interest mapped
according to the first type of relaxation T.sub.1. Two voxels having
different times of relaxation T.sub.1 are therefore represented by two
different gray values. This also applies to the second image produced in
accordance with the second coordinate, T.sub.2, and the third image
obtained from the proton density .rho. (the "proton image"). As discussed
above, the contrast between neighboring voxels having similar signal
characteristics is poor as the contrast depends on the distance chosen in
the signal space. Furthermore, the analysis of three images requires
special skills as the user is required to synthesize three types of images
in order to draw conclusions for a diagnosis. In order to facilitate the
reading of those images, a method has been disclosed entitled Hybrid Color
MR Imaging Display and disclosed in the article: Weiss, et al. "Hybrid
Color MR Imaging Display," AJR: 149: October 1987. In this method, two
images are synthesized into one image. A two-dimensional resolvable
contrast scale uses the pixel intensity from one image and the pixel
luminance of the other image. The pixel intensities from one image are
assigned varying special hues whereas the luminance of these hues is
derived from the intensities of the corresponding pixels of the second
specially aligned image. This technique, however, does not produce a sharp
contrast between neighboring voxels of similar intensities but made of
different subregions.
It should be noted that in the imaging techniques proposed so far,
smoothing operations are used to diminish spurious effects that may be
present in the digital image as a result of a poor sampling system or
transmission channel. In other words, these techniques already assume that
a gray level image has been obtained by projecting or by segmenting the
region to be processed in accordance with the signal characteristics of
the other voxels.
SUMMARY OF THE INVENTION
The present invention discloses a method for processing an image comprising
a plurality of pixels representing a region of interest partitioned into a
plurality of voxels. This method comprises the step of providing a
plurality of multidimensional signals, each of the signals being
associated with each of the voxels, the plurality of voxels being mapped
into the plurality of signals in a signal space by a mapping M, the step
of computing a density function representative of the frequency of
occurrence of each of the multidimensional signals in the signal space and
the step of assigning a tone value to each of the plurality of voxels in
accordance with the density function.
In accordance to the present invention in a second aspect, there is
disclosed a method for processing an image comprising a plurality of
pixels representing a region of interest partitioned into a plurality of
voxels, comprising the step of providing a plurality of multidimensional
signals, each of the multidimensional signals comprising a plurality of
parameters containing information about the region of interest, each of
the signals being further associated with each of the voxels, the
plurality of voxels being mapped into the plurality of signals in a signal
space by a mapping M, the mapping M being a multidimensional random
variable having a density function. The method further comprises the step
of computing the density function representative of the frequency of
occurrence of each of the multidimensional signals in the signal space,
the step of convolving the density function with a resolution function so
as to provide a real valued probability density function defined for each
of the plurality of voxels, the step of assigning a tone value to each of
the plurality of voxels corresponding to the value of the probability
density function for each of the voxels and the step of displaying a tone
level image wherein each voxel is represented by a pixel with the assigned
tone value. Preferably, the convolving step comprises the step of
convolving a Gaussian function of mean zero and variance .sigma. with the
density function. The Gaussian ideally has a variance .sigma. equal to or
greater than the variance .nu. of the density function of the signal
mapping M. The variance .sigma. of the Gaussian is preferably a multiple
of the variance .nu. of the density function of the signal mapping M. The
variance .sigma. advantageously defines the resolution of the image in the
signal space, whereby high values of the variance .nu. with respect to the
variance .sigma. produce a low resolution image in the signal space and
whereby low values of the variance .nu. with respect to the variance
.sigma. produce a high resolution image in the signal space.
Desirably, the image has a contrast defined by the tone value and the
signal space is defined with a metric, the contrast between two voxels of
the plurality of voxels being independent on the metric in the signal
space.
The convolving step may also include the step of convolving a cosine
function, a pulse function having a width .sigma., a sinc function or an
exponential function with the density function of the mapping M. More
generally, the resolution function may be a filter function.
In the method of the present invention, the density function is preferably
given by the following formula: E1 ? ?
##STR1##
wherein .delta. is the Delta function.
The convolution product is preferably performed using Fourier transforms.
It is advantageous that the convolution product is performed using Fast
Fourier transforms.
In the method of the present invention, the probability density function is
preferably given by the following equation: H.sub..sigma.
(s)=G.sub..sigma. (s) * (H.sub.o (s), wherein * denoted the convolution
product, G.sub..sigma. (s) represents the distribution function and
H.sub.o (s) represents the density function of the mapping M.
The image produced by the method of the present invention may be a medical
image. The signals may be NMR signals, obtained by computer tomography or
obtained from X-ray projections of the region of interest. The tone value
is preferably a grey value. It can also be a color. The step of assigning
a tone value may be performed by segmentation.
The convolving step is also possibly performed by using a multidimensional
resolution function so as to obtain a single image incorporating the
information from all of the plurality of parameters. The convolving step
may also be performed by using a plurality of unidimensional resolution
functions so as to obtain a plurality of images corresponding to each of
the parameters of the multidimensional signals in the signal space.
According to the invention in a second aspect, there is disclosed a method
for processing an image comprising a plurality of pixels representing a
region of interest partitioned into a plurality of voxels. This method
comprises the steps of providing a plurality of multidimensional signals,
each of the signals being associated with each of the voxels, the
plurality of voxels being mapped into the plurality of signals in a signal
space by a mapping M, the mapping M being a multidimensional random
variable and defining a probability measure in the signal space, the step
of computing the Radon-Nikodym derivative of the probability measure of
the mapping M, the Radon-Nikodym derivative defining a probability density
function assigning a tone value to each of the plurality of voxels
corresponding to the value of the probability density function for each of
the voxels and the step of displaying a tone level image wherein each
voxel is represented by a pixel with the assigned tone value.
The present invention also discloses an apparatus for processing an image
comprising a plurality of pixels representing a region of interest
partitioned into a plurality of voxels, the apparatus comprising an input
imaging device for providing a plurality of multidimensional signals, each
of the signals being associated with each of the voxels, the plurality of
voxels being mapped into the plurality of signals in a signal space by a
mapping M, the mapping M being a multidimensional random variable and
defining a probability measure in the signal space, means for computing
the density function representative of the frequency of occurrence of each
of the multidimensional signals in the signal space, means for convolving
the density function with a resolution function so as to provide a real
valued probability density function defined for each of the plurality of
voxels, means for assigning a tone value to each of the plurality of
voxels corresponding to the value of the probability density function for
each of the voxels and means for displaying a tone level image wherein
each voxel is represented by a pixel with the assigned tone value.
According to the invention in another aspect, there is disclosed an
apparatus for processing an image comprising a plurality of pixels
representing a region of interest partitioned into a plurality of voxels,
the apparatus comprising an input imaging device for providing a plurality
of multidimensional signals, each of the signals being associated with
each of the voxels, the plurality of voxels being mapped into the
plurality of signals in a signal space by a mapping M, the mapping M being
a multidimensional random variable and defining a probability measure in
the signal spac | | |
