SINGULARITY ANALYSIS FOR IMAGE PROCESSING AND ANOMALY ENHANCEMENT. Qiuming CHENG

SINGULARITY ANALYSIS FOR IMAGE PROCESSING AND ANOMALY ENHANCEMENT Qiuming CHENG Department of Earth and Atmospheric Science, Department of Geography, York University, Toronto, 4700 Keele Street, Ont. M3J 1P3, Canada, e-mail: qiuming@yorku.ca ABSTRACT: Local structural property of an image is often considered in image enhancement with various high/low pass filters. Spatial association indexes (autocorrelation, covariance, and variogram) have been commonly used to characterize the local structure of surfaces for data interpolation in kriging. Singularity is another index measuring the scaling invariant property of measures from a multifractal point view. Comparing with spatial association indexes, the singularity has not yet been generally known by the community of image processing and geostatistics. Spatial association and scaling invariant are two different aspects of local structure of surfaces. Both must be taken into account in data interpolation and image enhancement. Recent study of multifractal modeling has shown that the local singularity exponent involved in multifractal modeling can quantify the local scaling invariant property characterizing the concave/convex properties of the neighborhood values. In dealing with geochemical maps these properties may correspond to depletion and enrichment. A method proposed by the author incorporating both the singularity and spatial association has been used for image enhancement and data interpolation. A case study of the geochemical maps created from concentration values of trace elements As, Pb, Zn, and Cu in lake sediment samples from the southwestern Nova Scotia, Canada, was used to demonstrate the application of the method in geochemical anomaly enhancement. It has shown that the areas with significant singularity (enriched element concentration) are related either to gold mineralization and the occurrences of known deposits or to the local structures such as granitoid phase changes and intersections of linear structures. 32% of the known mineral deposits and occurrences in the area fall in those areas with enrichments of all four elements As, Cu, Pb and Zn and 97% in the areas with at least one element enrichment. 1 INTRODUCTION Singularity involved in the multifractal modeling for measuring the scaling invariant property of multifractal measures can be considered as an index for characterizing the local structural property of patterns on an image such as local enriched or depleted anomalies of trace elements shown on geochemical maps. Quantification of these local properties is often essential for image processing and anomaly enhancement. In this paper, the concept of singularity will be reviewed in the context of multifractal modeling. Then its physical meaning and application will be discussed from a point of view of both general image processing and geochemical anomaly analysis. Finally, a case study of anomaly analysis for gold deposit related multi-elements (Au, Cu, Pb, and Zn) concentration values from 1948 lake sediment samples in the southwestern Nova Scotia, Canada, has been used to demonstrate the application of the method. 2 SINGULARITY The singularity in the multifractal context characterizes how the statistical behavior varies as

measuring scale changes. Assume the mean concentration value Ž(r) calculated from the neighborhood values within a window of linear size r. For a multifractal measure or geocemical map showing multifractal property, the values of Ž(r) calculated for various window sizes r follow power-low relationships with r, Ž(r) r - E, where stands for proportional to, E = 2 for 2-D and E = 1 for 1-D problems, respectively. The following discussion will be based on 2-D problem (E = 2) only but the similar discussions can be applied to any 1-D situations. The power-low relationships hold in certain ranges of r. In some locations Ž(r) is independent of the size r, implying (x) = 2. In other locations Ž(r) may, however, depend on r and (x) 2. The former case is considered as nonsingular but the latter as singular locations. Further we differentiate the positive singularity (x) < 2 from the negative singularity (x) > 2. The mean values Ž(r) calculated from the positive singular areas increase, whereas Ž(r) from the negative singular areas decrease as reducing the window size. The index (x) can be estimated from the values Ž(r) calculated at different sizes r by means of least square on log-log paper. The singularity index has the following properties (Cheng, 1999): x) = 2, iff Ž(r) = constant, independent of vicinity size r; (x) iff Ž(r) r is an decreasing function of r which implies the convex property of Z(x) around the location x; and (x) iff Ž(r) r is an increasing function of r which indicates the concave property of Z(X) around the location x. 3 SINGULARITY CALCULATED ON GEOCHEMICAL MAPS Applying the concept of singularity to analysis a geochemical map, the positive singularity ( < 2) usually corresponds to enriched or elevated element concentration values, whereas the negative singularity to depleted values. Therefore, the estimation of singularities from a geochemcial map can be applied to characterize the patterns with singular element concentration values which might provide useful information for interpreting geochemical anomalies related to mineralization or local structure activities. From an application point of view, the following two methods can be used to estimate the singularity. 3.1 Windows Method To estimate the local singularity from a geochemical map, a window-based method can be used as follows: For any given location on the map, it involves defining a set of sliding windows (square, circle and rectangle) with seccessive window sizes, r min = r 1 < r 2 < < r n = r max. For each window size, calculate the average concentration value Ž(r i ). These values Ž(r i ) (i = 1,, n) will show a linear trend with r i on a log-log paper. The slope estimated from the linear relationship can be considered as the estimation of 2. Similar treatments with the sliding windows to all locations of the geochemical map can create a singularity distribution map. The uncertainty related to the estimation of the singularity index can be also recorded. Only will the singularities with small uncertainty be further interpreted. The scale range [r min, r max ] can be determined by observing scaling properties of the power-law functions and the consideration of the scales of the local structures of interest. The left bound r min is often limited by the resolution of the data. Different ranges of [r min, r max ] determined ensuring distinct power-law functions may provide different singularities, for example, small-scale singularity may reflect local anomalies associated with mineralization, whereas large-scale singularity represents regional background variability. 3.2 Contour Method The above discussion shows that the windows method involves regular windows in estimation of singularity. The fix shaped windows may not be suitable for dealing with heterogeneities of

geochemical anomalies. As an alternative, a contour method can be used to estimate the singularities at some locations with large negative or positive singularities on the geochemical map. The areas with significant singularity are of general interest in mineral exploration. The contour method can be implemented as follows: To estimate the singularity at a location within a set of successive closed contours, one can calculate the average values Ž(r) within each contour. The linear size of the contour can be taken as square root of the area of the contour ( A). The power-law function between values Ž(r) and r will be equivalent to Ž(r) A ½ -1. The singularity can then be estimated by least square applied to log Ž(r) against log A. The singularity obtained in this way will have no accurate location on the map rather than associated with the set of contours used for the calculation. The contour method is only applicable to some special locations where similarly shaped contours can be used to define Ž(r). Large contours containing several small closed contours may show a different power-law relationship from which the large-scale singularity may be calculated. The large-scale and small-scale singularities may provide multi-scale texture information for anomaly assessment. 4 DISTRIBUTION OF SINGULARITY INDEX The singularity index usually has finite values around 2. For a conservative multifractal measure, the dimension of the set with = 2 is close to 2 (box-counting dimension) which means that the areas with nonsingular values occupy most part of the map. The dimensions of the other areas with 2 are given by the fractal spectrum function f( ) < 2. 5 SINGULARITY ANALYSIS FOR IDENTIFICATION OF GOLD DEPOSIT ASSOCIATED GEOCHEMICAL ANOMALIES Singularity analysis has been applied to the geochemical maps created from the element concentration values of 1948 lake sediment samples in the southwestern Nova Scotia, Canada. The geology of the study area is illustrated in Fig. 1. The study area ( 4000 km 2 ) is mainly underlain Cambro-Ordovicien low-middle grade metamorphosed sedimentary rocks and Devonian granitoid rocks. The South Mountain Bathlith (SMB) is a complex of multi-phase granites covering nearly one-third of the entire study area. A number of Au, U, W, and Sn deposits have been found in the area. About 45 Au mineral deposits are shown as dots in Fig. 1. More detailed discussion of the geology and geological controlling features on the spatial distribution of Au deposits can be found in Xu & Cheng (2000, 2001). The values of As, Cu, Pb, and Zn from the 1948 lake sediment samples (shown in Fig. 2) have been mapped both by the ordinary kriging and by the multifractal data interpolation method developed by Cheng (1999, 2000, 2001). Fig. 3 shows the result from As values by means of ordinary kriging with spherical model with a search distance 8 km and maximum interpolation point 16. Fig. 4 illustrates the distribution of -values estimated by means of windows method (square window) from Fig. 3 with a maximum size r max = 15km (as half-side of the square) or 30km (as the size of the square window). It can be seen that patterns with < 2 are mainly distributed either in the south of SMB as linear patterns with NW-SE orientation or aggregated around the contacts of SMB, especially in those places where faults or transition zones of difference granitoid phases exist. Some of the clusters with low -values show strong spatial correlation with the spatial locations of Au deposits. This should not be surprised since low -value may indicate the area with the enrichment of geochemical values that might due to mineralization in this study area. The values of the correlation coefficients calculated from the linear fitting between the values log Ž(r) and log r for r = 2,, 15 km are ranging from 0.97 to 1, implying significant linear relationships exist between log Ž(r) and log r for all the locations. Similarily,

Figure 1. Simplified geology in southwestern Nova Scotia, Canada. Figure 2. Locations of 1948 lake sediment geochemical samples.

the singularity maps were obtained for Cu, Pb and Zn maps using windows method with similar window parameters. The results are shown in Figs. 5 to 8, respectively. Figs. 5 and 7 illustrate the results obtained by kriging and Figs. 6 and 8 show the singularity values estimated for Cu and Pb, respectively. The patterns with on Figs. 4, 6, and 8 clearly highlight not only the linear geochemical anomalies in the south of SMB but also the areas within and around the SMB with faults or transition linear features of granitoid phase changes. To illustrate the spatial associations between the known deposit locations and the patterns of singularities calculated for As, Cu, Pb and Zn, two types of statistics have been conducted. First, a GIS analysis was carried to append the singularity values to the locations of deposits so that the deposits can be plotted as scatter plots in Fig. 9. The red dots located in the low left quadrants of all three charts represent those 32% deposits showing enrichments of all four elements As, Cu, Pb and Zn. Only are 7% deposits located in those areas with > 2 on all four singularity maps, in other words, 93% of the deposits show at least one element enrichment. The weights of evidence method (Bonham-Carter, 1994) has also been applied to calculate the spatial correlation between the deposit locations and the patterns of -values. Table 1 shows the results obtained between the locations of 113 mineral deposits and occurrences and the singularity values calculated for Cu. The contrast (C) and the student value (C/S(C)) calculated for the ascending cumulative classes demonstrate that the locations of the 113 points are statistically significantly associated with the patterns with < 2, for instance, the areas with < 2 on the Cu geochemical map accounts less than 50% of the total area but contain 66 out of 107 points (6 points with in missing singularity values). Calculated C = 0.6307 and C/S(C) = 3.16. Similar conclusions can be reached from the other results obtained for As, Pb and Zn by means of weights of evidence method. Kriging (As) Figure 3. Kriging map of As. Detailed parameters can be found in the text. Black outlines represent the granitoid complex (SMB). Dots represent gold mineral deposits.

-value (As) Red: < < 2 Blue: >2 White: =2 Figure 4. Estimated singularity values for As. Kring (Cu) Figure 5. Kriging map for Cu.

(Cu) Figure 6. Estimated -values for Cu. Kriging (Pb)) Figure 7. Kriging map for Pb.

(Pb)) Figure 8. Estimated -values for Pb Lower left area with enrichments of all four elements (Cu, Pb As, Zn) As, Zn) contains 32% of deposits Figure 9. Chart showing locations of mineral deposits and occurrences in singularity domains. Crosses represent no singularity regions. Red dots represent deposits falling within low left quadrants on all three charts.

Table 1 Results obtained by means of weights of evidence method to singularity values calculated for Cu and the locations of mineral 113 mineral deposits and occurrences. Singularity Area #points W+ S(W+) W- S(W-) CONTRAST S(C) C/S(C) Missing value 1355.5000 6 1.468-1.796 275.0000 5 1.2321 0.4513-0.0341 0.0993 1.2662 0.4621 2.7400 1.796-1.831 453.5000 8 1.2013 0.3567-0.0550 0.1008 1.2563 0.3707 3.3892 1.831-1.867 751.7500 9 0.8079 0.3353-0.0496 0.1013 0.8576 0.3503 2.4481 1.867-1.902 1297.7500 10 0.3630 0.3175-0.0309 0.1018 0.3939 0.3334 1.1815 1.902-1.938 2352.5000 20 0.4621 0.2246-0.0816 0.1075 0.5437 0.2490 2.1841 1.938-1.973 4450.2500 30 0.2283 0.1832-0.0765 0.1142 0.3048 0.2159 1.4119 1.973-2.000 9206.5000 66 0.2902 0.1235-0.3404 0.1565 0.6307 0.1994 3.1634 2.000-2.044 14318.5000 85 0.1004 0.1088-0.3140 0.2136 0.4144 0.2397 1.7286 2.044-2.08 17602.0000 98 0.0358 0.1013-0.3245 0.3340 0.3604 0.3490 1.0326 2.08-2.115 18996.5000 103 0.0092 0.0988-0.2121 0.5011 0.2213 0.5107 0.4334 2.115-2.151 19614.0000 107 2.151-2.186 19842.0000 107 2.186-2.222 19901.5000 107 2.222-2.935 19916.0000 107 6 CONCLUSIONS The singularity index measures the local scaling properties of geochemical maps. The areas with positive singularity ( < 2) may correspond to the areas where the element concentration values are elevated due to mineralization or other local geological processes, whereas the areas with negative singularity ( > 2) to the regions with depleted element concentration values. The areas with no singularity normally dominant the geochemical map with background concentration values. The case study conducted in the paper from the southwestern Nova Scotia has demonstrated that the singularity analysis introduced in the paper may provide useful information for characterizing local anomalies caused by mineralization and other local geological features. REFERENCES Bonham-carter, G. F., 1994, Geographic Information System for Geosciences: Modelling with GIS. Pergamon Press, Oxford. Cheng, Q., 1999. Multifractal interpolation: in S.J. Lippard, A. Naess and R. Sinding-Larsen (eds.) Proceedings of the Firth Annual Conference of the International Association for Mathematical Geology, Trondheim, Norway, v. 1, 245-250. Cheng, Q., 2000. Interpolation by means of multiftractal, kriging and moving average techniques, in the Proceedings of GAC/MAC meeting GeoCanada2000, May 29 to June, 2, 2000, Calgary. http://www.gisworld.org/gac-gis/geo2000.htm Cheng, Q., 2001. Multifractal and geostatistical methods for exploration geochemical anomaly texture and singularity Analysis: Journal of Earth Science, v. 26, no. 2, p. 161 166 (in Chinese with English abstract). Xu, Y., and Cheng, Q., 2000. Geochemical and geophysical data processing aided by multiftractal-spectrum filters for GIS-based mineral exploration, Journal of China University of Geosciences, v. 11., no. 2., p. 128-130. Xu, Y., and Cheng, Q., 2001. A multifractal filter technique for geochemical data analysis from Nova Scotia, Canada, Journal of Geochemistry: Exploration, Environment and analysis (in press).