U.S. patent application number 10/152462 was filed with the patent office on 2002-10-31 for method and system for fast detection of lines in medical images.
This patent application is currently assigned to R2 TECHNOLOGY, INC.. Invention is credited to Roehrig, Jimmy R., Schneider, Alexander C..
Application Number | 20020159622 10/152462 |
Document ID | / |
Family ID | 22203993 |
Filed Date | 2002-10-31 |
United States Patent
Application |
20020159622 |
Kind Code |
A1 |
Schneider, Alexander C. ; et
al. |
October 31, 2002 |
Method and system for fast detection of lines in medical images
Abstract
A method an apparatus for detecting lines in medical images is
disclosed, wherein a direction image array and a line image array
are formed by filtering a digital image with a single-peaked
filter, convolving the resultant array with second order difference
operators oriented along the horizontal, vertical, and diagonal
axes, and computing the direction image arrays and line image
arrays as direct scalar functions of the results of the second
order difference operations. Advantageously, line detection based
on the use of four line operator functions along the horizontal,
vertical, and diagonal directions in accordance with the preferred
embodiments actually results in fewer computations than line
detection based on the use of three line operator functions. In
particular, because of the special symmetries involved, 3.times.3
second order difference operators may be effectively used.
Moreover, the number of computations associated with the second
order difference operations may be achieved with simple register
shifts, additions, and subtractions, yielding an overall line
detection process that is significantly less computationally
intensive than prior art algorithms. Also according to a preferred
embodiment, computational complexity is reduced by selecting a
separable single-peaked filter, and sequentially convolving the
digital image with the component kernels of the separable
single-peaked filter.
Inventors: |
Schneider, Alexander C.;
(Mountain View, CA) ; Roehrig, Jimmy R.; (Palo
Alto, CA) |
Correspondence
Address: |
PENNIE & EDMONDS LLP
1155 Avenue of Americas
New York
NY
10036-2711
US
|
Assignee: |
R2 TECHNOLOGY, INC.
|
Family ID: |
22203993 |
Appl. No.: |
10/152462 |
Filed: |
May 21, 2002 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10152462 |
May 21, 2002 |
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09087245 |
May 28, 1998 |
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6404908 |
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Current U.S.
Class: |
382/128 ;
382/199 |
Current CPC
Class: |
G06K 9/4609 20130101;
G06T 2207/30068 20130101; G06T 7/12 20170101; G06T 2207/10116
20130101; G06V 10/443 20220101 |
Class at
Publication: |
382/128 ;
382/199 |
International
Class: |
G06K 009/48 |
Claims
What is claimed is:
1. A method for detecting lines in a digital image, comprising the
steps of: filtering said digital image to produce a filtered image
array; convolving said filtered image array with a plurality of
second order difference operators designed to extract second order
directional derivative information from said filtered image array
in a predetermined set of directions; processing information
resulting from said step of convolving to produce a line image;
wherein said predetermined set of directions is selected to
correspond to an aspect ratio of said second order difference
operators.
2. The method of claim 1, wherein said second order difference
operators are square kernels, and wherein said predetermined set of
directions includes the directions of 0, 45, 90, and 135
degrees.
3. The method of claim 2, wherein said second order difference
operators are 3.times.3 kernels.
4. The method of claim 3, said step of filtering said digital image
array comprising the steps of: selecting a single-peaked filter
kernel; and convolving said digital mammogram image with said
single-peaked filter kernel.
5. The method of claim 4, wherein said single-peaked filter kernel
is a separable function comprising the convolution of a first one
dimensional kernel and a second one dimensional kernel, and wherein
said step of convolving said digital mammogram image with said
single-peaked filter kernel comprises the steps of convolving said
digital mammogram image with said first one dimensional kernel and
said second one dimensional kernel.
6. The method of claim 5, wherein said single-peaked filter kernel
is a Gaussian.
7. The method of claim 6, wherein said step of convolving said
filtered image array comprises the steps of: convolving said
filtered image array with 3.times.3 second order difference
operators designed to extract second order derivative information
along the 45 degree and 135 degree directions; and subsequent to
said step convolving said filtered image array with 3.times.3
second order difference operators designed to extract second order
derivative information along the 45 degree and 135 degree
directions, multiplying the results of said step by a constant
correction factor to accommodate for more widely spaced sampling
along the diagonals.
8. A method for detecting lines in a digital image, comprising the
steps of: selecting a spatial scale parameter, said spatial scale
parameter corresponding to a desired range of line widths for
detection; convolving said digital image with a first one
dimensional kernel and a second one dimensional kernel to produce a
filtered image array, said first one dimensional kernel and said
second one dimensional kernel each having a size related to said
spatial scale parameter; producing a line image based on
second-order spatial derivatives of said filtered image array;
wherein said line image is produced from said digital image using a
number of computations that is substantially proportional to the
spatial scale parameter such that, as the spatial scale parameter
is increased, said number of computations increases at a rate that
is less than the rate of increase of the square of the spatial
scale parameter.
9. The method of claim 8, said step of producing a line image based
on second-order spatial derivatives of said filtered image array
further comprising the steps of: convolving said filtered image
array with a plurality of second order difference operators
designed to extract second order directional derivative information
from said filtered image array in a predetermined set of
directions; and processing information resulting from said step of
convolving to produce a line image; wherein said predetermined set
of directions includes directions along the diagonals of the
digital mammogram image.
10. The method of claim 9, wherein said second order difference
operators are 3.times.3 kernels.
11. The method of claim 10, wherein said first one dimensional
kernel and said second one dimensional kernel are single-peaked
functions each having an odd number of elements.
12. The method of claim 11, wherein said first one dimensional
kernel and said second one dimensional kernel are Gaussians.
13. A method for detecting lines in a digital image, comprising the
steps of: selecting a spatial scale parameter, said spatial scale
parameter corresponding to a desired range of line widths for
detection; convolving said digital image with a first one
dimensional kernel and a second one dimensional kernel to produce a
filtered image array, said first one dimensional kernel and said
second one dimensional kernel each having a size related to said
spatial scale parameter; separately convolving said filtered image
array with a first, second, and third second order difference
operator to produce a first, second, and third resulting array,
respectively; computing a direction image array comprising, at each
pixel, a first predetermined scalar function of corresponding pixel
values in said first, second, and third resulting arrays; computing
a line intensity function array comprising, at each pixel, a second
predetermined scalar function of corresponding pixel values in said
first, second, and third resulting arrays; and computing a line
image array using information in said line intensity function
array.
14. The method of claim 13, wherein said first, second, and third
second order difference operators each comprise a 3.times.3
matrix.
15. The method of claim 14, wherein said first second order
difference operator comprises the difference between a horizontal
second order difference operator and a vertical difference
operator.
16. The method of claim 15, wherein said second order difference
operator comprises the difference between a first diagonal second
order difference operator and a second diagonal second order
difference operator.
17. The method of claim 16, wherein said third second order
difference operator is a Laplacian.
18. The method of claim 17, wherein said first predetermined scalar
function comprises the arctangent of the quotient of said
corresponding pixel value in said second resulting array divided by
said corresponding pixel value in said first resulting array.
19. The method of claim 18, wherein said second predetermined
scalar function comprises the sum of two times the corresponding
pixel value in said third resulting array plus the square root of
the sum of the squares of the corresponding pixel value in said
first resulting array and the corresponding pixel value in said
second resulting array.
20. The method of claim 19, wherein said step of computing a line
image array using information in said line intensity function array
comprises the step of using a modified thresholding process based
on a histogram of said line intensity function.
21. A computer-readable medium which can be used for directing an
apparatus to detect lines in a digital image, comprising: means for
directing said apparatus to filter said image to produce a filtered
array; means for directing said apparatus to convolve said filtered
image array with a plurality of second order difference operators
designed to extract second order directional derivative information
from said filtered image array in a predetermined set of
directions; means for directing said apparatus to process
information resulting from said step of convolving to produce a
line image; wherein said predetermined set of directions is
selected to correspond to an aspect ratio of said second order
difference operators.
22. The computer-readable medium of claim 21, wherein said second
order difference operators are square kernels, and wherein said
predetermined set of directions includes the directions of 0, 45,
90, and 135 degrees.
23. The computer-readable medium of claim 22, wherein said second
order difference operators are 3.times.3 kernels.
24. The computer-readable medium of claim 23, said means for
directing said apparatus to filter said image to produce a filtered
array further comprising means for directing said apparatus to
convolve said digital mammogram image with a single-peaked filter
kernel.
25. The computer-readable medium of claim 23, said means for
directing said apparatus to filter said image to produce a filtered
array further comprising means for directing said apparatus to
convolve said digital mammogram image with a separable
single-peaked filter kernel by successively convolving said digital
image with a first one dimensional component kernel and a second
one dimensional component kernel of said separable single-peaked
filter kernel.
26. The computer-readable medium of claim 25, wherein said
separable single-peaked filter kernel is a Gaussian.
27. An apparatus for detecting lines in digital images, said
apparatus comprising: a first memory for storing a digital image; a
first convolution device capable of convolving said digital image
with a first one dimensional kernel and a second one dimensional
kernel to produce a filtered image array, said first one
dimensional kernel and said second one dimensional kernel each
having a size related to the size of lines being detected; a second
convolution device capable of separately convolving said filtered
image array with a first, a second, and a third second order
difference operator to produce a first, second, and third resulting
array, respectively; a first processing device capable of computing
a direction image array comprising, at each pixel, a first
predetermined scalar function of corresponding pixel values in said
first, second, and third resulting arrays; a second processing
device capable of computing a line intensity function array
comprising, at each pixel, a second predetermined scalar function
of corresponding pixel values in said first, second, and third
resulting arrays; and a third processing device capable of
computing a line image array using information in said line
intensity function array.
28. The method of claim 27, wherein said first, second, and third
second order difference operators each comprise a 3.times.3
matrix.
29. The method of claim 28, wherein said first second order
difference operator comprises the difference between a horizontal
second order difference operator and a vertical difference
operator.
30. The method of claim 29, wherein said second second order
difference operator comprises the difference between a first
diagonal second order difference operator and a second diagonal
second order difference operator.
31. The method of claim 30, wherein said third second order
difference operator is a Laplacian.
32. The method of claim 31, wherein said first predetermined scalar
function comprises the arctangent of the quotient of said
corresponding pixel value in said second resulting array divided by
said corresponding pixel value in said first resulting array.
33. The method of claim 32, wherein said second predetermined
scalar function comprises the sum of two times the corresponding
pixel value in said third resulting array plus the square root of
the sum of the squares of the corresponding pixel value in said
first resulting array and the corresponding pixel value in said
second resulting array.
Description
FIELD OF THE INVENTION
[0001] The present invention relates to the field of computer aided
analysis of medical images. In particular, the present invention
relates to a fast method for detecting lines in medical images.
BACKGROUND OF THE INVENTION
[0002] Line detection is an important first step in many medical
image processing algorithms. For example, line detection is an
important early step of the algorithm disclosed in U.S. patent
application Ser. No. 08/676,660, entitled "Method and Apparatus for
Fast Detection of Spiculated Lesions in Digital Mammograms," filed
Jul. 19, 1996, the contents of which are hereby incorporated by
reference into the present application. Generally speaking, if the
execution time of the line detection step can be shortened, then
the execution time of the overall medical image processing
algorithm employing that line detection step can be shortened.
[0003] In order to clearly illustrate the features and advantages
of the preferred embodiments, the present disclosure will describe
the line detection algorithms of both the prior art and the
preferred embodiments in the context of the computer-assisted
diagnosis system of U.S. patent application Ser. No. 08/676,660,
supra. Importantly, however, the scope of the preferred embodiments
is not so limited, the features and advantages of the preferred
embodiments being applicable to a variety of image processing
applications.
[0004] FIG. 1 shows steps performed by a computer-assisted
diagnosis unit similar to that described in U.S. patent application
Ser. No. 08/676,660, which is adapted to detect abnormal
spiculations or lesions in digital mammograms. At step 102, an
x-ray mammogram is scanned in and digitized into a digital
mammogram. The digital mammogram may be, for example, a
4000.times.5000 array of 12-bit gray scale pixel values. Such a
digital mammogram would generally correspond to a typical
8".times.10" x-ray mammogram which has been digitized at 50 microns
(0.05 mm) per pixel.
[0005] At step 104, which is generally an optional step, the
digital mammogram image is locally averaged, using steps known in
the art, down to a smaller size corresponding, for example, to a
200 micron (0.2 mm) spatial resolution. The resulting digital
mammogram image that is processed by subsequent steps is thus
approximately 1000.times.1250 pixels. As is known in the art, a
digital mammogram may be processed at different resolutions
depending on the type of features being detected. If, for example,
the scale of interest is near the order of magnitude 1 mm-10 mm,
i.e., if lines on the order of 1 mm-10 mm are being detected, it is
neither efficient nor necessary to process a full 50-micron (0.05
mm) resolution digital mammogram. Instead, the digital mammogram is
processed at a lesser resolution such as 200 microns (0.2 mm) per
pixel.
[0006] Generally speaking, it is to be appreciated that the
advantages and features of the preferred embodiments disclosed
infra are applicable independent of the size and spatial resolution
of the digital mammogram image that is processed. Nevertheless, for
clarity of disclosure, and without limiting the scope of the
preferred embodiments, the digital mammogram images in the present
disclosure, which will be denoted by the symbol I, will be
M.times.N arrays of 12-bit gray scale pixel values, with M and N
having exemplary values of 1000 and 1250, respectively.
[0007] At step 106, line and direction detection is performed on
the digital mammogram image I. At this step, an M.times.N line
image L(i, j) and an M.times.N direction image .theta..sub.max(i,
j) are generated from the digital mammogram image I. The M.times.N
line image L(i, j) generated at step 106 comprises, for each pixel
(i, j), line information in the form of a "1" if that pixel has a
line passing through it, and a "0" otherwise. The M.times.N
direction image .theta..sub.max(i, j) comprises, for those pixels
(i, j) having a line image value of "1", the estimated direction of
the tangent to the line passing through the pixel (i, j).
Alternatively, of course, the direction image .theta..sub.max(i, j)
may be adjusted by 90 degrees to correspond to the direction
orthogonal to the line passing through the pixel (i, j).
[0008] At step 108, information in the line and direction images is
processed for determining the locations and relative priority of
spiculations in the digital mammogram image I. The early detection
of spiculated lesions ("spiculations") in mammograms is of
particular importance because a spiculated breast tumor has a
relatively high probability of being malignant.
[0009] Finally, at step 110, the locations and relative priorities
of suspicious spiculated lesions are output to a display device for
viewing by a radiologist, thus drawing his or her attention to
those areas. The radiologist may then closely examine the
corresponding locations on the actual film x-ray mammogram. In this
manner, the possibility of missed diagnosis due to human error is
reduced.
[0010] One of the desired characteristics of a
spiculation-detecting CAD system is high speed to allow processing
of more x-ray mammograms in less time. As indicated by the steps of
FIG. 1, if the execution time of the line and direction detection
step 106 can be shortened, then the execution time of the overall
mammogram spiculation detection algorithm can be shortened.
[0011] A first prior art method for generating line and direction
images is generally disclosed in Gonzales and Wintz, Digital Image
Processing (1987) at 333-34. This approach uses banks of filters,
each filter being "tuned" to detect lines in a certain direction.
Generally speaking, this "tuning" is achieved by making each filter
kernel resemble a second-order directional derivative operator in
that direction. Each filter kernel is separately convolved with the
digital mammogram image I. Then, at each pixel (i, j), line
orientation can be estimated by selecting the filter having the
highest output at (i, j), and line magnitude may be estimated from
that output and other filter outputs. The method can be generalized
to lines having pixel widths greater than 1 in a multiscale
representation shown in Daugman, "Complete Discrete 2-D Gabor
Transforms by Neural Networks for Image Analysis and Compression,"
IEEE Trans. ASSP, Vol. 36, pp. 1169-79 (1988).
[0012] The above filter-bank algorithms are computationally
intensive, generally requiring a separate convolution operation for
each orientation-selective filter in the filter bank. Additionally,
the accuracy of the angle estimate depends on the number of filters
in the filter bank, and thus there is an implicit tradeoff between
the size of the filter bank (and thus total computational cost) and
the accuracy of angle estimation.
[0013] A second prior art method of generating line and direction
images is described in Karssemeijer, "Recognition of Stellate
Lesions in Digital Mammograms," Digital Mammography: Proceedings of
the 2nd International Workshop on Digital Mammography, York,
England, (Jul. 10-12, 1994) at 211-19, and in Karssemeijer,
"Detection of Stellate Distortions in Mammograms using Scale Space
Operators," Information Processing in Medical Imaging 335-46
(Bizais et al., eds. 1995) at 335-46. A mathematical foundation for
the Karssemeijer approach is found in Koenderink and van Doorn,
"Generic Neighborhood Operators," IEEE Transactions on Pattern
Analysis and Machine Intelligence, Vol. 14, No. 6 (June 1992) at
597-605. The contents of each of the above two Karssemeijer
references and the above Koenderink reference are hereby
incorporated by reference into the present application.
[0014] The Karssemeijer algorithm uses scale space theory to
provide an accurate and more efficient method of line detection
relative to the filter-bank method. More precisely, at a given
level of spatial scale .sigma., Karssemeijer requires the
convolution of only three kernels with the digital mammogram image
I, the angle estimation at a pixel (i, j) then being derived as a
trigonometric function of the three convolution results at (i,
j).
[0015] FIG. 2 shows steps for computing line and direction images
in accordance with the Karssemeijer algorithm. At step 202, a
spatial scale parameter a and a filter kernel size N.sub.k are
selected. The spatial scale parameter o dictates the width, in
pixels, of a Gaussian kernel G(r,.sigma.), the equation for which
is shown in Eq. (1):
G(r,.sigma.)=(1/2.pi..sigma..sup.2)exp(-r.sup.2/2.sigma..sup.2)
(1)
[0016] At step 202, the filter kernel size N.sub.k, in pixels, is
generally chosen to be large enough to contain the Gaussian kernel
G(r,.sigma.) in digital matrix form, it being understood that the
function G(r,.sigma.) becomes quite small very quickly. Generally
speaking, the spatial scale parameter .sigma. corresponds, in an
order-of-magnitude sense, to the size of the lines being detected.
By way of example only, and not by way of limitation, for detecting
1 mm-10 mm lines in fibrous breast tissue in a 1000.times.1250
digital mammogram at 200 micron (0.2 mm) resolution, the value of
.sigma. may be selected as 1.5 pixels and the filter kernel size
N.sub.k may be selected as 11 pixels. For detecting different size
lines or for greater certainty of results, the algorithm or
portions thereof may be repeated using different values for a and
the kernel size.
[0017] At step 204, three filter kernels K.sub..sigma.(0),
K.sub..sigma.(60), and K.sub..sigma.(120) are formed as the second
order directional derivatives of the Gaussian kernel G(r,.sigma.)
at 0 degrees, 60 degrees, and 120 degrees, respectively. The three
filter kernels K.sub..sigma.(0), K.sub..sigma.(60), and
K.sub..sigma.(120) are each of size N.sub.k, each filter kernel
thus containing N.sub.k.times.N.sub.k elements.
[0018] At step 206, the digital mammogram image I is separately
convolved with each of the three filter kernels K.sub..sigma.(0),
K.sub..sigma.(60), and K(120) to produce three line operator
functions W.sub..sigma.(0), W.sub..sigma.(60), and
W.sub..sigma.(120), respectively, as shown in Eq. (2):
W.sub..sigma.(0)=I*K.sub..sigma.(0)W.sub..sigma.(60)=I*K.sub..sigma.(60)W.-
sub..sigma.(120)=I*K.sub..sigma.(120) (2)
[0019] Each of the line operator functions W.sub..sigma.(0),
W.sub..sigma.(60), and W.sub..sigma.(120) is, of course, a
two-dimension array that is slightly larger than the original
M.times.N digital mammogram image array I due to the size N.sub.k
of the filter kernels.
[0020] Subsequent steps of the Karssemeijer algorithm are based on
a relation shown in Koenderink, supra, which shows that an
estimation function W.sub..sigma.(.theta.) may be formed as a
combination of the line operator functions W.sub..sigma.(0),
W.sub..sigma.(60), and W.sub..sigma.(120) as defined in equation
(3):
W.sub..sigma.(.theta.)=(1/3)(1+2
cos(2.theta.))W.sub..sigma.(0)+(1/3)(1- cos(2.theta.)+({square
root}3)sin(2.theta.))W.sub..sigma.(60)+(1/3)(1-
cos(2.theta.)-({square root}3)sin(2.theta.))W.sub..sigma.(120)
(3)
[0021] As indicated by the above definition, the estimation
function W.sub..sigma.(.theta.) is a function of three variables,
the first two variables being pixel coordinates (i, j) and the
third variable being an angle .theta.. For each pixel location (i,
j), the estimation function W.sub..sigma.(.theta.) represents a
measurement of line strength at pixel (i, j) in the direction
perpendicular to .theta.. According to the Karssemeijer method, an
analytical expression for the extrema of W.sub..sigma.(.theta.)
with respect to .theta., denoted .theta..sub.min,max at a given
pixel (i, j) is given by Eq. (4):
.theta..sub.min,max=1/2[arc tan{({square
root}3)(W.sub..sigma.(60)-W.sub.9- 4
(120))/(W.sub..sigma.(60)+W.sub..sigma.(120)-2W.sub..sigma.(0))}.+-..pi.-
] (4)
[0022] Thus, at step 208, the expression of Eq. (4) is computed for
each pixel based on the values of W.sub..sigma.(0),
W.sub..sigma.(60), and W.sub..sigma.(120) that were computed at
step 206. Of the two solutions to equation (4), the direction
.theta..sub.max is then selected as the solution that yields the
larger magnitude for W.sub..sigma.(.theta.) at that pixel, denoted
W.sub..sigma.(.theta..sub.max). Thus, at step 208, an array
.theta..sub.max(i, j) is formed that constitutes the direction
image corresponding to the digital mammogram image I. As an outcome
of this process, a corresponding two-dimensional array of line
intensities corresponding to the maximum direction .theta..sub.max
at each pixel is formed, denoted as the line intensity function
W.sub..sigma.(.theta..sub.- max).
[0023] At step 210, a line image L(i, j) is formed using
information derived from the line intensity function
W.sub..sigma.(.theta..sub.max) that was inherently generated during
step 208. The array L(i, j) is formed from
W.sub..sigma.(.theta..sub.max) using known methods such as a simple
thresholding process or a modified thresholding process based on a
histogram of W.sub..sigma.(.theta..sub.max). With the completion of
the line image array L(i, j) and the direction image array
.theta..sub.max(i, j), the line detection process is complete.
[0024] Optionally, in the Karssemeijer algorithm a plurality of
spatial scale values .sigma.1, .sigma.2, . . . , .sigma.n may be
selected at step 202. The steps 204-210 are then separately carried
out for each of the spatial scale values (.sigma.1, .sigma.2, . . .
, .sigma.n. For a given pixel (i, j), the value of
.theta..sub.max(i, j) is selected to correspond to the largest
value among W.sub..sigma.1(.theta..sub.max1),
W.sub..sigma.2(.theta..sub.max2), . . . ,
W.sub..sigma.n(.theta..sub.maxn- ). The line image L(i, j) is
formed by thresholding an array corresponding to largest value
among W.sub..sigma.1(.theta..sub.max1),
W.sub..sigma.2(.theta..sub.max2), . . . ,
W.sub..sigma.n(.theta..sub.maxn- ) at each pixel.
[0025] Although it is generally more computationally efficient than
the filter-bank method, the prior art Karssemeijer algorithm has
computational disadvantages. In particular, for a given spatial
scale parameter .sigma., the Karssemeijer algorithm requires three
separate convolutions of N.sub.k.times.N.sub.k kernels with the
M.times.N digital mammogram image I. Each convolution, in turn,
requires approximately M.multidot.N.multidot.(N.sub.k).sup.2
multiplication and addition operations, which becomes
computationally expensive as the kernel size N.sub.k, which is
proportional to the spatial scale parameter .sigma., grows. Thus,
for a constant digital mammogram image size, the computational
intensity of the Karssemeijer algorithm generally grows according
to the square of the scale of interest.
[0026] Accordingly, it would be desirable to provide a line
detection algorithm for use in a medical imaging system that is
less computationally intensive, and therefore faster, than the
above prior art algorithms.
[0027] It would further be desirable to provide a line detection
algorithm for use in a medical imaging system that is capable of
operating at multiple spatial scales for detecting lines of varying
widths.
[0028] It would be even further desirable to provide a line
detection algorithm for use in a medical imaging system in which,
as the scale of interest grows, the computational intensity grows
at a rate less than the rate of growth of the square of the scale
of interest.
SUMMARY OF THE INVENTION
[0029] These and other objects are provided for by a-method and
apparatus for detecting lines in a medical imaging system by
filtering the digital image with a single-peaked filter, convolving
the resultant array with second order difference operators oriented
along the horizontal, vertical, and diagonal axes, and computing
direction image arrays and line image arrays as direct scalar
functions of the results of the second order difference operations.
Advantageously, it has been found that line detection based on the
use of four line operator functions can actually require fewer
computations than line detection based on the use of three line
operator functions, if the four line operator functions correspond
to the special orientations of 0, 45, 90, and 135 degrees. Stated
another way, it has been found that the number of required
computations is significantly reduced where the aspect ratio of the
second order difference operators corresponds to the angular
distribution of the line operator functions. Thus, where the second
order difference operators are square kernels, having an aspect
ratio of unity, the preferred directions of four line operator
functions is at 0, 45, 90, and 135 degrees.
[0030] In a preferred embodiment, a spatial scale parameter is
selected that corresponds to a desired range of line widths for
detection. The digital image is then filtered with a single-peaked
filter having a size related to the spatial scale parameter, to
produce a filtered image array. The filtered image array is
separately convolved with second order difference operators at 0,
45, 90, and 135 degrees. The direction image array and the line
image array are then computed at each pixel as scalar functions of
the elements of the arrays resulting from these convolutions.
Because of the special symmetries involved, the second order
difference operators may be 3.times.3 kernels. Moreover, the number
of computations associated with the second order difference
operations may be achieved with simple register shifts, additions,
and subtractions, yielding an overall line detection process that
is significantly less computationally intensive than prior art
algorithms.
[0031] In another preferred embodiment, the digital image is first
convolved with a separable single-peaked filter kernel, such as a
Gaussian. Because a separable function may be expressed as the
convolution of a first one dimensional kernel and a second one
dimensional kernel, the convolution with the separable
single-peaked filter kernel is achieved by successive convolutions
with a first one dimensional kernel and a second one dimensional
kernel, which significantly reduces computation time in generating
the filtered image array. The filtered image array is then
convolved with three 3.times.3 second order difference operators,
the first such operator comprising the difference between a
horizontal second order difference operator and a vertical
difference operator, the second such operator comprising the
difference between a first diagonal second order difference
operator and a second diagonal second order difference operator,
and the third such operator being a Laplacian operator. Because of
the special symmetries associated with the selection of line
operator functions at 0, 45, 90, and 135 degrees, the direction
image array and the line image array are then computed at each
pixel as even simpler scalar functions of the elements of the
arrays resulting from the three convolutions.
[0032] Thus, line detection algorithms in accordance with the
preferred embodiments are capable of generating line and direction
images using significantly fewer computations than prior art
algorithms by taking advantage of the separability of Gaussians and
other symmetric filter kernels, while also taking advantage of
discovered computational simplifications that result from the
consideration of four line operator functions oriented in the
horizontal, vertical, and diagonal directions.
BRIEF DESCRIPTION OF THE DRAWINGS
[0033] FIG. 1 shows steps taken by a computer-aided diagnosis
("CAD") system for detecting spiculations in digital mammograms in
accordance with the prior art.
[0034] FIG. 2 shows line detection steps taken by the CAD system of
FIG. 1.
[0035] FIG. 3 shows line detection steps according to a preferred
embodiment.
[0036] FIG. 4 shows steps for convolution with second order
directional derivative operators in accordance with a preferred
embodiment.
[0037] FIG. 5 shows line detection steps according to another
preferred embodiment.
DETAILED DESCRIPTION
[0038] FIG. 3 shows steps of a line detection algorithm in
accordance with a preferred embodiment. At step 302, a spatial
scale parameter .theta. and a filter kernel size N.sub.k are
selected in manner similar to that of step 202 of FIG. 2. However,
in a line detection system according to a preferred embodiment, it
is possible to make these factors larger than with the prior art
system of FIG. 2 while not increasing the computational intensity
of the algorithm. Alternatively, in a line detection system
according to a preferred embodiment, these factors may remain the
same as with the prior art system of FIG. 2 and the computational
intensity of the algorithm will be reduced. As a further
alternative, in a line detection system according to a preferred
embodiment, it is possible to detect lines using a greater number
of different spatial scales of interest .sigma. while not
increasing the computational intensity of the algorithm.
[0039] At step 304, the digital mammogram image I is convolved with
a two-dimensional single-peaked filter F having dimensions
N.sub.k.times.N.sub.k to form a filtered image array I.sub.F as
shown in Eq. (5):
I.sub.F=I*F (5)
[0040] By single-peaked filter, it is meant that the filter F is a
function with a single maximum point or single maximum region.
Examples of such a filter include the Gaussian, but may also
include other filter kernels such as a Butterworth filter, an
inverted triangle or parabola, or a flat "pillbox" function. It has
been found, however, that a Gaussian filter is, the most
preferable. The size of the single-peaked filter F is dictated by
the spatial scale parameter .sigma.. For example, where a Gaussian
filter is used, .sigma. is the standard deviation of the Gaussian,
and where a flat pillbox function is used, .sigma. corresponds to
the radius of the pillbox. In subsequent steps it is assumed that a
Gaussian filter is used, although the algorithm may be adapted by
one skilled in the art to use other filters.
[0041] At step 306, the filtered image array I.sub.F is then
separately convolved with second order directional derivative
operators. In accordance with a preferred embodiment, it is
computationally advantageous to compute four directional
derivatives at 0, 45, 90, and 135 degrees by convolving filtered
image array I.sub.F with second order directional derivative
operators D.sub.2(0), D.sub.2(45), D.sub.2(90), and D.sub.2(135) to
produce the line operator functions W.sub..sigma.(0),
W.sub..sigma.(45), W.sub..sigma.(90), and W.sub..sigma.(135),
respectively, as shown in Eqs. (6a)-(6d).
W.sub..sigma.(0)=I.sub.F*D.sub.2(0) (6a)
W.sub..sigma.(45)=I.sub.F*D.sub.2(45) (6b)
W.sub..sigma.(90)=I.sub.F*D.sub.2(90) (6c)
W.sub..sigma.(135)=I.sub.F*D.sub.2(135) (6d)
[0042] Advantageously, because the particular directions of 0, 45,
90, and 135 degrees are chosen, these directional derivative
operators are permitted to consist of the small 3.times.3 kernels
shown in Eqs. (7a)-(7d): 1 0 0 0 D 2 ( 0 ) = - 1 2 - 1 0 0 0 (7a) 0
0 - 1 D 2 ( 45 ) = 0 2 0 - 1 0 0 (7b) 0 - 1 0 D 2 ( 90 ) = 0 2 0 0
- 1 0 (7c) - 1 0 0 D 2 ( 135 ) = 0 2 0 0 0 - 1 (7d)
[0043] The above 3.times.3 second order directional derivative
operators are preferred, as they result in fewer computations than
larger second order directional derivative operators while still
providing a good estimate of the second order directional
derivative when convolved with the filtered image array I.sub.F.
However, the scope of the preferred embodiments is not necessarily
so limited, it being understood that larger operators for
estimating the second order directional derivatives may be used if
a larger number of computations is determined to be acceptable. For
a minimal number of computations in accordance with a preferred
embodiment, however, 3.times.3 kernels are used.
[0044] Subsequent steps are based on an estimation function
W.sub..sigma.(.theta.) that can be formed from the arrays
W.sub..sigma.(0), W.sub..sigma.(45), W.sub..sigma.(90), and
W.sub..sigma.(135) by adapting the formulas in Koenderink, supra,
for four estimators spaced at intervals of 45 degrees. The
resulting formula is shown below in Eq. (8).
W.sub..sigma.(.theta.)=1/4{(1+2 cos(2.theta.))W.sub..sigma.(0)+(1+2
sin(2.theta.))W.sub..sigma.(45)+(1-2
cos(2.theta.))W.sub..sigma.(90)+(1-2
sin(2.theta.))W.sub..sigma.(135)} (8)
[0045] It has been found that the extrema of the estimation
function W.sub..sigma.(.theta.) with respect to .theta., denoted
.theta..sub.min,max at a given pixel (i, j) is given by Eq.
(9):
.theta..sub.min,max=1/2[a
tan{(W.sub..sigma.(45)-W.sub..sigma.(135))/(W.su-
b..sigma.(0)-W.sub..sigma.(90))}.+-..pi.] (9)
[0046] At step 308, the expression of Eq. (9) is computed for each
pixel. Of the two solutions to equation (4), the direction
.theta..sub.max is then selected as the solution that yields the
larger magnitude for W.sub..sigma.(.theta.) at that pixel, denoted
as the line intensity W.sub..sigma.(.theta..sub.max). Thus, at step
308, an array .theta..sub.max(i, j) is formed that constitutes the
direction image corresponding to the digital mammogram image I. As
an outcome of this process, a corresponding two-dimensional array
of line intensities corresponding to the maximum direction
.theta..sub.max at each pixel is formed, denoted as the line
intensity function W.sub..sigma.(.theta..sub.- max).
[0047] At step 310, a line image array L(i, j) is formed using
information derived from the line intensity function
W.sub..sigma.(.theta..sub.max) that was inherently generated during
step 308. The line image array L(i, j) is formed from the line
intensity function W.sub..sigma.(.theta..sub.m- ax) using known
methods such as a simple thresholding process or a modified
thresholding process based on a histogram of the line intensity
function W.sub..sigma.(.theta..sub.max). With the completion of the
line image array L(i, j) and the direction image array
.theta..sub.max(i, j), the line detection process is complete.
[0048] FIG. 4 illustrates unique computational steps corresponding
to the step 306 of FIG. 3. At step 306, the filtered image array
I.sub.F is convolved with the second order directional derivative
operators D.sub.2(0), D.sub.2(45), D.sub.2(90), and D.sub.2(135)
shown in Eq. (7). An advantage of the use of the small 3.times.3
kernels D.sub.2(0), D.sub.2(45), D.sub.2(90), and D.sub.2(135)
evidences itself in the convolution operations corresponding to
step 306. In particular, because each of the directional derivative
operators has only 3 nonzero elements -1, 2, and -1, general
multiplies are not necessary at all in step 306, as the
multiplication by 2 just corresponds to a single left bitwise
register shift and the multiplications by -1 are simply sign
inversions. Indeed, each convolution operation of Eq. (6) can be
simply carried out at each pixel by a single bitwise left register
shift followed by two subtractions of neighboring pixel values from
the shifted result.
[0049] Thus, at step 402 each pixel in the filtered image array
I.sub.F is doubled to produce the doubled filtered image array
2I.sub.F. This can be achieved through a multiplication by 2 or, as
discussed above, a single bitwise left register shift. At step 404,
at each pixel (i, j) in the array 2I.sub.F, the value of
I.sub.F(i-1,j) is subtracted, and at step 406, the value of
I.sub.F(i+1,j) is subtracted, the result being equal to the desired
convolution result I.sub.F*D.sub.2(0) at pixel (i, j). Similarly,
at step 408, at each pixel (i, j) in the array 2I.sub.F, the value
of I.sub.F(i-1,j-1) is subtracted, and at step 410, the value of
I.sub.F(i+1,j+1) is subtracted, the result being equal to the
desired convolution result I.sub.F*D.sub.2(45) at pixel (i, j).
Similarly, at step 412, at each pixel (i, j) in the array 2I.sub.F,
the value of I.sub.F(i, j-1) is subtracted, and at step 414, the
value of I.sub.F(i, j+1) is subtracted, the result being equal to
the desired convolution result I.sub.F*D.sub.2(90) at pixel (i, j).
Finally, at step 416, at each pixel (i, j) in the array 2I.sub.F,
the value of I.sub.F(i+1,j-1) is subtracted, and at step 418, the
value of I.sub.F(i-1,j+1) is subtracted, the result being equal to
the desired convolution result I.sub.F*D.sub.2(135) at pixel (i,
j). The steps 406-418 are preferably carried out in the parallel
fashion shown in FIG. 4 but can generally be carried out in any
order.
[0050] Thus, it is to be appreciated that in the embodiment of
FIGS. 3 and 4 a line detection algorithm is executed using four
line operator functions W.sub..sigma.(0), W.sub..sigma.(45),
W.sub..sigma.(90), and W.sub..sigma.(135) while at the same time
using fewer computations than the Karssemeijer algorithm of FIG. 2,
which uses only three line operator functions W.sub..sigma.(0),
W.sub..sigma.(60), W.sub..sigma.(120). In accordance with a
preferred embodiment, the algorithm of FIGS. 3 and 4 takes
advantage of the interchangeability of the derivative and
convolution operations while also taking advantage of the finding
that second order directional derivative operators in each of the
four directions 0, 45, 90, and 135 degrees may be implemented using
small 3.times.3 kernels each having only three nonzero elements -1,
2, and -1. In the Karssemeijer algorithm of FIG. 2, there are three
convolutions of the M.times.N digital mammogram image I with the
N.sub.k.times.N.sub.k kernels, requiring approximately
3.multidot.(N.sub.k).sup.2.multidot.M.mu- ltidot.N multiplications
and adds to derive the three line estimator functions
W.sub..sigma.(0), W.sub..sigma.(60), and W.sub..sigma.(120).
However, in the embodiment of FIGS. 3 and 4, the computation of the
four line estimator functions W.sub.94 (0), W.sub..sigma.(45),
W.sub..sigma.(90), and W.sub..sigma.(135) requires a first
convolution requiring (N.sub.k).sup.2.multidot.M.multidot.N
multiplications, followed by M.multidot.N doubling operations and
8.multidot.M.multidot.N subtractions, which is a very significant
computational advantage. The remaining portions of the different
algorithms take approximately the same amount of computations once
the line estimator functions are computed.
[0051] For illustrative purposes in comparing the algorithm of
FIGS. 3 and 4 with the prior art Karssemeijer algorithm of FIG. 2,
let us assume that the operations of addition, subtraction, and
register-shifting operation take 10 clock cycles each, while the
process of multiplication takes 30 clock cycles. Let us further
assume that an exemplary digital mammogram of
M.times.N=1000.times.1250 is used and that N.sub.k is 11. For
comparison purposes, it is most useful to look at the operations
associated with the required convolutions, as they require the
majority of computational time. For this set of parameters, the
Karssemeijer algorithm would require
3(11).sup.2(1000)(1250)(30+10)=18.2 billion clock cycles to compute
the three line estimator functions W.sub..sigma.(0),
W.sub..sigma.(60), and W.sub..sigma.(120). In contrast, the
algorithm of FIGS. 3 and 4 would require only
(11).sup.2(1000)(1250)(30+10)+(1250)(100-
0)(10)+8(1250)(1000)(10)=6.2 billion clock cycles to generate the
four line operator functions W.sub..sigma.(0), W.sub..sigma.(45),
W.sub..sigma.(90), and W.sub..sigma.(135), a significant
computational advantage.
[0052] FIG. 5 shows steps of a line detection algorithm in
accordance with another preferred embodiment. It has been found
that the algorithm of FIGS. 3 and 4 can be made even more
computationally efficient where the single-peaked filter kernel F
is selected to be separable. Generally speaking, a separable kernel
can be expressed as a convolution of two kernels of lesser
dimensions, such as one-dimensional kernels. Thus, the
N.sub.k.times.N.sub.k filter kernel F(i, j) is separable where it
can be formed as a convolution of an N.sub.k.times.1 kernel
F.sub.x(i) and a 1.times.N.sub.k kernel F.sub.y(j), i.e., F(i,
j)=F.sub.x(i)*F.sub.y(j). As known in the art, an N.sub.k.times.1
kernel is analogous to a row vector of length N.sub.k while a
1.times.N.sub.k kernel is analogous to a column vector of length
N.sub.k.
[0053] Although a variety of single-peaked functions are within the
scope of the preferred embodiments, the most optimal function has
been found to be the Gaussian function of Eq. (1), supra. For
purposes of the embodiment of FIG. 5, and without limiting the
scope of the preferred embodiments, the filter kernel notation F
will be replaced by the notation G to indicate that a Gaussian
filter is being used: 2 G = ( 1 / 2 2 ) exp ( - x 2 / 2 2 ) exp ( -
y 2 / 2 2 ) = G x * G y ( 10 ) G x = [ g x , 0 g x , 1 g x , 2 g x
, Nk - 1 ] g y , 0 g y , 1 ( 11 ) G y = g y , 3 g y , Nk - 1 ( 12
)
[0054] At step 502, the parameters .sigma. and N.sub.k are selected
in a manner similar to step 302 of FIG. 3. It is preferable for
N.sub.k to be selected as an odd number, so that a one-dimensional
Gaussian kernel of length N.sub.k may be symmetric about its
central element. At step 504, the M.times.N digital mammogram image
I is convolved with the Gaussian N.sub.k.times.1 kernel G.sub.x to
produce an intermediate array I.sub.x:
I.sub.x=G.sub.x*I (13)
[0055] In accordance with a preferred embodiment, the sigma of the
one-dimensional Gaussian kernel G.sub.x is the spatial scale
parameter a selected at step 502. The intermediate array I.sub.x
resulting from step 504 is a two-dimensional array having
dimensions of approximately (M+2N.sub.k).times.N.
[0056] At step 506, the intermediate array I.sub.x is convolved
with the Gaussian 1.times.N.sub.k kernel G.sub.y to produce a
Gaussian-filtered image array I.sub.G:
I.sub.G=I.sub.x*G.sub.y (14)
[0057] In accordance with a preferred embodiment, the sigma of the
one-dimensional Gaussian kernel G.sub.y is also the spatial scale
parameter a selected at step 502. The filtered image array I.sub.G
resulting from step 506 is a two-dimensional array having
dimensions of approximately (M+2N.sub.k).times.(N+2N.sub.k).
Advantageously, because of the separability property of the
two-dimensional Gaussian, the filtered image array I.sub.G
resulting from step 506 is identical to the result of a complete
two-dimensional convolution of an N.sub.k.times.N.sub.k Gaussian
kernel and the digital mammogram image I. However, the number of
multiplications and additions is reduced to
2.multidot.N.sub.k.multidot.M- .multidot.N instead of
(N.sub.k).sup.2.multidot.M.multidot.N.
[0058] Even more advantageously, in the situation where N.sub.k is
selected to be an odd number and the one-dimensional Gaussian
kernels are therefore symmetric about a central element, the number
of multiplications is reduced even further. This computational
reduction can be achieved because, if N.sub.k is odd, then the
component one dimensional kernels G.sub.x and G.sub.y are each
symmetric about a central peak element. Because of this relation,
the image values corresponding to symmetric kernel locations can be
added prior to multiplication by those kernel values, thereby
reducing by half the number of required multiplications during the
computations of Eqs. (13) and (14). Accordingly, in a preferred
embodiment in which N.sub.k is selected to be an odd number, the
number of multiplications associated with the required convolutions
is approximately N.sub.k.multidot.M.multid- ot.N and the number of
additions is approximately 2.multidot.N.sub.k.multi-
dot.M.multidot.N.
[0059] In addition to the computational savings over the embodiment
of FIGS. 3 and 4 due to filter separability, it has also been found
that the algorithm of FIGS. 3 and 4 may be made even more efficient
by taking advantage of the special symmetry of the spatial
derivative operators at 0, 45, 90, and 135 in performing operations
corresponding to steps 306-310. In particular, it has been found
that for each pixel (i, j), the solution for the direction image
.theta..sub.max and the line intensity function
W.sub..sigma.(.theta..sub.max) can be simplified to the following
formulas of Eqs. (15)-(16):
W.sub..sigma.(.theta..sub.max)=1/2(L+{square
root}(A.sup.2+D.sup.2)) (15)
.theta..sub.max=1/2a tan(D/A) (16)
[0060] In the above formulas, the array L is defined as
follows:
L=W.sub..sigma.(0)+W.sub..sigma.(90)=I.sub.G*D.sub.2(0)+I.sub.G*D.sub.2(90-
)=I.sub.G*[D.sub.2(0)+D.sub.2(90)] (17)
[0061] 3 0 - 1 0 L = I G * - 1 4 - 1 0 - 1 0 ( 18 )
[0062] As known in the art, the array L is the result of the
convolution of I.sub.G with a Laplacian operator. Furthermore, the
array A in Eqs. (15) and (16) is defined as follows:
A=W.sub..sigma.(0)-W.sub..sigma.(90)=I.sub.G*D.sub.2(0)-I.sub.G*D.sub.2(90-
)=I.sub.G*[D.sub.2(0)-D.sub.2(90)] (19)
[0063] 4 0 1 0 A = I G * - 1 0 - 1 0 1 0 ( 20 )
[0064] Finally, the array D in Eqs. (15) and (16) is defined as
follows:
D=W.sub..sigma.(45)-W.sub..sigma.(135)=I.sub.G*D.sub.2(45)-I.sub.G*D.sub.2-
(135)=I.sub.G*[D.sub.2(45)-D.sub.2(135)] (21)
[0065] 5 1 0 - 1 D = I G * 0 0 0 - 1 0 1 ( 22 )
[0066] Accordingly, at step 508 the convolution of Eq. (20) is
performed on the filtered image array I.sub.G that results from the
previous step 506 to produce the array A. At step 510, the
convolution of Eq. (22) is performed on the filtered image array
I.sub.G to produce the array D, and at step 512, the convolution of
Eq. (18) is performed to produce the array L. Since they are
independent of each other, the steps 508-512 may be performed in
parallel or in any order. At step 514, the line intensity function
W.sub..sigma.(.theta..sub.max) is formed directly from the arrays
L, A, and D in accordance with Eq. (15). Subsequent to step 514, at
step 516 the line image array L(i, j) is formed from the line
intensity function W.sub..sigma.(.theta..sub.max) using known
methods such as a simple thresholding process or a modified
thresholding process based on a histogram of the line intensity
function W.sub..sigma.(.theta..sub.max).
[0067] Finally, at step 518, the direction image array
.theta..sub.max(i, j) is formed from the arrays D and A in
accordance with Eq. (16). Advantageously, according to the
preferred embodiment of FIG. 5, the step 518 of computing the
direction image array .theta..sub.max(i, j) and the steps 514-516
of generating the line image array L(i, j) may be performed
independently of each other and in any order. Stated another way,
according to the preferred embodiment of FIG. 5, it is not
necessary to actually compute the elements of the direction image
.theta..sub.max(i, j) in order to evaluate the line intensity
estimator function W.sub.94 (.theta..sub.max) at any pixel. This is
in contrast to the algorithms described in FIG. 2 and FIGS. 3 and
4, where it is first necessary to compute the direction image
.theta..sub.max(i, j) in order to be able to evaluate the line
intensity estimator function W.sub..sigma.(.theta.) at the maximum
angle .theta..sub.max.
[0068] It is readily apparent that in the preferred embodiment of
FIG. 5, steps 512, 514, and 516 may be omitted altogether if
downstream medical image processing algorithms only require
knowledge of the direction image array .theta..sub.max(i, j).
Alternatively, the step 518 may be omitted altogether if downstream
medical image processing algorithms only require knowledge of the
line image array L(i, j). Thus, computational independence of the
direction image array .theta..sub.max(i, j) and the line image
array L(i, j) in the preferred embodiment of FIG. 5 allows for
increased computational efficiency when only one or the other of
the direction image array .theta..sub.max(i, j) and the line image
array L(i, j) is required by downstream algorithms.
[0069] The preferred embodiment of FIG. 5 is even less
computationally complex than the algorithm of FIG. 3 and 4. In
particular, to generate the filtered image array I.sub.G there is
required only approximately N.sub.k.multidot.M.multidot.N
multiplications and 2.multidot.N.sub.k.mult- idot.M.multidot.N
additions. To generate the array A from the filtered image array
I.sub.G, there is required 2.multidot.M.multidot.N additions and
M.multidot.N subtractions. Likewise, to generate the array D from
the filtered image array I.sub.G, there is required
2.multidot.M.multidot.N additions and M.multidot.N subtractions.
Finally, to generate L from the filtered image array I.sub.G, there
is required M.multidot.N bitwise left register shift of two
positions (corresponding to a multiplication by 4), followed by
4.multidot.M.multidot.N subtractions. Accordingly, to generate the
arrays A, D, and L from the digital mammogram image I, there is
required only 2.multidot.N.sub.k.multidot.M.multidot.N
multiplications, 2.multidot.N.sub.kM.multidot.N additions,
4.multidot.M.multidot.N additions, 4.multidot.M.multidot.N
subtractions, and M.multidot.N bitwise register shifts.
[0070] For illustrative purposes in comparing the algorithms, let
us again assume the operational parameters assumed previously: that
addition, subtraction, and register-shifting operation take 10
clock cycles each; that multiplication takes 30 clock cycles; that
M.times.N=1000.times.1250- ; and that N.sub.k is 11. As computed
previously, the Karssemeijer algorithm would require 18.2 billion
clock cycles to compute the three line estimator functions
W.sub..sigma.(0), W.sub..sigma.(60), and W.sub..sigma.(120), while
the algorithm of FIGS. 3 and 4 would require about 6.2 billion
clock cycles to generate the four line operator functions
W.sub..sigma.(0), W.sub..sigma.(45), W.sub..sigma.(90), and
W.sub..sigma.(135), a significant computational advantage. However,
using the results of the previous paragraph, the algorithm of FIG.
5 would require only
(11)(1000)(1250)(30)+2(11)(1000)(1250)(10)+(4)(1000)(1250)(1-
0)+(4)(1000)(1250)(10)+(1000)(1250)(10)=0.8 billion clock cycles to
produce the arrays A, D, and L. For the preferred embodiment of
FIG. 5, the reduction in computation becomes even more dramatic as
the scale of interest (reflected by the size of the kernel size
N.sub.k) grows larger, because the number of computations only
increases linearly with N.sub.k. It is to be appreciated that the
above numerical example is a rough estimate and is for illustrative
purposes only to clarify the features and advantages of the present
invention, and is not intended to limit the scope of the preferred
embodiments.
[0071] Optionally, in the preferred embodiment of FIGS. 3-5, a
plurality of spatial scale values .sigma.1, .sigma.2, . . . ,
.sigma.n may be selected at step 302 or 502. The remainder of the
steps of the embodiments of FIGS. 3-5 are then separately carried
out for each of the spatial scale values .sigma.1, .sigma.2, . . .
, .sigma.n. For a given pixel (i, j), the value of the direction
image array .theta..sub.max(i, j) is selected to correspond to the
largest value among W.sub..sigma.1(.theta..sub.max1),
W.sub..sigma.2(.theta..sub.max2), . . . ,
W.sub..sigma.n(.theta..sub.maxn). The line image array L(i, j) is
formed by thresholding an array corresponding to largest value
among W.sub..sigma.1(.theta..sub.max1),
W.sub..sigma.2(.theta..sub.max2), . . . ,
W.sub..sigma.n(.theta..sub.maxn) at each pixel.
[0072] As another option, which may be used separately or in
combination with the above option of using multiple spatial scale
values, a plurality of filter kernel sizes N.sub.k1, N.sub.k2, . .
. , N.sub.kn ay be selected at step 302 or 502. The remainder of
the steps of the embodiments of FIGS. 3-5 are then separately
carried out for each of the filter kernel sizes N.sub.k1, N.sub.k2,
. . . , N.sub.kn. For a given pixel (i, j), the value of the
direction image array .theta..sub.max(i, j) is selected to
correspond to the largest one of the different
W.sub..sigma.(.theta..sub.max) values yielded for the different
values of filter kernel size N.sub.k. The line image array L(i, j)
is formed by thresholding an array corresponding to largest value
among the different W.sub..sigma.(.theta..sub.max) values yielded
by the different values of filter kernel size N.sub.k. By way of
example and not by way of limitation, it has been found that with
reference to the previously disclosed system for detecting lines in
fibrous breast tissue in a 1000.times.1250 digital mammogram at 200
micron resolution, results are good when the pair of combinations
(N.sub.k=11, .sigma.=1.5) and (N.sub.k=7, .sigma.=0.9) are
used.
[0073] The preferred embodiments disclosed in FIGS. 3-5 require a
corrective algorithm to normalize the responses of certain portions
of the algorithms associated with directional second order
derivatives in diagonal directions. In particular, the responses of
Eqs. (6b), (6d), and (22) require normalization because the
filtered image is being sampled at more widely displaced points,
resulting in a response that is too large by a constant factor. In
the preferred algorithms that use a Gaussian filter G at step 304
of FIG. 3 or steps 504-506 of FIG. 5, a constant correction factor
"p" is determined as shown in Eqs. (23)-(25):
p=SQRT{.SIGMA.(K.sub.A(i,j)).sup.2/.SIGMA.(K.sub.D(i,j)).sup.2}
(23)
[0074] 6 0 1 0 K A = G * - 1 0 - 1 0 1 0 ( 24 ) 7 1 0 - 1 K D = G *
0 0 0 - 1 0 1 ( 25 )
[0075] In the general case where the digital mammogram image I is
convolved with a single-peaked filter F at step 304 of FIG. 3 or
steps 504-506 of FIG. 5, the constant correction actor p is
determined by using F instead of G in Eqs. (24) and (25).
[0076] Importantly, the constant correction factor p does not
actually affect the number of computations in the convolutions of
Eqs. (6b), (6d), and (22), but rather is incorporated into later
parts of the algorithm. In particular, in the algorithm of FIG. 3,
the constant correction factor p is incorporated by substituting,
for each instance of W.sub..sigma.(45) and W.sub..sigma.(135) in
Eqs. (8) and (9), and step 308, the quantities pW.sub..sigma.(45)
and pW.sub..sigma.(135), respectively. In the algorithm of FIG. 5,
the constant correction factor p is incorporated by substituting,
for each instance of D in Eqs. (15) and (16), and steps 514 and
518, the quantity pD. Accordingly, the computational efficiency of
the preferred embodiments is maintained in terms of the reduced
number and complexity of required convolutions.
[0077] A computational simplification in the implementation of the
constant correction factor p is found where the size of the spatial
scale parameter 6 corresponds to a relatively large number of
pixels, e.g. on the order of 11 pixels or greater. In this
situation the constant correction factor p approaches the value of
1/2, the sampling distance going up by a factor of {square root}2
and the magnitude of the second derivative estimate going up by the
square of the sampling distance. In such case, multiplication by
the constant correction factor p is achieved by a simple bitwise
right register shift.
[0078] As disclosed above, a method and system for line detection
in medical images according to the preferred embodiments contains
several advantages. The preferred embodiments share the
homogeneity, isotropy, and other desirable scale-space properties
associated with the Karssemeijer method. However, as described
above, the preferred embodiments significantly reduce the number of
required computations. Indeed, for one of the preferred
embodiments, running time increases only linearly with the scale of
interest, thus typically requiring an order of magnitude fewer
operations in order to produce equivalent results. For applications
in which processing time is a constraint, this makes the use of
higher resolution images in order to improve line detection
accuracy more practical.
[0079] While preferred embodiments of the invention have been
described, these descriptions are merely illustrative and are not
intended to limit the present invention. For example, although the
component kernels of the separable single-peaked filter function
are described above as one-dimensional kernels, the selection of
appropriate two-dimensional kernels as component kernels of the
single-peaked filter function can also result in computational
efficiencies, where one of the dimensions is smaller than the
initial size of the single-peaked filter function. As another
example, although the embodiments of the invention described above
were in the context of medical imaging systems, those skilled in
the art will recognize that the disclosed methods and structures
are readily adaptable for broader image processing applications.
Examples include the fields of optical sensing, robotics, vehicular
guidance and control systems, synthetic vision, or generally any
system requiring the generation of line images or direction images
from an input image.
* * * * *