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38 3. Fundamental Geo-Raster Operations Arrays are represented as functions mapping n-dimensional points from discrete Euclidean space to values. The spatial domain of an array is defined as a finite set of n-dimensional points in Euclidean space forming a hypercube with boundaries parallel to the coordinate system axes. Let X ⊆ Z d be a spatial domain and F a value set i.e., a homogeneous algebra. Then, an F-valued d-dimensional array over the spatial domain X(multi-dimensional array) is defined as: a : X → F (i.e., a ∈ F X ), a = {(x, a(x)) : x ∈ X, a(x) ∈ F } The array elements a(x) are referred to as cells. Auxiliary function sdom(a) denotes the spatial domain of some array a. 3.1.1 Constructor The MARRAY array constructor allows arrays to be defined by indicating a spatial domain and an expression evaluated for each cell position of the array. An iteration variable bound to a spatial domain is available in the cell expression so that the cell value depends on its position. Let X be a spatial domain, F a value set, and v a free identifier. Let e v be an expression with result type F containing zero or more free occurrences of v as placeholder(s) for an expression with result type X. Then, an array over spatial domain X with base type F is constructed through: MARRAY X,v (e v ) = {(x, a(x)) : a(x) = e x , x ∈ X} A straightforward application of MARRAY is spatio-temporal sub-setting by simply changing its domain. Example: For some 2-D grey-scale image a, its cutout to domain [x0:x1,y0:y1] (assumed to lie inside the array) is given by: MARRAY [x0:x1,y0:y1],p (a[p]) Similarly, trimming produces a cutout of an array of lower volume, but unchanged dimensionality, and section cuts out a hyperplane with reduced dimensionality. We can also change an array’s values by changing the e v expression. In the simplest case this expression takes the cell value and modifies it. The following expression adds the values in the cells of two raster images, regardless of their extent and dimension: a + b = MARRAY X,p (a[p] + b[p]) If we allow the use of all operations known on the base algebra, i.e., on the pixel type, we immediately obtain a cohort of the following useful operations.
3.2 Geo-Raster Operations 39 3.1.2 Condenser The COND array condenser (aggregator) takes the values of an array’s cells and combines them through some commutative and associative operation, thereby obtaining a scalar value. For some v free identifier, spatial domain X = x 1 , ..., x n , x i ∈ Z d consisting of n points, and e a,v an expression of result type F containing occurrences of an array a and identifier v, the condense of a by o is defined as: COND o,X,v (e a,v ) := O x∈X e a,x = e a,x1 o...oe a,xn Example: Let a be the image as defined in above. The average over all pixel intensities in a is then given by: ∑ COND +,sdom(a),p (a) = a[x]/(m ∗ n) 3.1.3 Sorter x∈[1:m,1:n] The SORT array sorter proceeds along a selected dimension to reorder the corresponding hyperslices. Functional sort s rearranges a given array along a specified dimension s without changing its value set or spatial domain. To that end, an order-generating function is provided that associates a sequence position to each (d- 1)-dimensional hyperslice. Note that function f s,a has all degrees of freedom to assess any of a’s cell values for determining the measure value of a hyperslice on hand - it can be a particular cell value in the current hyperslice, the average of all hyperslice values, or the value of one or more neighboring slices. Note that the sort operator includes the relational group by. The language is recursive in the array expression e v and hence allows arbitrary nesting of expressions. In the sequel we use the abbreviations introduced above for nested expressions. 3.2 Geo-Raster Operations This section presents a set of fundamental operations for Geo-raster data. These operations have been selected based on an exhaustive literature review of classification schemes, international standards, and best practices [2, 19, 27, 32, 35, 45, 46, 47, 49]. By examining the Array Algebra operators involved in the computation of the operations, we identify those that require data summarization (aggregation) and therefore may benefit from pre-aggregation. Queries were executed in a raster database management system (RasDaMan), and formulated according to the syntax of an SQL-based query language for multidimensional raster databases based on Array Algebra, namely, rasql. 3.2.1 Mathematical Operations The following groups of mathematical operators are distinguished: arithmetic, trigonometric, boolean and relational. They operate at cell level and can be applied
Page 1: Applying OLAP Pre-Aggregation Techn
Page 5 and 6: Acknowledgments I would like to exp
Page 7 and 8: Abstract Large multidimensional arr
Page 9 and 10: Contents 1 Introduction and Problem
Page 11 and 12: List of Figures 2.1 3D Array . . .
Page 13 and 14: List of Tables 3.1 UNO and FAO Suit
Page 15 and 16: Chapter 1 Introduction and Problem
Page 17 and 18: Relevant and complementary question
Page 19 and 20: 1.2 Publications Related to this Th
Page 21 and 22: Chapter 2 Background and Related Wo
Page 23 and 24: 2.1 Array Databases 17 Figure 2.2 s
Page 25 and 26: 2.1 Array Databases 19 toward the s
Page 27 and 28: 2.1 Array Databases 21 • Bilinear
Page 29 and 30: 2.1 Array Databases 23 given image
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Page 39 and 40: 2.3 Discussion 33 spatial-vector da
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Page 49 and 50: 3.2 Geo-Raster Operations 43 Table
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Page 53 and 54: 3.2 Geo-Raster Operations 47 (a) Or
Page 55 and 56: 3.2 Geo-Raster Operations 49 Query
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Page 65 and 66: 3.2 Geo-Raster Operations 59 Local
Page 67 and 68: 3.3 Summary 61 Slicing The slicing
Page 69 and 70: Chapter 4 Answering Basic Aggregate
Page 71 and 72: 4.1 Framework 65 pre-aggregated res
Page 73 and 74: 4.2 Cost Model 67 By partitioning t
Page 75 and 76: 4.2 Cost Model 69 Cost of independe
Page 77 and 78: 4.3 Implementation 71 Algorithm 1 Q
Page 79 and 80: 4.4 Experimental Results 73 Query E
Page 81 and 82: 4.5 Summary 75 pre-aggregates: inde
Page 83 and 84: Chapter 5 Pre-Aggregation Support B
Page 85 and 86: 5.2 Conceptual Framework 79 Figure
Page 87 and 88: 5.2 Conceptual Framework 81 Benefit
Page 89 and 90: 5.4 Answering Scaling Operations Us
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5.5 Experimental Results 89 (a) Sel
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5.5 Experimental Results 91 vectors
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5.5 Experimental Results 93 root no
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5.5 Experimental Results 95 Figure
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5.6 Summary 101 we considered non-u
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Chapter 6 Conclusion One of the big
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6.1 Future Work 105 more non-spatio
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Bibliography [1] Blakeley J. A., La
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BIBLIOGRAPHY 109 [22] Moon B., Vega
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BIBLIOGRAPHY 111 [47] ESRI Inc. Arc
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BIBLIOGRAPHY 113 [73] Stefanovic N.
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BIBLIOGRAPHY 115 [97] Kotidis Y. an
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