aWolson Image Analysis Unit, Department of Medical Biophysics, Stopford Building, University of Manchester, Oxford Road, Manchester, M13 9PT, email: tcp@sv1.smb.man.ac.uk
bThe Nightingale Centre, Withington Hospital, Nell Lane, Manchester.
Malignant breast lesions in x-ray mammograms are often characterised by abnormal patterns of linear structures. Architectural distortions and stellate lesions are examples of patterns frequently presenting with an appearance of radiating lines. Attempts to automatically detect these abnormalities have generally concentrated on features of known importance, such as radiating linear structure concurrency, spread of focus and radial distance. We describe a generic representation of patterns of oriented lines that is both complete and uncommitted. Our representation places no emphasis on the features known to be important, yet clearly incorporates them. Statistical models, based on factor analysis, are trained on representations extracted from stellate lesions to detect abnormal patterns of linear structures. We present an application of the technique to a set of 129 mammograms containing 29 stellate lesions. In addition, we describe how directional recursive median filtering can be applied at a number of scales and how the resulting scale orientation signatures can be used to train a model to detect the central masses of stellate lesions. Results are provided for a set of 53 mammograms containing 26 stellate lesions. The combined results of the two techniques are presented and simple methods of combination are discussed. Selecting a single operating point for mass detection in combination with the oriented pattern technique improves the overall accuracy to a sensitivity of 100% at zero false positives per image for structures of diameter 16mm and above.
Breast cancer is a leading cause of fatality in women, with approximately 1 in 12 women affected by the disease during their lifetime1. Mass screening of women using x-ray mammography is currently the most effective method of early detection of the disease; this is essential for successful treatment2. Malignant breast lesions in x-ray mammograms are often characterised by abnormal patterns of linear structures. Architectural distortions and stellate lesions are examples of patterns frequently presenting with an appearance of radiating lines3. These malignancies can be extremely subtle and a significant proportion of them are undetected by radiologists. Attempts to detect stellate lesions automatically have generally concentrated on features of known importance, such as radiating linear structure concurrency, spread of focus and radial distance4,5,6. Previous experiments show that radiologists' performance can be improved by prompting the possible locations of abnormalities7. Recent work provides quantitive targets for clinically useful algorithm performance8. Most published techniques provide useful insight into the problem but fall well below the level of prompting accuracy that leads to an improvement in radiologists' performance. This implies that existing methods are failing to fully exploit the available image evidence. We describe two alternative approaches. The first uses a compact statistical representation of pattern structure, based on factor analysis. A more detailed description can be found in [9]. We present the results of an experiment in which the technique was applied to a set of 129 mammograms. The second approach is based on directional recursive median filtering10 (DRMF). We present the results of applying the technique to a set of 53 mammograms containing 26 stellate lesions. As the two methods describe different features of a stellate lesion they have been developed separately. However, the combination of the techniques provides a complete generic description of a stellate lesion; with a central mass and surrounding distorted pattern of linear structures. Using simplistic methods of combining the results from both techniques a sensitivity of 100% is obtained with zero false positives per image for lesions >16mm.
We extract linear structures from digitised mammograms by the application of a multi-scale oriented line detector; this provides a line-strength, orientation and scale at each pixel11.A generic representation of oriented line patterns is obtained by constructing observation vectors from the orientation values of pixels in a square window scanned over the image. Thus any pattern of lines with the same window size can be represented. Building statistical models of orientation data involves the calculation of mean angles. Ambiguous results are obtained if the orientations are represented as simple angles12. An alternative representation was therefore utilised. The orientation of each pixel was doubled and represented as a 2D unit vector. Thus, orientations can be properly summed and the vectors representing lines at right angles cancelled. Each oriented pattern window is represented by a single observation vector comprising the 2D unit vectors corresponding to the orientations of pixels within the window. The size of the observation vector is thus equal to twice the number of sampled pixels in the window. In our experiments the window size was set to 512 pixels square (21.5mm) which was approximately equal to the mean size of the lesions in the test/training set. If the orientations of the pixels within the window were to be statistically modelled the covariance matrix would contain approximately 3x1011 elements. We therefore adopted a more practical approach by sampling within the window on a square grid with step size 32 pixels.
We wish to model mammographic oriented patterns without making any assumptions about pattern structure, thus avoiding bias toward any particular pattern whilst encapsulating all possible patterns. We require the model to find the underlying structure via training, rather than assume one on an ad-hoc basis. Mammograms often contain only subtle evidence of abnormal systematic pattern structure, especially for patients with early signs of cancer. Factor analysis provides a technique for separating systematic variation from specific (random) variation13. Factor analysis can be summarised by considering an observation vector as being made up of a part which is peculiar to the observation (called specific or error) and a part which is a function of some fundamental variables of the population data. The factor model is given by (1):
xi = Lf +U + m (1)
S = LLT + Y (2)
Observation vectors xi are modelled by a linear combination L of factor scores f, which are sometimes called common factors as they describe a common property of several observations, a specific factor U corresponding to each of the observation variables, and the average observation over the population m. The coefficients of the linear combination of factor scores (L) are called factor loadings. Since it is impossible to distinguish between measurement errors and the specific factors, the vector U is simply termed the error. The factor model is trained by constructing the covariance matrix (S) of the population and solving (2) via maximum likelihood estimation13 for L and Y. The solution assumes f and U are normally distributed. The error covariance matrix (Y) is positive definite and diagonal. The common factor scores (fi) for an example observation (xi) can be obtained from the weighted least squares estimator given by.
fi = (LT Y-1 L)-1 LT Y-1 (x i - m) (3)
To measure the lesion detection performance of the technique, a set of 129 mammograms digitised to a resolution of 42microns/pixel was used. The set comprised a chronological sequence of 29 screening mammograms containing stellate lesions and 100 normal mammograms. For each stellate lesion a detailed annotation by expert radiologist was obtained in the form of a polygon delineating the extent of the central mass of the lesion. A tenth of the pixels within each annotation were randomly selected as the centres of oriented pattern windows to provide observation vectors for training the factor models. A covariance matrix of factor scores (Cg) and a mean vector of factor scores (mg) was constructed from the training set on a leave-one-mammogram-out basis. A factor model was built from the whole training set and used to obtain factor score vectors estimated from patterns randomly positioned in 30 normal mammograms. Thus, two unbiased covariance matrices (Cg, g=1,2) and mean estimated factor score vectors (mg, g=1,2) were obtained, representing the distribution of lesion (g=1) and normal (g=2) patterns respectively. The experiment proceeded on a leave-one-mammogram-out basis. The factor model built from the full training set of 29 mammograms was applied to the 100 normal mammograms. The factor models built on a leave-one-out basis were applied to their corresponding omitted mammograms. A factor model was applied to a mammogram by extracting observation vectors from oriented patterns centred at each pixel within the breast area on a 0.7mm grid. For each observation, a vector of 10 factor scores (fi) was estimated from (3). Classification of each point on the 0.7mm grid was performed by calculating an odds ratio of class probabilities. The mahalanobis distance (Dg) from each parameterised sample (fi) to each class mean (mg) was calculated at each grid point using the class covariance matrix Cg via (4). Assuming normally distributed data, two probability densities were calculated via (5), thus, allowing an odds-ratio (odds(x)) to be calculated from (6).
Dg = (fi - mg)T Cg-1 (fi - mg) (4)
Following the procedure outlined above an odds ratio image was obtained for each of the 129 mammograms from which regions classified as lesion could be extracted. The odds-ratio images were gaussian smoothed (sigma=1.1mm) before region extraction to reduce the effects of noise. Two criteria were used to investigate the performance of the technique. The first enables a comparison with other published techniques as it is the method most commonly used : the fraction of true positives and the number of false positives per image were determined whilst varying a threshold applied to the odds ratio images; this resulted in a set of Free Receiver Operating Characteristic (FROC) curves, each one with a minimum region size threshold, below which all detected regions and lesions were ignored. For each odds-ratio threshold a true positive detection was generated if a region overlapped the lesion annotation, otherwise a false positive was generated. This criterion is similar to those adopted by Brake and Karssemeijer14The second criterion relates to the motivation for abnormality detection, namely to automatically produce a prompt. Just as a radiologist will lose faith in an inaccurate system8, the size of the prompt, which relates to the localisation accuracy of the algorithm, will affect the ability of the system to aid the radiologist. Whilst the use of traditional detection FROC curves enable published methods to be compared, when viewed in isolation the traditional FROC curve presents results with a favourable bias. The false positive area must be taken into account. A large prompt ( in the extreme case the whole of the breast) with respect to a small lesion gives a biased view of algorithm accuracy as traditionally it counts as a single false positive. For this reason, the sensitivity of our technique for a single minimum region size was plotted against the false positive area divided by the area of a circular prompt with a fixed diameter. This effectively measures the number of prompts that will fit into a falsely detected region. Previous experiments designed to determine the efficacy of prompting used a circular prompt with a diameter of 20mm7,8, we therefore present results with the same prompt size.
Figure 1 : Example of applying our oriented pattern technique to a single mammogram chosen for display purposes. (a) original mammogram, (b) resulting odds ratio image.
Figure 2 : Detection (a) and Prompt Size (b) FROC curves resulting from the application of the oriented pattern algorithm to a set of 129 mammograms containing 29 stellate lesions.
Results from previous experiments designed to determine the lesion detection perfor mance required by automated systems8 suggest that a false to true positive ratio of <1.5 is required before an improvement in radiologist performance is achieved. At this rate, sensitivities of 77%, 89% and 93% are achieved for minimum region sizes of 8mm, 12mm and 16mm respectively (Figure 2a). These results demonstrate the suitability of employing our technique to produce prompts for stellate lesions. An operating point can be selected for a minimum region size of 12mm to produce 80% sensitivity with only 0.24 false positives per image. This corresponds to a false/true positive ratio of 1.07 for this test set. It must be noted, however, that even though the test set is reasonably large, the proportion of images containing abnormalities (22%) in the test set is not representative of the set of films viewed in a typical film reading session in a breast screening centre. Typically, around 10% of patients are recalled for further investigation15 and each patient has two or more films. As the detection of cancers with a central mass of approximately 1cm leads to an improvement in patient prognosis16 the minimum region size of 12mm diameter was used to produce the prompt size FROC curve (Figure 2b). At 100% sensitivity the equivalent of 1.1 prompts with diameter 20mm were obtained. An operating point can be selected to achieve 92.5% sensitivity with only 0.2 false prompts per image. This corresponds to a false prompt to true positive ratio of 0.89, which is well within the target ratio of 1.5. The results compare favorably with other methods currently reported in the literature5,6,7,17. Our technique achieves a lower false positive rate at similar sensitivity without first utilising noise equalisation or mammogram normalisation. However, an increase in specificity will be required before an improvement in radiologists' performance is expected8.
The Recursive Median Filter (RMF) is one of a class of filters known as sieves (closely related to morphological operators18) that recursively remove image peaks or troughs of less than a chosen size19. By applying sieves of increasing scale to an image, then subtracting the output images of adjacent sieve scales, it is possible to isolate image features of a specific size. In 1-D, for every position at which the RMF is centred, the output is the median grey-level within some local neighbourhood determined by the size of the filter. As the filter is scanned along the signal the updated values of pixels that have already been visited are used to determine the median, rather than using the original grey-levels, hence the filter is recursive. For 2-D images, a 1-D RMF can be applied at any chosen orientation by covering the image with lines at this orientation and ensuring that every pixel belongs only to one line20. Thus, Recursive Median Filtering can be applied at several orientations to provide a scale-orientation signature at each pixel (hence the term Directional RMF). The signature is a 2-D array with scale and orientation axes. The values contained in the array represent the change in grey-level at the pixel, resulting from applying a filter at the scale and orientation corresponding to the position in the array. The array is constructed by recursively filtering an image with fixed orientation median filters and updating the grey-level changes for each scale. The process then iterates through each orientation. The characteristic pattern of the scale-orientation signatures resulting from various structures can be used to discriminate between the different types of structures. For example, blob-like structures produce sparse signatures that are orientation independent with similar values at a range of scales. Linear structures produce radically different signatures: they typically show a distribution that peaks at the orientation of the linear structure and has maximum and minimum scale values related to the length and width of the structure.
A set of 53 mammograms containing 26 stellate lesions, was used to determine the ability of the DRMF signature to represent the central mass of stellate lesions. The mammograms were digitised with a spatial resolution of 24microns/pixel then reduced by a factor of 8 by Gaussian smoothing and sub-sampling. The DRMF signature arrays were arranged into 12 orientation columns and 11 scale rows. The scale values ranged from 1 to 64 pixels and followed a log1.5 scale. Each signature therefore contained 132 elements. A training population was constructed by selecting signatures centred within the central mass of each of the 26 lesions. An equivalent number of signatures were randomly extracted from the normal tissue in the strips across the mammograms containing the lesions. To provide a more compact representation the dimensionality of the signature was reduced by performing Principal Component Analysis (PCA) on a leave-one-mam mogram-out basis in the same way as the factor models were produced from the oriented patterns (Section 2.3). The PCA models were applied to the training set in a leave-one-mammogram-out fashion to produce a population of reduced dimension parametric vectors (bi). A single covariance matrix (C) was constructed from the population and two mean vectors (mg) representing the mean parametric description of a lesion signature (g=1) and the mean parametric description of a normal tissue signature (g=2). The DRMF PCA models were applied to a mammogram by extracting the signature at each pixel and transforming the signature into parametric space via the appropriate PCA model. The Mahalanobis distances from the resulting parametric vector to each group mean (mg, g=1,2) were calculated via the pooled covariance matrix (C) and used to calculate the group probability densities (p(bi)g, g=1,2) and finally an odds ratio using equations (4,5,6). We used a pooled covariance matrix (C) due to the near singularity of the covariance matrix representing the distribution of non-mass observations. Figure 3 provides an example of applying the DRMF PCA technique to the same image used to present the application of the factor model (section 2.4).
Figure 3 : (a) original (b) odds ratio image after an application of the DRMF PCA technique
Figure 4 : Detection (a) and Prompt Size (b) FROC curves resulting from the application of the DRMF PCA central mass detection algorithm to a set of 53 mammograms containing 26 stellate lesions.
At the false to true positive ratio of 1.5 the DRMF PCA technique achieved sensitivities of 0%, 24%, 68% for minimum regions sizes of 8mm, 12mm and 16mm respectively. An operating point can be selected to produce 83% sensitivity with 1.6 false positives per image for a minimum region size of 16mm. This corresponds to a false/true positive ratio of 3.3 which is approximately double the target ratio of 1.5. An improvement in specificity is therefore required . However, other published mass detection results show typical values of two21 and three22 false positives per image at a sensitivity of 80%. It can be seen from Figure 4a that zero sensitivity results when the threshold is raised to the point where no false positives are detected. This is caused by the similarity of the mass signatures to those describing normal blob-like structures. The prevalence of normal blob-like structures is related to the type and density of the mammogram3. Figure 4b demonstrates the tendency for the DRMF PCA technique to produce large false positive prompts, a fact which remains hidden using the traditional detection FROC curves. At a sensitivity of 80% the technique produces 2.8 false positives (Figure 4a) which translates to 4.2 false positive prompts with a prompt size of 20mm (Figure 4b). This is caused by the technique detecting all mass-like structures, some of which cover a large area. This contrasts with the oriented pattern technique in which abnormal patterns are more likely to be localised. The next section describes the combination of the two techniques.
The two generic representations and statistical models described in the previous sections model the separate parts of a stellate lesion. The DRMF PCA technique models the central mass and the oriented pattern factor analysis technique models the surrounding distorted pattern of linear structures. Thus, the combination of the two techniques should provide a complete generic description of stellate lesions. Both techniques resulted in the construction of odds ratio images. To combine the lesion evidence produced by the two methods the odds ratio images obtained for a given mammogram were smoothed (sigma=1mm) and then multiplied together.
Figure 5 : Detection (a) and Prompt Size (b) FROC curves resulting from the combined odds ratio images to a set of 53 mammograms containing 26 stellate lesions.
Figure 5a shows the detection FROC curves resulting from the combination of odds ratio images for a set of 53 mammograms containing 26 stellate lesions. Figure 5b shows the prompt size FROC curves for a minimum region/lesion size of 12mm diameter. The combination of the odds ratio images by multiplication of values does not lead to an overall accuracy improvement. Table 1 gives the specificity of each technique for a sen sitivity of 80%. The combined odds ratio images improve the accuracy from 2.8 using the DRMF PCA model to 1.6 false positives per image for lesions >=12mm. However, the specificity is reduced when compared to the oriented pattern technique (0.25 false positives per image). This is caused by the differing sensitivities of the two methods at the same threshold. In the case where one technique fails to detect a lesion, the combined results will also show this reduced sensitivity. Hence, for a given sensitivity the effect will be an increased number of false positives.
| 8mm | 12mm | 16mm | |
| Oriented pattern & factor analysis | 0.41 | 0.25 | 0.006 |
| DRMF PCA mass detection | 6.3 | 2.8 | 1.5 |
| Combination by multiplication | 2.4 | 1.6 | 0.35 |
Clearly a more sophisticated method of combining results is required. To provide an insight into the level of accuracy that can be achieved the odds ratio images obtained via the DRMF PCA model were used as mask images by selecting the corresponding threshold of a single operating point from the method's FROC curve. The mask images were then combined with the oriented pattern odds ratio images to produce the FROC curves given in Figure 6.
Figure 6 : Detection (a) and prompt size (b) FROC curves obtained using the oriented pattern odds ratio images masked by the DRMF mass detection odds ratio images after a threshold.
Figure 6a demonstrates that combining the results can improve the overall accuracy under certain conditions. For lesions >=16mm an improvement in sensitivity at zero false positives from 75% using the oriented pattern technique to 100% is achieved. For lesions >=8mm improvement from 39% to 43% occurs, however, a reduction in performance from 54% to 45% is obtained for lesions >=12mm. These results suggest that it may be possible for a more sophisticated method of combining the two techniques, that fully exploits their relative merits, to improve the overall accuracy. Figure 6b demonstrates an overall improvement in false positive area with this simplistic method of combining the results. An operating point can be selected at which a sensitivity of 90% is obtained with the equivalent area of 0.12 false positive prompts with diameter 20mm.
We have described two generic approaches which are complete and uncommitted. The techniques separately model the central mass of a lesion and the surrounding pattern (dis torted by the presence of the carcinoma) of linear structures. In combination, they provide a complete generic representation of a stellate lesion. The technique of modelling oriented line patterns via a generic pattern representation in conjunction with factor analysis has previously been shown to incorporate the more obvious features (concurrency, spread etc.) and the more complicated and less obvious systematic pattern variations via the the loadings of the factor model9. We have described an application of this technique to a set of 129 mammograms, 29 of which comprise a chronological screening sequence of mammograms containing a single stellate lesion and the remaining 100 extracted chronologically from a screening sequence of mammograms containing no abnormalities. The accuracy of this technique compares favourably with those current in the literature4,5,6,17. Algorithms capable of detecting lesions with an effective diameter of >=1 cm are likely to improve radiologists' performance provided the required level of accuracy is achieved8,16. We have shown that the oriented pattern method produces a specificity of 0.24 false positives per image at a sensitivity of 80% for lesions of diameter >=12 mm (for lesions of diameter >=16 mm the specificity improves to 0.006 false positives per image). At this operating point the ratio of false/true positives is 1.07 which falls well inside the target8 of 1.5. Additional results were presented in the form of prompt size FROC curves which provide a measure of the localisation accuracy by dividing the total false positive area by the area of a fixed sized circular prompt. Previous experiments performed to determine the target accuracy for prompting systems7 utilised a fixed prompt size of 20 mm, consequently FROC curves were presented for a prompt of this size. We have shown that 92.5% sensitivity can be achieved with the equivalent of 0.2 false prompts of diameter 20 mm. These results do not account for the 2-D separation of false positive regions, however, they do present less biased results than the traditional detection FROC curve; a large prompt with respect to a small lesion (for example the extreme of prompting the whole of the breast region) gives a biased view of algorithm accuracy since in the standard approach it counts as a single false positive. Dividing the false positive area by a prompt size provides a crude way of reducing the bias. We are currently developing methods to determine system accuracy in a more realistic manner. The technique of modelling the central mass of stellate lesions via Directional Recursive Median Filter (DRMF) signatures has been previously applied to a set of mammograms containing stellate lesions10. We have briefly described signature dimensionality reduc tion via Principal Component Analysis (PCA). The application of the DRMF PCA mass detection method to a set of 53 mammograms has been presented. The mammogram set contained chronological sequences of 26 mammograms containing a stellate lesion and 27 mammograms without an abnormality. We have shown that this technique is capable of a mass detection sensitivity of 83% with 1.6 false positives per image for lesions >=16 mm in diameter. The specificity of the technique is low in comparison to other published methods of mass detection21,22, however, the ratio of false/true positives is approximately double the target ratio of 1.5. The DRMF PCA mass detection results demonstrate the benefit of displaying system accuracy via a prompt size FROC curve. The regions falsely prompted by the technique were often large : a prompt size of 20 mm diameter resulted in 4.2 false prompts for the detection of lesions >=12 mm at 80% sensitivity. The cause of this result was the similarity of the lesion central masses with normal mass-like struc tures that were present at much larger scales. The direct combination of the two methods by multiplying their resulting odds ratio images did not lead to an increase in specificity as they separately modelled different types of image evidence. It is clear that a more sophisticated approach to evidence combination is required. To provide insight into the level of accuracy that might be achieved the DRMF PCA odds ratio images were used as mask images via a threshold chosen from the mass detection FROC curves. The combination of the mask images with the oriented pattern odds ratio images produced FROC curves that showed a slight improvement for lesions >=8mm and an improvement to 100% sensitivity for lesions >=16mm with zero false positives. However, the results for lesions >=12mm suffered a reduction in specificity. The methods of odds ratio combination that led to the results presented in the paper were simplistic and somewhat crude. We intend to investigate a more thorough Bayesian approach. In addition, a further improvement of overall accuracy may be expected by applying noise equalisation and grey-level normalisation to the original mammograms as existing techniques achieve improved performance in this way.
REFERENCES
[1] "Breast Cancer. How to help yourself", Macmillan breast cancer campaign, 1994
[2] Forrest A, Aitken R, "Mammography Screening for Breast Cancer", Annual Review in Medicine 41:pp117-132,1990
[3] Tabar L, Dean P, "The Mammographic Teaching Atlas", Georg Thieme Verlag, 1985.
[4] Astley a, Taylor C, Boggis C, Asbury D, Wilson M, "Cue Generation & Combination for Mammographic Screening", Chapter 15 of Visual Search 2, Eds: Brogan D, Gale A, Carr C, pub: Taylor & Francis, London 1993
[5] Kegelmeyer W, Pruneda J, Bourland P, Hills A, Riggs M, Nipper M, "Computer-aided mammographic screening for spiculated lesions", Radiology vol 191, pp331-337, 1994
[6] Karssemeijer N, "Detection of stellate distortions in mammograms using scale space operators", Proc.IPMI 95, pp335-346, Kluwer Academic Publishers, 1995
[7] Hutt I, Asley S, Taylor C, Boggis C, "Acceptable error rates for computer generated prompts in mammographs", (abstract) in Proc.Radiology UK 96 p164.
[8] Hutt I, "The Computer-aided Detection of Abnormalities in Digital Mammograms" Ph.D. Thesis, Manchester University U.K., 1996
[9] Parr T, Taylor C, Astley S, Boggis C, "Statistical Modelling of Oriented Line Patterns in Mammograms", Proc. SPIE Medical Imaging Conf. Newport Beach, Calif. U.S.A, Feb 1997
[10] Zwiggelaar R, Schumm J, Taylor C, "The Detection of Abnormal Masses in Mammograms", in Proc. Medical Image Understanding and Analysis `97, Oxford U.K., July 1997.
[11] Zwiggelaar R, Parr T, Taylor C, "Finding oriented line patterns in digital mammographic images", in Proc. of 7th BMVC Edinburgh 1996.
[12] Mardia K, "Statistics of Directional Data", 1st edition 1972, Academic Press series Probability and Mathematical Statistics.
[13] Anderson T, "An introduction to multivariate statistical analysis", 2nd edition, Wiley Series in probability and mathematical statistics, Jojn Wiley & Sons, 1984
[14] Brake G, Karssemeijer N, "Detection criteria for evaluation of computer aided diagnosis systems", 18th Int. IEEE EMBS Conf. pp 243, 1996
[15] Manchester Breast Screening Results for year ending March 1993, from North West Regional Training Centre,for Breast Screening, Withington Hospital, Manchester U.K.
[16] Dean P, "Overview of breast cancer screening", Excerpta Medica 1119, pp19-26. 1996
[17] Zhang M, Giger M, Vyborny C, Doi K, "Mammographic texture analysis for the detection of spiculated lesions", Excerpta Medica 1119, pp347-351, 1996
[18] Soille P, Breen E, Jones R, "Recursive implementation of erosions and dilations along discrete lines at arbitrary angles", IEEE Tans. on PAMI, 18(5) pp562-567, 1996.
[19] Bangham J, Ling P, Young R,"Multiscale recursive medians, scale-space and transformations with applications to image processing", IEEE Trans. on Image Processing, 5(6) pp1043-1048, 1996.
[20] Press W, Teukolsky s, Vetterling W, Flannery B, "Numerical Recipes in C: The Art of Scientific Computing", Cambridge University Press, 2nd edition, 1992.
[21] Petrick N, Chan H, Sahiner S, Helvie M, Goodsitt M, Adler D, "Computer-aided breast mass detection: false positive reduction using breast tissue composition", Excerpta Medica 1119, pp373-378, 1996.
[22] Zouras W, Giger M, Lu P, Wolverton D, Vyborny C, Doi K, "Investigation of a temporal subtraction scheme for computerized detection of breast masses in mammograms", Excerpta Medica 1119, pp411-415 1996.