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 computerassisted 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]
[0004]FIG. 1 shows steps performed by a computerassisted 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 xray mammogram is scanned in and digitized into a digital mammogram. The digital mammogram may be, for example, a 4000×5000 array of 12bit gray scale pixel values. Such a digital mammogram would generally correspond to a typical 8″×10″ xray 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 1000×1250 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 mm10 mm, i.e., if lines on the order of 1 mm10 mm are being detected, it is neither efficient nor necessary to process a full 50micron (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×N arrays of 12bit 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×N line image L(i, j) and an M×N direction image θ_{max}(i, j) are generated from the digital mammogram image I. The M×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×N direction image θ_{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 θ_{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 xray mammogram. In this manner, the possibility of missed diagnosis due to human error is reduced.

[0010]
One of the desired characteristics of a spiculationdetecting CAD system is high speed to allow processing of more xray 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 33334. 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 secondorder 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 2D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. ASSP, Vol. 36, pp. 116979 (1988).

[0012]
The above filterbank algorithms are computationally intensive, generally requiring a separate convolution operation for each orientationselective 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. 1012, 1994) at 21119, and in Karssemeijer, “Detection of Stellate Distortions in Mammograms using Scale Space Operators,” Information Processing in Medical Imaging 33546 (Bizais et al., eds. 1995) at 33546. 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 597605. 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 filterbank method. More precisely, at a given level of spatial scale σ, 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]
[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_{k }are selected. The spatial scale parameter o dictates the width, in pixels, of a Gaussian kernel G(r,σ), the equation for which is shown in Eq. (1):

G(r,σ)=(½πσ^{2})exp(−r ^{2}/2σ^{2}) (1)

[0016]
At step 202, the filter kernel size N_{k}, in pixels, is generally chosen to be large enough to contain the Gaussian kernel G(r,σ) in digital matrix form, it being understood that the function G(r,σ) becomes quite small very quickly. Generally speaking, the spatial scale parameter σ corresponds, in an orderofmagnitude sense, to the size of the lines being detected. By way of example only, and not by way of limitation, for detecting 1 mm10 mm lines in fibrous breast tissue in a 1000×1250 digital mammogram at 200 micron (0.2 mm) resolution, the value of σ may be selected as 1.5 pixels and the filter kernel size N_{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_{σ}(0), K_{σ}(60), and K_{σ}(120) are formed as the second order directional derivatives of the Gaussian kernel G(r,σ) at 0 degrees, 60 degrees, and 120 degrees, respectively. The three filter kernels K_{σ}(0), K_{σ}(60), and K_{σ}(120) are each of size N_{k}, each filter kernel thus containing N_{k×}N_{k }elements.

[0018]
At step 206, the digital mammogram image I is separately convolved with each of the three filter kernels K_{σ}(0), K_{σ}(60), and K(120) to produce three line operator functions W_{σ}(0), W_{σ}(60), and W_{σ}(120), respectively, as shown in Eq. (2):

W _{σ}(0)=I*K _{σ}(0)W _{σ}(60)=I*K _{σ}(60)W _{σ}(120)=I*K _{σ}(120) (2)

[0019]
Each of the line operator functions W_{σ}(0), W_{σ}(60), and W_{σ}(120) is, of course, a twodimension array that is slightly larger than the original M×N digital mammogram image array I due to the size N_{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_{σ}(θ) may be formed as a combination of the line operator functions W_{σ}(0), W_{σ}(60), and W_{σ}(120) as defined in equation (3):

W _{σ}(θ)=(⅓)(1+2 cos(2θ))W _{σ}(0)+(⅓)(1− cos(2θ)+({square root}3)sin(2θ))W _{σ}(60)+(⅓)(1− cos(2θ)−({square root}3)sin(2θ))W_{σ}(120) (3)

[0021]
As indicated by the above definition, the estimation function W_{σ}(θ) is a function of three variables, the first two variables being pixel coordinates (i, j) and the third variable being an angle θ. For each pixel location (i, j), the estimation function W_{σ}(θ) represents a measurement of line strength at pixel (i, j) in the direction perpendicular to θ. According to the Karssemeijer method, an analytical expression for the extrema of W_{σ}(θ) with respect to θ, denoted θ_{min,max }at a given pixel (i, j) is given by Eq. (4):

θ_{min,max}=½[arc tan{({square root}3)(W _{σ}(60)−W _{94 }(120))/(W _{σ}(60)+W _{σ}(120)−2W _{σ}(0))}±π] (4)

[0022]
Thus, at step 208, the expression of Eq. (4) is computed for each pixel based on the values of W_{σ}(0), W_{σ}(60), and W_{σ}(120) that were computed at step 206. Of the two solutions to equation (4), the direction θ_{max }is then selected as the solution that yields the larger magnitude for W_{σ}(θ) at that pixel, denoted W_{σ}(θ_{max}). Thus, at step 208, an array θ_{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 twodimensional array of line intensities corresponding to the maximum direction θ_{max }at each pixel is formed, denoted as the line intensity function W_{σ}(θ_{max}).

[0023]
At step 210, a line image L(i, j) is formed using information derived from the line intensity function W_{σ}(θ_{max}) that was inherently generated during step 208. The array L(i, j) is formed from W_{σ}(θ_{max}) using known methods such as a simple thresholding process or a modified thresholding process based on a histogram of W_{σ}(θ_{max}). With the completion of the line image array L(i, j) and the direction image array θ_{max}(i, j), the line detection process is complete.

[0024]
Optionally, in the Karssemeijer algorithm a plurality of spatial scale values σ1, σ2, . . . , σn may be selected at step 202. The steps 204210 are then separately carried out for each of the spatial scale values (σ1, σ2, . . . , σn. For a given pixel (i, j), the value of θ_{max}(i, j) is selected to correspond to the largest value among W_{σ1}(θ_{max1}), W_{σ2}(θ_{max2}), . . . , W_{σn}(θ_{maxn}). The line image L(i, j) is formed by thresholding an array corresponding to largest value among W_{σ1}(θ_{max1}), W_{σ2}(θ_{max2}), . . . , W_{σn}(θ_{maxn}) at each pixel.

[0025]
Although it is generally more computationally efficient than the filterbank method, the prior art Karssemeijer algorithm has computational disadvantages. In particular, for a given spatial scale parameter σ, the Karssemeijer algorithm requires three separate convolutions of N_{k}×N_{k }kernels with the M×N digital mammogram image I. Each convolution, in turn, requires approximately M·N·(N_{k})^{2 }multiplication and addition operations, which becomes computationally expensive as the kernel size N_{k}, which is proportional to the spatial scale parameter σ, 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 amethod and apparatus for detecting lines in a medical imaging system by filtering the digital image with a singlepeaked 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 singlepeaked 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 3×3 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 singlepeaked 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 singlepeaked 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 3×3 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]
[0033]FIG. 1 shows steps taken by a computeraided diagnosis (“CAD”) system for detecting spiculations in digital mammograms in accordance with the prior art.

[0034]
[0034]FIG. 2 shows line detection steps taken by the CAD system of FIG. 1.

[0035]
[0035]FIG. 3 shows line detection steps according to a preferred embodiment.

[0036]
[0036]FIG. 4 shows steps for convolution with second order directional derivative operators in accordance with a preferred embodiment.

[0037]
[0037]FIG. 5 shows line detection steps according to another preferred embodiment.
DETAILED DESCRIPTION

[0038]
[0038]FIG. 3 shows steps of a line detection algorithm in accordance with a preferred embodiment. At step 302, a spatial scale parameter θ and a filter kernel size N_{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 σ while not increasing the computational intensity of the algorithm.

[0039]
At step 304, the digital mammogram image I is convolved with a twodimensional singlepeaked filter F having dimensions N_{k}×N_{k }to form a filtered image array I_{F }as shown in Eq. (5):

I _{F} =I*F (5)

[0040]
By singlepeaked 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 singlepeaked filter F is dictated by the spatial scale parameter σ. For example, where a Gaussian filter is used, σ is the standard deviation of the Gaussian, and where a flat pillbox function is used, σ 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_{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_{F }with second order directional derivative operators D_{2}(0), D_{2}(45), D_{2}(90), and D_{2}(135) to produce the line operator functions W_{σ}(0), W_{σ}(45), W_{σ}(90), and W_{σ}(135), respectively, as shown in Eqs. (6a)(6d).

W _{σ}(0)=I _{F} *D _{2}(0) (6a)

W _{σ}(45)=I _{F} *D _{2}(45) (6b)

W _{σ}(90)=I _{F} *D _{2}(90) (6c)

W _{σ}(135)=I _{F} *D _{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 3×3 kernels shown in Eqs. (7a)(7d):
$\begin{array}{cc}\begin{array}{ccccc}0\ue89e\text{\hspace{1em}}& 0& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {D}_{2}\ue8a0\left(0\right)& =& 1& 2& 1\\ 0\ue89e\text{\hspace{1em}}& 0& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \text{(7a)}\\ \begin{array}{ccccc}0\ue89e\text{\hspace{1em}}& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {D}_{2}\ue8a0\left(45\right)& =& 0& 2& 0\\ 1\ue89e\text{\hspace{1em}}& 0& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \text{(7b)}\\ \begin{array}{ccccc}0\ue89e\text{\hspace{1em}}& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {D}_{2}\ue8a0\left(90\right)& =& 0& 2& 0\\ 0\ue89e\text{\hspace{1em}}& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \text{(7c)}\\ \begin{array}{ccccc}1\ue89e\text{\hspace{1em}}& 0& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {D}_{2}\ue8a0\left(135\right)& =& 0& 2& 0\\ 0\ue89e\text{\hspace{1em}}& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \text{(7d)}\end{array}$

[0043]
The above 3×3 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_{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, 3×3 kernels are used.

[0044]
Subsequent steps are based on an estimation function W_{σ}(θ) that can be formed from the arrays W_{σ}(0), W_{σ}(45), W_{σ}(90), and W_{σ}(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 _{σ}(θ)=¼{(1+2 cos(2θ))W _{σ}(0)+(1+2 sin(2θ))W _{σ}(45)+(1−2 cos(2θ))W _{σ}(90)+(1−2 sin(2θ))W _{σ}(135)} (8)

[0045]
It has been found that the extrema of the estimation function W_{σ}(θ) with respect to θ, denoted θ_{min,max }at a given pixel (i, j) is given by Eq. (9):

θ _{min,max}=½[a tan{(W _{σ}(45)−W _{σ}(135))/(W _{σ}(0)−W _{σ}(90))}±π] (9)

[0046]
At step 308, the expression of Eq. (9) is computed for each pixel. Of the two solutions to equation (4), the direction θ_{max }is then selected as the solution that yields the larger magnitude for W_{σ}(θ) at that pixel, denoted as the line intensity W_{σ}(θ_{max}). Thus, at step 308, an array θ_{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 twodimensional array of line intensities corresponding to the maximum direction θ_{max }at each pixel is formed, denoted as the line intensity function W_{σ}(θ_{max}).

[0047]
At step 310, a line image array L(i, j) is formed using information derived from the line intensity function W_{σ}(θ_{max}) that was inherently generated during step 308. The line image array L(i, j) is formed from the line intensity function W_{σ}(θ_{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_{σ}(θ_{max}). With the completion of the line image array L(i, j) and the direction image array θ_{max}(i, j), the line detection process is complete.

[0048]
[0048]FIG. 4 illustrates unique computational steps corresponding to the step 306 of FIG. 3. At step 306, the filtered image array I_{F }is convolved with the second order directional derivative operators D_{2}(0), D_{2}(45), D_{2}(90), and D_{2}(135) shown in Eq. (7). An advantage of the use of the small 3×3 kernels D_{2}(0), D_{2}(45), D_{2}(90), and D_{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_{F }is doubled to produce the doubled filtered image array 2I_{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_{F}, the value of I_{F}(i−1,j) is subtracted, and at step 406, the value of I_{F}(i+1,j) is subtracted, the result being equal to the desired convolution result I_{F}*D_{2}(0) at pixel (i, j). Similarly, at step 408, at each pixel (i, j) in the array 2I_{F}, the value of I_{F}(i−1,j−1) is subtracted, and at step 410, the value of I_{F}(i+1,j+1) is subtracted, the result being equal to the desired convolution result I_{F}*D_{2}(45) at pixel (i, j). Similarly, at step 412, at each pixel (i, j) in the array 2I_{F}, the value of I_{F}(i, j−1) is subtracted, and at step 414, the value of I_{F}(i, j+1) is subtracted, the result being equal to the desired convolution result I_{F}*D_{2}(90) at pixel (i, j). Finally, at step 416, at each pixel (i, j) in the array 2I_{F}, the value of I_{F}(i+1,j−1) is subtracted, and at step 418, the value of I_{F}(i−1,j+1) is subtracted, the result being equal to the desired convolution result I_{F}*D_{2}(135) at pixel (i, j). The steps 406418 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_{σ}(0), W_{σ}(45), W_{σ}(90), and W_{σ}(135) while at the same time using fewer computations than the Karssemeijer algorithm of FIG. 2, which uses only three line operator functions W_{σ}(0), W_{σ}(60), W_{σ}(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 3×3 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×N digital mammogram image I with the N_{k}×N_{k }kernels, requiring approximately 3·(N_{k})^{2}·M·N multiplications and adds to derive the three line estimator functions W_{σ}(0), W_{σ}(60), and W_{σ}(120). However, in the embodiment of FIGS. 3 and 4, the computation of the four line estimator functions W_{94 }(0), W_{σ}(45), W_{σ}(90), and W_{σ}(135) requires a first convolution requiring (N_{k})^{2}·M·N multiplications, followed by M·N doubling operations and 8·M·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 registershifting 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×N=1000×1250 is used and that N_{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)^{2}(1000)(1250)(30+10)=18.2 billion clock cycles to compute the three line estimator functions W_{σ}(0), W_{σ}(60), and W_{σ}(120). In contrast, the algorithm of FIGS. 3 and 4 would require only (11)^{2}(1000)(1250)(30+10)+(1250)(1000)(10)+8(1250)(1000)(10)=6.2 billion clock cycles to generate the four line operator functions W_{σ}(0), W_{σ}(45), W_{σ}(90), and W_{σ}(135), a significant computational advantage.

[0052]
[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 singlepeaked 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 onedimensional kernels. Thus, the N_{k}×N_{k }filter kernel F(i, j) is separable where it can be formed as a convolution of an N_{k}×1 kernel F_{x}(i) and a 1×N_{k }kernel F_{y}(j), i.e., F(i, j)=F_{x}(i)*F_{y}(j). As known in the art, an N_{k}×1 kernel is analogous to a row vector of length N_{k }while a 1×N_{k }kernel is analogous to a column vector of length N_{k}.

[0053]
Although a variety of singlepeaked 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:
$\begin{array}{cc}\begin{array}{c}G=\left(1/2\ue89e{\mathrm{\pi \sigma}}^{2}\right)\ue89e\mathrm{exp}\ue8a0\left({x}^{2}/2\ue89e{\sigma}^{2}\right)\ue89e\mathrm{exp}\ue8a0\left({y}^{2}/2\ue89e{\sigma}^{2}\right)\\ ={G}_{x}*{G}_{y}\end{array}& \left(10\right)\\ \begin{array}{c}{G}_{x}=\text{\hspace{1em}}\ue89e\left[{g}_{x,0}\ue89e\text{\hspace{1em}}\ue89e{g}_{x,1}\ue89e\text{\hspace{1em}}\ue89e{g}_{x,2}\ue89e\text{\hspace{1em}}\ue89e\cdots \ue89e\text{\hspace{1em}}\ue89e{g}_{x,\mathrm{Nk}1}\right]\\ \text{\hspace{1em}}\ue89e{g}_{y,0}\\ \text{\hspace{1em}}\ue89e{g}_{y,1}\end{array}& \left(11\right)\\ \begin{array}{c}{G}_{y}=\text{\hspace{1em}}\ue89e{g}_{y,3}\\ \text{\hspace{1em}}\ue89e\vdots \\ \text{\hspace{1em}}\ue89e{g}_{y,\mathrm{Nk}1}\end{array}& \left(12\right)\end{array}$

[0054]
At step 502, the parameters σ and N_{k }are selected in a manner similar to step 302 of FIG. 3. It is preferable for N_{k }to be selected as an odd number, so that a onedimensional Gaussian kernel of length N_{k }may be symmetric about its central element. At step 504, the M×N digital mammogram image I is convolved with the Gaussian N_{k}×1 kernel G_{x }to produce an intermediate array I_{x}:

I _{x} =G _{x} *I (13)

[0055]
In accordance with a preferred embodiment, the sigma of the onedimensional Gaussian kernel G_{x }is the spatial scale parameter a selected at step 502. The intermediate array I_{x }resulting from step 504 is a twodimensional array having dimensions of approximately (M+2N_{k})×N.

[0056]
At step 506, the intermediate array I_{x }is convolved with the Gaussian 1×N_{k }kernel G_{y }to produce a Gaussianfiltered image array I_{G}:

I _{G} =I _{x} *G _{y} (14)

[0057]
In accordance with a preferred embodiment, the sigma of the onedimensional Gaussian kernel G_{y }is also the spatial scale parameter a selected at step 502. The filtered image array I_{G }resulting from step 506 is a twodimensional array having dimensions of approximately (M+2N_{k})×(N+2N_{k}). Advantageously, because of the separability property of the twodimensional Gaussian, the filtered image array I_{G }resulting from step 506 is identical to the result of a complete twodimensional convolution of an N_{k}×N_{k }Gaussian kernel and the digital mammogram image I. However, the number of multiplications and additions is reduced to 2·N_{k}·M·N instead of (N_{k})^{2}·M·N.

[0058]
Even more advantageously, in the situation where N_{k }is selected to be an odd number and the onedimensional 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_{k }is odd, then the component one dimensional kernels G_{x }and G_{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_{k }is selected to be an odd number, the number of multiplications associated with the required convolutions is approximately N_{k}·M·N and the number of additions is approximately 2·N_{k}·M·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 306310. In particular, it has been found that for each pixel (i, j), the solution for the direction image θ_{max }and the line intensity function W_{σ}(θ_{max}) can be simplified to the following formulas of Eqs. (15)(16):

W _{σ}(θ_{max})=½(L+{square root}(A ^{2} +D ^{2})) (15)

θ_{max}=½a tan(D/A) (16)

[0060]
In the above formulas, the array L is defined as follows:

L=W _{σ(}0)+W _{σ}(90)=I _{G} *D _{2}(0)+I _{G} *D _{2}(90)=I_{G} *[D _{2}(0)+D_{2}(90)] (17)

[0061]
[0061]
$\begin{array}{cc}\begin{array}{cccccc}\text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ L=& {I}_{G}& *& 1& 4& 1\\ \text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(18\right)\end{array}$

[0062]
As known in the art, the array L is the result of the convolution of I_{G }with a Laplacian operator. Furthermore, the array A in Eqs. (15) and (16) is defined as follows:

A=W _{σ}(0)−W _{σ}(90)=I _{G} *D _{2}(0)−I _{G} *D _{2}(90)=I _{G} *[D _{2}(0)−D_{2}(90)] (19)

[0063]
[0063]
$\begin{array}{cc}\begin{array}{cccccc}\text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ A=& {I}_{G}& *& 1& 0& 1\\ \text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(20\right)\end{array}$

[0064]
Finally, the array D in Eqs. (15) and (16) is defined as follows:

D=W _{σ}(45)−W _{σ}(135)=I _{G} *D _{2}(45)−I _{G} *D _{2}(135)=I _{G} *[D _{2}(45)−D _{2}(135)] (21)

[0065]
[0065]
$\begin{array}{cc}\begin{array}{cccccc}\text{\hspace{1em}}& 1& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ D=& {I}_{G}& *& 0& 0& 0\\ \text{\hspace{1em}}& 1& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(22\right)\end{array}$

[0066]
Accordingly, at step 508 the convolution of Eq. (20) is performed on the filtered image array I_{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_{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 508512 may be performed in parallel or in any order. At step 514, the line intensity function W_{σ}(θ_{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_{σ}(θ_{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_{σ}(θ_{max}).

[0067]
Finally, at step 518, the direction image array θ_{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 θ_{max}(i, j) and the steps 514516 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 θ_{max}(i, j) in order to evaluate the line intensity estimator function W_{94 }(θ_{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 θ_{max}(i, j) in order to be able to evaluate the line intensity estimator function W_{σ}(θ) at the maximum angle θ_{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 θ_{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 θ_{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 θ_{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_{G }there is required only approximately N_{k}·M·N multiplications and 2·N_{k}·M·N additions. To generate the array A from the filtered image array I_{G}, there is required 2·M·N additions and M·N subtractions. Likewise, to generate the array D from the filtered image array I_{G}, there is required 2·M·N additions and M·N subtractions. Finally, to generate L from the filtered image array I_{G}, there is required M·N bitwise left register shift of two positions (corresponding to a multiplication by 4), followed by 4·M·N subtractions. Accordingly, to generate the arrays A, D, and L from the digital mammogram image I, there is required only 2·N_{k}·M·N multiplications, 2·N_{k}M·N additions, 4·M·N additions, 4·M·N subtractions, and M·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 registershifting operation take 10 clock cycles each; that multiplication takes 30 clock cycles; that M×N=1000×1250; and that N_{k }is 11. As computed previously, the Karssemeijer algorithm would require 18.2 billion clock cycles to compute the three line estimator functions W_{σ}(0), W_{σ}(60), and W_{σ}(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_{σ}(0), W_{σ}(45), W_{σ}(90), and W_{σ}(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)(10)+(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_{k}) grows larger, because the number of computations only increases linearly with N_{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. 35, a plurality of spatial scale values σ1, σ2, . . . , σn may be selected at step 302 or 502. The remainder of the steps of the embodiments of FIGS. 35 are then separately carried out for each of the spatial scale values σ1, σ2, . . . , σn. For a given pixel (i, j), the value of the direction image array θ_{max}(i, j) is selected to correspond to the largest value among W_{σ1}(θ_{max1}), W_{σ2}(θ_{max2}), . . . , W_{σn}(θ_{maxn}). The line image array L(i, j) is formed by thresholding an array corresponding to largest value among W_{σ1}(θ_{max1}), W_{σ2}(θ_{max2}), . . . , W_{σn}(θ_{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_{k1}, N_{k2}, . . . , N_{kn }ay be selected at step 302 or 502. The remainder of the steps of the embodiments of FIGS. 35 are then separately carried out for each of the filter kernel sizes N_{k1}, N_{k2}, . . . , N_{kn}. For a given pixel (i, j), the value of the direction image array θ_{max}(i, j) is selected to correspond to the largest one of the different W_{σ}(θ_{max}) values yielded for the different values of filter kernel size N_{k}. The line image array L(i, j) is formed by thresholding an array corresponding to largest value among the different W_{σ}(θ_{max}) values yielded by the different values of filter kernel size N_{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 1000×1250 digital mammogram at 200 micron resolution, results are good when the pair of combinations (N_{k}=11, σ=1.5) and (N_{k}=7, σ=0.9) are used.

[0073]
The preferred embodiments disclosed in FIGS. 35 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 504506 of FIG. 5, a constant correction factor “p” is determined as shown in Eqs. (23)(25):

p=SQRT{Σ(K _{A}(i,j))^{2}/Σ(K _{D}(i,j))^{2}} (23)

[0074]
[0074]
$\begin{array}{cc}\begin{array}{cccccc}\text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {K}_{A}=& G& *& 1& 0& 1\\ \text{\hspace{1em}}& 0& 1& 0& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(24\right)\end{array}$ $\begin{array}{cc}\begin{array}{cccccc}\text{\hspace{1em}}& 1& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {K}_{D}=& G& *& 0& 0& 0\\ \text{\hspace{1em}}& 1& 0& 1& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(25\right)\end{array}$

[0075]
In the general case where the digital mammogram image I is convolved with a singlepeaked filter F at step 304 of FIG. 3 or steps 504506 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_{σ}(45) and W_{σ}(135) in Eqs. (8) and (9), and step 308, the quantities pW_{σ}(45) and pW_{σ}(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 ½, 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 scalespace 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 singlepeaked filter function are described above as onedimensional kernels, the selection of appropriate twodimensional kernels as component kernels of the singlepeaked filter function can also result in computational efficiencies, where one of the dimensions is smaller than the initial size of the singlepeaked 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.