Formation of a near Optimal Solution for Raster Region having Min/Max Cell Value

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1 th International Conference on Frontiers of Information Technology Formation of a near Optimal Solution for Raster Region having Min/Max Cell Value Muhammad Waqas Raja Institute of Geographical Information Systems, National University of Science and Technology, Islamabad, Pakistan waqas09@igis.nust.edu.pk Rizwan Bulbul Institute of Geographical Information Systems, National University of Science and Technology, Islamabad, Pakistan bulbul@igis.nust.edu.pk Abstract Region selection problems from raster images in Geographic Information Systems (GIS) involve lots of computational complexity for which designing of the optimized algorithms is required. A region with maximum sum of an attribute value is useful in different decision making problems, for which no efficient solution has yet been found. For selecting a region of such interest, there are lots of unique regions in both 4 and 8-Adjacency region rules. In case of 4-Adjacency rule, for selecting a 10-cell region, more than 288 thousand unique regions can be found among one will be a region with maximum sum value. The scenario suggests that the problem falls in the class of NP-Hard decision problems, so the problem is complex in nature as we deal with the computation terms of the algorithm to be designed in polynomial time. This research contains a new dynamic programming approach for the problem, whose focus is on the efficiency for finding approximate solutions. Experiments have suggested that with the data input of images having higher degree of spatial autocorrelation, the algorithm is quite useful and practical in nature, as it was tested with real life data and has extracted maximum value regions with thousands of cells in a sufficient time. Index Terms Region Selection, NP-Complete Decision Problems, Spatial Autocorrelation, Decision Support Systems I. INTRODUCTION Geographical Information Systems (GIS) primarily deals with two types of data; they are called vector and raster data. Vector data analysis is usually simpler in terms of computation complexity. While raster data involves large scale data sets and problems most of the times are multidimensional, multi-directional and discrete in nature, for which optimal solutions are required. Different cell function algorithms with optimal solutions are already adopted in many Raster data Analysis software s in GIS and Remote Sensing (RS). Such cell functions for raster cartography modeling are Local, Focal, Zonal, Global and Application functions. An exact solution for a similar cell function having a single-attribute raster layer for a region containing a Maximum or Minimum total cell sum value for connected set of user specified cells is yet to be established in the standard functionality of GIS software s. All cells are said to be in a Region, if and only if they are adjacent and connected to each other. In simple words if there is a path between every pair of cells, then these cells will contain a region. This pertains to an optimization problem because raster data are usually constituted of millions of pixels (cells) and for max/min region selection, we need to inspect every single cell and possible number of connected set of specified cells combinations will remain unknown. In computational complexity theory, this problem falls in NP- Complete class of decision problems whose solution can be solved in polynomial time. This research will produce an approximate solution to the NP-Complete problem which will be tested on real data and comparisons with other algorithms for same problem will be conducted on same data sets. In a raster image having each cell containing a brightness value, a region for a specified number of cells can be extracted. Applying 4-Adjacency rule, the possible number of unique combinations in that case can be formulized as: : For all k>=2, where k is the specified number of cells to be contained in a region. For region size of 50 cells, more than 32 million unique combinations can be found among which one will be of prime importance, i.e. Maximum value region. So a rather simpler, yet computationally complex problem lies in the field of RS, GIS and Image processing for which an efficient solution needs to be implemented. Raster images have spatial autocorrelation; closer cells are usually correlated, so if there is a cloud patch or if there is snowing area, it is certain that their cells are most probably connected and adjacent to each other. Keeping spatial autocorrelation in mind a dynamic programming approach is developed using Stack data structure with In-place Online Insertion sort technique, which has proven to be quite optimal as compared with other approaches. Real life data was used with this algorithm and results were quite accurate. A raster image with 250*250 resolution sizes and a Maximum value region size of around 10,000 cells was calculated in just 41 seconds, where most of the other algorithms failed in terms of efficiency as they took some hours to form out an optimal region. Stack data structure follows the policy of FIFO (First in first out) which is used in the proposed algorithm as at each iteration, a cell containing highest probability of being included in the region is always on top of stack and it is popped out and being added to the region and strategy goes on until and unless we reach the specified number of /12 $ IEEE DOI /FIT

2 cells. The algorithm is online, as it doesn t take into account of the whole problem size. In-place insertion sort is quick, as we are dealing with already sorted array and pushing a new cell into its proper place, and Insertion sort is stable and adoptive, which is best for the problems where we have an array in sorted order. The algorithm was tested with 8-bit Bitmap images of environment and cloud data, and reasonable accurate results were found for maximum value regions of 10-50% of overall image resolution and the maximum layer was extracted efficiently. II. METHODS Formation of exact solution would require exhaustive search which is not possible in case of real-life data, as it will be extremely inefficient. So approximate solution the only possibility. Keeping these factors in mind, different approximate methods were designed. Each method has its own benefit and use with different levels of accuracy and efficiency in terms of computation time, explanation for each method is elaborated below. II.I NEIGHBORHOOD WINDOW METHOD This is a statistical method which takes advantage of incorporating windows over each cell s neighbor and using mean values for the selection of the window as a candidate for the maximum value region. The size of 3*2 neighbor window is arbitrarily chosen, as it less complicates the rule of connectedness for the region and exhibits a pretty good estimate of where the maximum value region could possibly fall in the overall space. Though still, it doesn t necessarily chose the best cells from each neighbor window. For illustration purposes we took a 9*9 grid size with an initial starting cell arbitrary chosen is shaded in black. We added four neighborhood windows evenly, starting from the initial cell as shown in Figure-1. Now following the pattern we can move on like this by placing windows evenly in same order while taking each neighborhood window as initial cell. Note that the value with each cell in the grid is indicating index for that cell. Figure 2. Radius for 3*2 Neighborhood window 47. So applying the strategy, the windows are placed over the search area with a size of half of the region size in single direction, which means there will be a total of 2*region size windows in all four directions at maximum, in case we don t reach at the end of the search space indexes. A case for a region size of 8 cells is shown in figure-3. Figure 3 all possible windows for a region size of 8 cells. When all windows are in position and their associated mean values are calculated, an exhaustive search can be applied using each mean value as an indicator for the selection of a window for maximum value region. Figure-4 is showing a maximum value region with each selected window s mean value that happened to be the best candidates. Figure 4 Window shaded in black having maximum means. So the maximum value region would probably fall in the area as outlined in figure-5. Although the method is quite reliable in terms of accuracy, but it s still premature to be used practically, as it takes big amount of computational cost in terms of time taken by the algorithm, especially when it comes in solving a real life problem. Figure 1 Neighborhood windows for cell 45 shaded in black. The radius for each neighborhood window is comprised of six cells including the cell itself as shaded in Figure-2. If selected, the cells in that window will become a part of the maximum value region, either all or some part of overall radius, depending upon their cell value difference as compared with their mean window value and other mean values from each window selected. Figure 5 Maximum Value Region for 50 cells Outlined. The method may not prove to be foolproof in terms of cell selection for the region, but the major strength of this method lies in estimation of the area where the maximum 378

3 value region possibly lies. Talking about the efficiency, the exhaustive search for watching out each single cell for unique region combinations is now replaced by six cells for each neighborhood window size, and the original formula for finding out unique number of region combinations for each cell is replaced by six. So the formula will be and the total number of neighbor windows for each side would be k/2, which gives k/2*4 for all four sides or simply 2k-number of total windows using radius of 3*2. Note that the formula for unique number of region combinations and total number of windows drawn is true when a cell or windows is inside the bounds of the problem space. Data structure behind implementation for this method could possibly be a tree, as the region size grows the tree grows as well which will require high maintenance. For improving efficiency with some trade off in accuracy, we can increase the window size. A sample window size is shown in Figure-6. Neighborhood window method is quite useful for having an idea of where the region with specified size having maximum or minimum total cell value could fall in the connected space. This method will be capable of producing some useful application in Environment, where each single attribute is accountable of every other connected cell. Figure 7 reducing search space method II.III REGION GROWING METHOD USING INPLACE INSERTION WITH STACKS A near optimal solution is proposed using greedy approach to the problem, which was experimented to be the best in terms of efficiency by far, in comparison with all other approaches for the same problem. III. COMPLEXITY Pseudo code for region growing method using in-place insertion with stacks is given below with complexity analysis of each process in terms of cost and time. TABLE 1: ALGORITHM COMPLEXITY TABLE Line/Process Cost Time For i=2 to K C1 K For each cell Adj(Si) C2 k-1 If cell not in AdjMatrix and RegMatrix C3 K ti i=2 Update AdjMatrix with CurrentCell C4 K ti-1 i=2 Figure 6 Increasing Neighborhood Window Method PUSH CurrentCell(Value,Index) at TOP of Stack C5 K ti-1 i=2 II.II REDUCING SEARCH SPACE METHOD A relatively different method can be thought of by reducing the overall search space and finding maximum connected cell attributes in that space- maximized subspace region. The original version of maximum value region although may not be solved applying this method, but this method will be useful in case the decision maker want to extract a maximized region sum value within a defined boundary of interest from an overall space. Perform InplaceInsertion for CurrentCell C6 K (ti-1)d i=2 Increment Counter C7 K ti-1 i=2 Set Top to Si C8 k-1 Update RegionSum, RegMatrix with TOP value C9 k-1 Pop Top Item from Stack C10 k-1 Decrement Counter C11 k-1 Populate RegMatrix on Grid C12 1 Algorithm in its run time is not much different from insertion sort, as it involves constant time single line operations from c1-c5, c7-c11 and the cost associated with line 6, i.e. In-place Insertion, would depend upon the 379

4 nature of data. In case of high spatial autocorrelation, it would become the best case insertion sort routine, while all other cases would still be considered as average cases. While the final line will just run once, but, as it populates/transforms the Region to Grid, so it will take time. For a region size of 32,000 pixels, this algorithm took 128seconds in total, in which 24seconds belonged to the grid transformation for last line. Irrespective of the grid transformation, the running time of this algorithm will be O(Ktd), where t is the total number of adjacent cells encountered and d is the number of total swaps performed. IV. EXPERIMENTS Wind Speed data with 30m resolution with floating-point data ranging 0-1 for each pixel was tested with the region growing algorithm with fair results. Original Image and the region extracted of 5000 pixel region for Max and Min algorithms are shown in figures below. Trainee Cell for both Max and Min algo. Figure-8. 30m WindSpeed DEM Selection for the start or trainee cell is specified by the user plus the size for calculating regions in terms of pixel numbers are also required by the user to input. In this example, user specifies 5,000 pixels for max and min regions to be calculted by the algorithm. The results for both cases are shown below. Figure-10. Minimum value region using 4-Adj Whiter areas are the brightest areas with higher pixel values, and darker are vice-versa. Naked eye suggests that the output given by the algorithm is quite accurate, as exact accuracy assessment is not possible because of the hardware and processing limitations. V. CONCLUSION The near optimal solution (region growing method) has proven to be the fastest in terms of computational time it requires in computing the maximum valued region. The algorithm was experimented with real-life data, where we used single layer environment data of wind speed, temperature, precipitation and 30meter DEM s of different areas. Results suggested that algorithm was quite accurate as the regions calculated were 20 to 40% of overall N through visual interpretations of the extracted regions using region growing method and has looked useful as shown in figures-3 to 7 with maximum, minimum and 4, 8 adjacency rules. Applicability of this algorithm is dependable of how a decision maker sees it. In figure-11 we used the directed region growing algorithm for finding an escape pattern using DEM. Though, considering the fact that fluctuation in accuracy of the algorithm would remain, when degree of spatial autocorrelation in the input data is low, yet the algorithm still is quite handy as it has computed regions of thousands of pixels from larger datasets provided in just a few seconds. REFERENCES Figure-9. Maximum value region using 4-Adj [1] T. Shirabe (2011), A heuristic for the Maximum value region problem in raster space. International Journal of Geographical Information Science. [2] Pablo Vanegas, Dirk Cattrysse and Jos Van Orshoven(2011), A Multiple Criteria Heuristic Solution Method for Locating Near to Optimal Contiguous and Compact Sites in Raster Maps. Geocomputation, sustainability and environmental planning 380

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