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, 3×3 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.

Description:
FIELD OF THE INVENTION 
     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 
     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. 
     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. 
     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 4000×5000 array of 12-bit gray scale pixel values. Such a digital mammogram would generally correspond to a typical 8″×10″ x-ray mammogram which has been digitized at 50 microns (0.05 mm) per pixel. 
     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 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. 
     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 12-bit gray scale pixel values, with M and N having exemplary values of 1000 and 1250, respectively. 
     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). 
     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. 
     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. 
     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. 
     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). 
     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. 
     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  2 nd 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 Doom, “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. 
     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 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) . 
     FIG. 2 shows steps for computing line and direction images in accordance with the Karssemeijer algorithm. At step  202 , a spatial scale parameter σ and a filter kernel size N k  are selected. The spatial scale parameter σ 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) 
       
     
     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 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 1000×1250 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 σ and the kernel size. 
     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. 
     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 σ (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) 
       
     
     Each of the line operator functions W σ (0), W σ (60), and W σ (120) is, of course, a two-dimensional 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. 
     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)+(3)sin(2θ)) W   σ (60)+(⅓)(1−cos(2θ)−(3)sin(2θ)) W   σ (120)  (3) 
       
     
     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{(3)( W   σ (60)− W   σ (120))/( W   σ (60)+ W   σ (120)−2 W   σ (0))}±π]  (4) 
       
     
     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 two-dimensional 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 ). 
     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. 
     Optionally, in the Karssemeijer algorithm a plurality of spatial scale values σ1, σ2, . . . , σn may be selected at step  202 . The steps  204 - 210  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. 
     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 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. 
     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. 
     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. 
     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 
     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. 
     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 3×3 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. 
     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 3×3 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. 
     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 
     FIG. 1 shows steps taken by a computer-aided diagnosis (“CAD”) system for detecting spiculations in digital mammograms in accordance with the prior art. 
     FIG. 2 shows line detection steps taken by the CAD system of FIG.  1 . 
     FIG. 3 shows line detection steps according to a preferred embodiment. 
     FIG. 4 shows steps for convolution with second order directional derivative operators in accordance with a preferred embodiment. 
     FIG. 5 shows line detection steps according to another preferred embodiment. 
    
    
     DETAILED DESCRIPTION 
     FIG. 3 shows steps of a line detection algorithm in accordance with a preferred embodiment. At step  302 , a spatial scale parameter σ 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. 
     At step  304 , the digital mammogram image I is convolved with a two-dimensional single-peaked 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) 
       
     
     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 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. 
     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 function 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) 
       
     
     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):                  D   2          (   0   )       =         0       0       0             -   1         2         -   1             0       0       0                 (7a)                   D   2          (   45   )       =         0       0         -   1             0       2       0             -   1         0       0                 (7b)                   D   2          (   90   )       =         0         -   1         0           0       2       0           0         -   1         0                 (7c)                   D   2          (   135   )       =           -   1         0       0           0       2       0           0       0         -   1                   (7d)                                
     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. 
     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) 
       
     
     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[arc tan{( W   σ (45)− W   σ (135))/( W   σ (0)− W   σ (90))}±π]  (9) 
       
     
     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 two-dimensional 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 ). 
     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. 
     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. 
     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  406 - 418  are preferably carried out in the parallel fashion shown in FIG. 4 but can generally be carried out in any order. 
     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 σ (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. 
     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×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. 
     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 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 . 
     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:                    G   =       (       1   /   2                   π                   σ   2       )                   exp                   (         -     x   2       /   2                     σ   2       )                   exp                   (         -     y   2       /   2                     σ   2       )                   =       G   x     *     G   y                     (   10   )                       G   x     =     [       g     x   ,   0                       g     x   ,   1                       g     x   ,   2                     ⋯                   g     x   ,       N                 k     -   1           ]                                   (   11   )                 G   y     =     [           g     y   ,   0                 g     y   ,   1                 g     y   ,   2               ⋮             g     y   ,       N                 k     -   1               ]             (   12   )                                
     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 one-dimensional 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) 
       
     
     In accordance with a preferred embodiment, the sigma of the one-dimensional Gaussian kernel G x  is the spatial scale parameter σ selected at step  502 . The intermediate array I x  resulting from step  504  is a two-dimensional array having dimensions of approximately (M+2N k )×N. 
     At step  506 , the intermediate array I x  is convolved with the Gaussian 1×N k  kernel G y  to produce a Gaussian-filtered image array I G : 
     
       
           I   G   =I   x   *G   y   (14) 
       
     
     In accordance with a preferred embodiment, the sigma of the one-dimensional Gaussian kernel G y  is also the spatial scale parameter σ selected at step  502 . The filtered image array I G  resulting from step  506  is a two-dimensional array having dimensions of approximately (M+2N k )×(N+2N k ). Advantageously, because of the separability property of the two-dimensional Gaussian, the filtered image array I G  resulting from step  506  is identical to the result of a complete two-dimensional 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. 
     Even more advantageously, in the situation where N 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 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. 
     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 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 +( A   2   +D   2 ))  (15) 
       
     
     
       
         θ max =½ a  tan( D/A )  (16) 
       
     
     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) 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     
                       I 
                       G 
                     
                     * 
                     
                       
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           4 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     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) 
       
     
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     
                       I 
                       G 
                     
                     * 
                     
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     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) 
       
     
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       I 
                       G 
                     
                     * 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     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  508 - 512  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 ). 
     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  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 θ max (i,j) in order to evaluate the line intensity estimator function W σ (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 . 
     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. 
     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. 
     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×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. 
     Optionally, in the preferred embodiment of FIGS. 3-5, 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. 3-5 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. 
     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 σn  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 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. 
     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 {Σ( K   A ( i,j )) 2 /Σ( K   D ( i,j )) 2 }  (23) 
     
       
         
           
             
               
                 
                   
                     K 
                     A 
                   
                   = 
                   
                     G 
                     * 
                     
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     K 
                     D 
                   
                   = 
                   
                     G 
                     * 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     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 factor p is determined by using F instead of G in Eqs. (24) and (25). 
     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. 
     A computational simplification in the implementation of the constant correction factor p is found where the size of the spatial scale parameter σ 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 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. 
     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. 
     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.