Circular Features - Semantic Scholar
ARTICLE IN PRESS
Computers & Geosciences 30 (2004) 101–105
The detection of circular features in irregularly spaced data$ G.R.J. Coopera,*, D.R. Cowanb a
School of Geosciences, University of the Witwatersrand, School of Geosciences, Private Bag 3, Johannesburg, WITS 2050, South Africa b Cowan Geodata Services, 12 Edna Road, Dalkeith, Western Australia, Australia Received 3 February 2003; received in revised form 24 July 2003; accepted 5 August 2003
Abstract Geophysical potential field data are usually interpolated onto a regular grid before data enhancement and interpretation. Unfortunately the inherent smoothing in the gridding process can be sufficient to distort or even hide small-amplitude anomalies that are nevertheless of economic importance. Circular features may correspond to anomalies from Kimberlite pipes or meteorite impact craters, and are therefore of considerable interest. The Hough transform is a useful tool for the detection of circular features in gridded data, but its sensitivity to the choice of radius means that it performs poorly when applied to ungridded data. A modified version of the Hough transform which works well on ungridded data is described here, and demonstrated on gravity data from South Africa. r 2003 Published by Elsevier Ltd. Keywords: Hough transform; Gravity data; Aeromagnetics
1. Introduction Geophysical potential field data such as aeromagnetic and gravity data are collected by both government Geological Surveys and mineral exploration companies for a wide range of purposes including geological mapping and mineral exploration. When collected on the ground using hand-held equipment, measurement sites are often restricted to the locations of roads due to inaccessible terrain and impenetrable vegetation, making a regular measurement grid impossible. Even when measured by airborne platforms where these problems do not exist, the data are still not collected on a regular grid since a combination of the sampling time of the sensors and the economics of flying a survey result in data that are sampled much more densely along the flight lines than across them. It is important to have data $
Code available from server at http://www.iamg.org/ CGEditor/index.htm *Corresponding author. E-mail addresses: [email protected] (G.R.J. Cooper), [email protected] (D.R. Cowan). URL: http://www.wits.ac.za/science/geophysics/gc.htm. 0098-3004/$ - see front matter r 2003 Published by Elsevier Ltd. doi:10.1016/j.cageo.2003.08.006
that lie on a regular grid for imaging and Fourier transform-based filtering purpose, so irregularly spaced data are usually interpolated onto a regular grid before further processing takes place. There are many interpolation methods, although those using some form of weighted cubic spline function such as in minimumcurvature gridding (Briggs, 1974) are currently popular for aeromagnetic and gravity data. Unfortunately the interpolation process produces a surface which is smoother than the actual potential field, and can also introduce distortions, particularly in regions where measurements are sparse. Subtle, low amplitude anomalies (such as those associated with the presence of mineral sands) can be lost. The development of techniques to process irregularly spaced data is therefore important.
2. The Hough transform for circular features in gridded data The Hough transform (Wang and Howarth, 1990) is a frequently used tool for the detection of features on images such as lines and circles. It has had many
ARTICLE IN PRESS 102
G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
applications in fields such as computer vision. The form used to locate circular features in binary image data selects each pixel in turn as a possible image centre, and then sums all pixel values that lie at a fixed circular radius from it. The transform will have a large value (that depends on the circle circumference) at circle centres, and a small or zero value otherwise. Fig. 1 shows the result of using the Hough transform to detect circular features in a synthetic binary dataset consisting of two circles of different radii. The radius of the circle being sought must be specified when the transform is applied, and unfortunately the results of the transform are sensitive to the value used. Fig. 2 shows graphs of a profile through the Hough transform response of the circular features shown in Fig. 1, using several different search radii. A 1-pixel error in the radius (10%) produces a reduction in the transform response of 92% (the small response at a position of approximately 80 units on the profile is due to the fact that a circle centred on that location with the current radius would touch part of the larger circular feature shown). Searching for circles using many different radii is computer intensive. A further complication is that when real-valued (as opposed to binary) data are used, then the radius of a circular feature is no longer such a simple concept. When dealing with irregularly spaced geophysical data, it is highly unlikely that measurement points will lie exactly on both the centre and perimeter of any circular features present. Since geophysical data is also real valued, the combination of the two factors means that the standard Hough transform performs poorly. Fig. 3b shows the result of using The Hough transform to search for circular features (of known radii) in the synthetic aeromagnetic dataset shown in Fig. 3a. Data collection in aeromagnetic surveys is typically highly anisotropic, with the data sampling interval along the flight lines being much shorter than the distance between lines. The large line separation reduces the likelihood of a measurement point being located on the centre of a circular feature.
3. The Hough transform for circular features in irregularly spaced data A different approach must be taken to detect circular features in irregularly spaced data. Instead of assuming that a given datapoint lies at the centre of a circular feature it is assumed to lie upon its perimeter. The value of the current point and each individual point that lies at a distance 2R (+/ a given tolerance) is averaged, and the result is placed at the mid-point location of the two points. This procedure is repeated for each point in the dataset, forming a new dataset with values at the midpoints of the pairs of datapoints used. This method does not require binary data, and the measured (real)
Fig. 1. Detecting circular features in gridded data using Hough transform. (a) Binary dataset consisting of two circles of radii 10 and 30 units, (b) result of using circular Hough transform to find circles of radius 30 units and (c) result of using circular Hough transform to find circles of radius 10 units. Dotted line A-B shows position of the profile through Hough transform response plotted in Fig. 2.
values of the field can be used directly. Since the circular features must have an amplitude that is different than the background field (otherwise there would be no circular feature present), then clusters of points in the
ARTICLE IN PRESS G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
103
Fig. 2. Profiles through Hough transform response of circular features shown in Fig. 1(a) using different search radii. (a) Search radius 10 units, (b) search radius 11 units, (c) search radius 12 units, (d) search radius 13 units and (e) search radius 14 units. Centre of circular feature is located at position of peak in Fig. 2a.
Fig. 3. Application of Hough transform to a synthetic aeromagnetic dataset. (a) Synthetic aeromagnetic dataset that contains two circular features. Spacing between flight lines is much greater than data sampling interval along lines, (b) output of standard Hough transform. Possible circle centres are white. Hough transform output may only lie on positions at which aeromagnetic field was measured and (c) output of modified Hough transform. Possible circle centres are white. Circle centres may lie anywhere within survey boundaries.
processed dataset with a higher (or lower) value than the background will form around the circle centres. A final image output is then produced by discretising the
processed dataset, i.e. splitting the survey area into a grid of cells of equal size and summing all the responses that lie within each cell. This image will have bright or
ARTICLE IN PRESS 104
G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
dark pixels (depending on whether the anomaly amplitude is smaller or greater than the background) at the circle centres. Fig. 4 demonstrates the principle of the method, and the result of its application to the synthetic aeromagnetic dataset in Fig. 3a can be seen in Fig. 3c. When Fig. 3c is compared with the output of the standard Hough transform shown in Fig. 3b the improvement is obvious. The circular features shown in Fig. 3a are simplified and unrealistic representations of actual geophysical
anomalies. In particular, the synthetic features have a precisely defined radius. Whereas it is possible to preprocess the data using some kind of edge detection (or other high-pass filtering method) to attempt to sharpen the edges of any circular features prior to the application of the Hough transform, a better method in practice is to modify the filter slightly. Given an anomaly which is a smoothly varying function of position, a circular feature will have values that are symmetric about its centre. Hence if some measure of the similarity of any two points (which by definition in the method described previously must lie an equal distance from the circle centre) is applied when the values of two possible circle perimeter points are being averaged (see Fig. 4) then the method will be able to detect more realistic anomalies. An exponential function was chosen to compare the similarity of the datapoints, though other functions could have been used. The irregular Hough transform output H12 (located at the mid-point of the two data point locations) from the comparison of a single pair of datapoints with values f1 and f2 is then H12 ¼
ðf1 þ f2 Þ kjf1 f2 j e ; 2
ð1Þ
where k is a sensitivity parameter. The larger the value of k; the more sensitive to the similarity of the two data points the algorithm will be. However, flat areas where data values are similar to one another will give a strong response using this weighting scheme, so the presence of a circular feature must be checked by examining the data near the centre to see if the values there are different from those further away. This approach can be effective, though computationally expensive. The modified transform function then becomes H12 ¼
Fig. 4. Locating circular features in irregularly spaced data. (a) Current point P is compared with all points a distance 2R+/ tolerance away. In each case a new point with average value of each pair of points is placed at midpoint of pair, (b) as procedure is repeated for other data points, density of new points with high or low values (compared to local data average) is built up around circle centre location and (c) after all original data points have been processed, new points are discretised. Value of pixels in new grid is sum of all values within it.
ðf1 þ f2 Þ kjf1 f2 j kjf1 fc j e e 2
ð2Þ
where fc is the value of the observed datapoint located closest to the (possible) circle centre. Fig. 5a shows a gravity dataset from South Africa of variable sampling density. A prominent circular feature known as the Trompsburg high (Buchmann, 1960) is present in the west of the image, and another, more diffuse circular anomaly of much smaller amplitude is barely visible to its northeast. Because the anomalies possess circular symmetry the modified transform output is not critically dependent on the search radius used, as long as this is smaller than the size of the circular feature.
4. Conclusions The circular form of the Hough transform has been modified to work with irregularly spaced data. This is
ARTICLE IN PRESS G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
105
Fig. 5. Application of modified Hough transform to gravity data from South Africa. (a) Gravity data. Image covers an area of approximately 450 500 km in size. Light colours indicate large values of gravity field and dark colours indicate smaller values. (b) Modified Hough transform output of data from 5(a). Light colours indicate most likely locations of centre of circular features.
useful since it allows subtle features to be detected that otherwise might be lost or distorted by the gridding process.
Acknowledgements The Council for Geoscience, Pretoria, is thanked for permission to use the gravity data shown in Fig. 5.
References Briggs, I.C., 1974. Machine contouring using minimum curvature. Geophysics 39, 39–48. Buchmann, J.P., 1960. Exploration of a geophysical anomaly at Trompsburg, Orange Free State, South Africa. Transactions of the Geological Society of South Africa 63, 1–10. Wang, J., Howarth, P.J., 1990. Use of the Hough transform in automated lineament detection. IEEE Transactions on Geoscience and Remote Sensing 28 (4), 561–566.
Computers & Geosciences 30 (2004) 101–105
The detection of circular features in irregularly spaced data$ G.R.J. Coopera,*, D.R. Cowanb a
School of Geosciences, University of the Witwatersrand, School of Geosciences, Private Bag 3, Johannesburg, WITS 2050, South Africa b Cowan Geodata Services, 12 Edna Road, Dalkeith, Western Australia, Australia Received 3 February 2003; received in revised form 24 July 2003; accepted 5 August 2003
Abstract Geophysical potential field data are usually interpolated onto a regular grid before data enhancement and interpretation. Unfortunately the inherent smoothing in the gridding process can be sufficient to distort or even hide small-amplitude anomalies that are nevertheless of economic importance. Circular features may correspond to anomalies from Kimberlite pipes or meteorite impact craters, and are therefore of considerable interest. The Hough transform is a useful tool for the detection of circular features in gridded data, but its sensitivity to the choice of radius means that it performs poorly when applied to ungridded data. A modified version of the Hough transform which works well on ungridded data is described here, and demonstrated on gravity data from South Africa. r 2003 Published by Elsevier Ltd. Keywords: Hough transform; Gravity data; Aeromagnetics
1. Introduction Geophysical potential field data such as aeromagnetic and gravity data are collected by both government Geological Surveys and mineral exploration companies for a wide range of purposes including geological mapping and mineral exploration. When collected on the ground using hand-held equipment, measurement sites are often restricted to the locations of roads due to inaccessible terrain and impenetrable vegetation, making a regular measurement grid impossible. Even when measured by airborne platforms where these problems do not exist, the data are still not collected on a regular grid since a combination of the sampling time of the sensors and the economics of flying a survey result in data that are sampled much more densely along the flight lines than across them. It is important to have data $
Code available from server at http://www.iamg.org/ CGEditor/index.htm *Corresponding author. E-mail addresses: [email protected] (G.R.J. Cooper), [email protected] (D.R. Cowan). URL: http://www.wits.ac.za/science/geophysics/gc.htm. 0098-3004/$ - see front matter r 2003 Published by Elsevier Ltd. doi:10.1016/j.cageo.2003.08.006
that lie on a regular grid for imaging and Fourier transform-based filtering purpose, so irregularly spaced data are usually interpolated onto a regular grid before further processing takes place. There are many interpolation methods, although those using some form of weighted cubic spline function such as in minimumcurvature gridding (Briggs, 1974) are currently popular for aeromagnetic and gravity data. Unfortunately the interpolation process produces a surface which is smoother than the actual potential field, and can also introduce distortions, particularly in regions where measurements are sparse. Subtle, low amplitude anomalies (such as those associated with the presence of mineral sands) can be lost. The development of techniques to process irregularly spaced data is therefore important.
2. The Hough transform for circular features in gridded data The Hough transform (Wang and Howarth, 1990) is a frequently used tool for the detection of features on images such as lines and circles. It has had many
ARTICLE IN PRESS 102
G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
applications in fields such as computer vision. The form used to locate circular features in binary image data selects each pixel in turn as a possible image centre, and then sums all pixel values that lie at a fixed circular radius from it. The transform will have a large value (that depends on the circle circumference) at circle centres, and a small or zero value otherwise. Fig. 1 shows the result of using the Hough transform to detect circular features in a synthetic binary dataset consisting of two circles of different radii. The radius of the circle being sought must be specified when the transform is applied, and unfortunately the results of the transform are sensitive to the value used. Fig. 2 shows graphs of a profile through the Hough transform response of the circular features shown in Fig. 1, using several different search radii. A 1-pixel error in the radius (10%) produces a reduction in the transform response of 92% (the small response at a position of approximately 80 units on the profile is due to the fact that a circle centred on that location with the current radius would touch part of the larger circular feature shown). Searching for circles using many different radii is computer intensive. A further complication is that when real-valued (as opposed to binary) data are used, then the radius of a circular feature is no longer such a simple concept. When dealing with irregularly spaced geophysical data, it is highly unlikely that measurement points will lie exactly on both the centre and perimeter of any circular features present. Since geophysical data is also real valued, the combination of the two factors means that the standard Hough transform performs poorly. Fig. 3b shows the result of using The Hough transform to search for circular features (of known radii) in the synthetic aeromagnetic dataset shown in Fig. 3a. Data collection in aeromagnetic surveys is typically highly anisotropic, with the data sampling interval along the flight lines being much shorter than the distance between lines. The large line separation reduces the likelihood of a measurement point being located on the centre of a circular feature.
3. The Hough transform for circular features in irregularly spaced data A different approach must be taken to detect circular features in irregularly spaced data. Instead of assuming that a given datapoint lies at the centre of a circular feature it is assumed to lie upon its perimeter. The value of the current point and each individual point that lies at a distance 2R (+/ a given tolerance) is averaged, and the result is placed at the mid-point location of the two points. This procedure is repeated for each point in the dataset, forming a new dataset with values at the midpoints of the pairs of datapoints used. This method does not require binary data, and the measured (real)
Fig. 1. Detecting circular features in gridded data using Hough transform. (a) Binary dataset consisting of two circles of radii 10 and 30 units, (b) result of using circular Hough transform to find circles of radius 30 units and (c) result of using circular Hough transform to find circles of radius 10 units. Dotted line A-B shows position of the profile through Hough transform response plotted in Fig. 2.
values of the field can be used directly. Since the circular features must have an amplitude that is different than the background field (otherwise there would be no circular feature present), then clusters of points in the
ARTICLE IN PRESS G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
103
Fig. 2. Profiles through Hough transform response of circular features shown in Fig. 1(a) using different search radii. (a) Search radius 10 units, (b) search radius 11 units, (c) search radius 12 units, (d) search radius 13 units and (e) search radius 14 units. Centre of circular feature is located at position of peak in Fig. 2a.
Fig. 3. Application of Hough transform to a synthetic aeromagnetic dataset. (a) Synthetic aeromagnetic dataset that contains two circular features. Spacing between flight lines is much greater than data sampling interval along lines, (b) output of standard Hough transform. Possible circle centres are white. Hough transform output may only lie on positions at which aeromagnetic field was measured and (c) output of modified Hough transform. Possible circle centres are white. Circle centres may lie anywhere within survey boundaries.
processed dataset with a higher (or lower) value than the background will form around the circle centres. A final image output is then produced by discretising the
processed dataset, i.e. splitting the survey area into a grid of cells of equal size and summing all the responses that lie within each cell. This image will have bright or
ARTICLE IN PRESS 104
G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
dark pixels (depending on whether the anomaly amplitude is smaller or greater than the background) at the circle centres. Fig. 4 demonstrates the principle of the method, and the result of its application to the synthetic aeromagnetic dataset in Fig. 3a can be seen in Fig. 3c. When Fig. 3c is compared with the output of the standard Hough transform shown in Fig. 3b the improvement is obvious. The circular features shown in Fig. 3a are simplified and unrealistic representations of actual geophysical
anomalies. In particular, the synthetic features have a precisely defined radius. Whereas it is possible to preprocess the data using some kind of edge detection (or other high-pass filtering method) to attempt to sharpen the edges of any circular features prior to the application of the Hough transform, a better method in practice is to modify the filter slightly. Given an anomaly which is a smoothly varying function of position, a circular feature will have values that are symmetric about its centre. Hence if some measure of the similarity of any two points (which by definition in the method described previously must lie an equal distance from the circle centre) is applied when the values of two possible circle perimeter points are being averaged (see Fig. 4) then the method will be able to detect more realistic anomalies. An exponential function was chosen to compare the similarity of the datapoints, though other functions could have been used. The irregular Hough transform output H12 (located at the mid-point of the two data point locations) from the comparison of a single pair of datapoints with values f1 and f2 is then H12 ¼
ðf1 þ f2 Þ kjf1 f2 j e ; 2
ð1Þ
where k is a sensitivity parameter. The larger the value of k; the more sensitive to the similarity of the two data points the algorithm will be. However, flat areas where data values are similar to one another will give a strong response using this weighting scheme, so the presence of a circular feature must be checked by examining the data near the centre to see if the values there are different from those further away. This approach can be effective, though computationally expensive. The modified transform function then becomes H12 ¼
Fig. 4. Locating circular features in irregularly spaced data. (a) Current point P is compared with all points a distance 2R+/ tolerance away. In each case a new point with average value of each pair of points is placed at midpoint of pair, (b) as procedure is repeated for other data points, density of new points with high or low values (compared to local data average) is built up around circle centre location and (c) after all original data points have been processed, new points are discretised. Value of pixels in new grid is sum of all values within it.
ðf1 þ f2 Þ kjf1 f2 j kjf1 fc j e e 2
ð2Þ
where fc is the value of the observed datapoint located closest to the (possible) circle centre. Fig. 5a shows a gravity dataset from South Africa of variable sampling density. A prominent circular feature known as the Trompsburg high (Buchmann, 1960) is present in the west of the image, and another, more diffuse circular anomaly of much smaller amplitude is barely visible to its northeast. Because the anomalies possess circular symmetry the modified transform output is not critically dependent on the search radius used, as long as this is smaller than the size of the circular feature.
4. Conclusions The circular form of the Hough transform has been modified to work with irregularly spaced data. This is
ARTICLE IN PRESS G.R.J. Cooper, D.R. Cowan / Computers & Geosciences 30 (2004) 101–105
105
Fig. 5. Application of modified Hough transform to gravity data from South Africa. (a) Gravity data. Image covers an area of approximately 450 500 km in size. Light colours indicate large values of gravity field and dark colours indicate smaller values. (b) Modified Hough transform output of data from 5(a). Light colours indicate most likely locations of centre of circular features.
useful since it allows subtle features to be detected that otherwise might be lost or distorted by the gridding process.
Acknowledgements The Council for Geoscience, Pretoria, is thanked for permission to use the gravity data shown in Fig. 5.
References Briggs, I.C., 1974. Machine contouring using minimum curvature. Geophysics 39, 39–48. Buchmann, J.P., 1960. Exploration of a geophysical anomaly at Trompsburg, Orange Free State, South Africa. Transactions of the Geological Society of South Africa 63, 1–10. Wang, J., Howarth, P.J., 1990. Use of the Hough transform in automated lineament detection. IEEE Transactions on Geoscience and Remote Sensing 28 (4), 561–566.
