Raster model. Alexandre Gonçalves, DECivil, IST

Transcription

1 Raster model 1. Resolution 2. Values and data types 3. Storage 4. Fitting rasters 5. Map algebra 6. Interpolation 7. Conversion vector raster 8. Vector vs. raster 1

2 Raster model Divides the space into a regular grid of cells in a specific order each cell has one assigned value each place is occupied by a single cell 2

3 Resolution spatial spectral (number of distinct values that can be stored) 3

4 Value and data types Values Integer Real Alphanumeric (coded) Data Continuous functions Categorical data 4

5 Pixel Generally a pixel has been assigned a single value maybe inadequate: a border can cross a pixel rules for deciding the classification Information for a area is usually composed by several layers 5

6 Structure and storage 6

7 Coordinates image coordinates real coordinates 7

8 Structure: Zones and Regions Zones are groups of cells that share the same value Regions are contiguous zones Null

9 Structure: Associated tables Only to integer-valued rasters Value and Count of zones (not of regions) always available Other attributes may be added Null

10 Affine transformation 10

11 Storage Row major ABBB ABBB AABB AAAB Column major AAAA BBAA BBBA BBBB Run Length Encoding (RLE) Row Major 1A3B1A3B2A1B3A1B RLE Column Major 4A2B2A3B1A4B 11

12 1) Linear Scan Storage 2) Interleaved Scan (TV) 3) Byte Offset 4) Zig Zag Order 5) Sub Block Scan Order 6a) Peano Order 6b) Peano Order Non-equal Cell Sizes 7) Hilbert Order 8) Morton Order 12

13 Storage quadtrees original 1ª partition 13

14 Storage quadtrees 2ª partition 3ª partition 14

15 Storage quadtrees 4ª partition final tree 15

16 Storage The generic model is implemented in diverse computational formats: GRID proprietary format ESRI JPEG, TIFF, MrSid standard format for display but not for analysis; generically need an additional file to get the location (chamado world file) Type Image File World File TIFF image.tif image.tfw Bitmap image.bmp image.bpw BIL image.bil image.blw JPEG image.jpg image.jpw 16

17 Fitting rasters Before the analysis, grids must be made compatible Need for resampling the grid Change in the projection system Change in the width of cells Georeferencing 17

18 Fitting rasters georeferencing Corrected image Original image (grid) Reference map Original image 18

19 1st Order u = a0 + a1x +a2y y = b0 + b1x +b2y Fitting rasters polynomial transformation (6 coefficients) 2nd Order (12 coefficients) 2 2 u = a0 + a1x + a2y + a3xy + a4x + a5 y y = b0 + b1x + b2y + b3xy + b4x + b5y 2 2 3rd Order (20 coefficients) u = a0 + a1x + a2y + a3xy + a4x + a5 y + a6x y + a7xy + a8x + a9y 2 y = b0 + b1x + b2y + b3xy + b4x + b5 y + b6x y + b7xy + b8x + b9y

20 Fitting rasters Pixel value in the corrected image is the value of the nearest pixel in the original image. Advantages Computationally simple Does not change original values Applies to nominal scales Disadvantages nearest neighbor Objects may displace up to half pixel Structures get a zigzag shape 20

21 Fitting rasters bilinear interpolation The value of the pixel in the corrected image is a weighted average value of the 4 closest pixels in the original image. Advantages Smoothes the image Disadvantages Smoothes the image Changes pixel values 21

22 Fitting rasters cubic convolution Pixel value in the corrected image is given by some weight of the 16 closest pixels in the original image. Advantages Less interpolation artifacts because the neighborhood is larger Disadvantages Computationally intensive Changes pixel values May extrapolate in places where local variation is large 22

23 Local functions Map algebra Focal functions Zonal functions Global functions 23

24 Local functions combine the values of one or more rasters to produce a new raster using the same cell positions in each one _ Null Null Null Null Null Null = Pop 2000 Pop 1990 Var Pop 24

25 Local functions Arithmetic operators Basic operators (+,-,*,/) available. Rounding and precedence. Boolean operators Logic operators (AND, OR, NOT) available. Output: 0=FALSE, 1=TRUE Input: 0=FALSE, ~0=TRUE 25

26 Local functions Null 0 2 Null Null Null Null Null AND = Null Null Null Null 0 2 Null Null 0 1 Null Null Null Null AND = Null Null Null

27 Local functions reclassify [0,2]=0 ]2,6]=1 ]6,9]=

28 Focal functions take the value from a cell s neighborhood

29 Focal functions Focal (normal) Block Nas operações focais existe Focal operations use a mobile uma janela móvel, i.e., as window, i.e., neighbourhoods vizinhanças sobrepõem-se. overlap Nas operações de Bloco as vizinhanças Block operations use non-overlapping são justapostas. neighbourhoods. The output is the same O valor de output é igual em todas as for all cells in a given block células de um dado bloco. 29

30 Focal functions possible neighborhoods 30

31 Focal functions Removing noise Errors/outliers Paul Bolstad, GIS Fundamentals 31

32 Focal functions Roughness index R cel ( X ij X cel i, j ) 2 32

33 Zonal functions Zonal functions are very similar to focal functions, except the neighbourhood has not a fixed shape, and can be defined by another grid Taking a raster, they calculate for each cell some function or statistic, using its value and that of all cells belonging to the same zone. Some zonal functions (type I) for which zones are defined by an isolated value, allow statistics or the quantification of geometric characteristics of input zones. Other zonal functions (type II) for which zones are defined through a second grid, allow the statistics or the filling of specific zones with values from the input grid. 33

34 Zonal functions similar to focal functions, but the neighborhood has no fixed shape, being defined by the distribution of values (type I) or by a second grid (type II) type I example: distance to the sea 34

35 zone layer Zonal functions A A Null X X G G A A X G G A A X A G G A X A A X X A Null result type II example: zonal sum

36 Global functions Position - Buffer - Triangulation - Voronoi diagram Position and value - Visibility maps - Interpolation 36

37 Global functions: Interpolation Set of methods to estimate unknown values of a function based on measured/known values Point data, but Phenomena extending to areas Transformations Point to area Polinomial Based on distance (IDW, Kriging, etc) Stochastic functions x a f(x) f(a) b? c d f(c) f(d) 37

38 Global functions : Interpolation Linear f^(b) = f(a)(b-a)/(c-a) + f(c)(c-b)/(c-a) Simple polynomials f(x,y) = S S a ik x i y k a ik are the coefficients Polynomial fitted by minimum mean square error method f(x,y) = S S a ik x i y k + e 38

39 Global functions: interpolation Point-to-area Calculation based on proximity only ex: Voronoi diagrams 39

40 Global functions: interpolation Distance-based Splines: Based on the nearest points Smooth surfaces Not exact in input points Triangulations: J f(x,y) = z = ax + by + c A OIJ A OJK z 1 = ax 1 + by 1 + c z 2 = ax 2 + by 2 + c z 3 = ax 3 + by 3 + c I O K A OIK All things are related, but nearby things are more related than distant things 40

41 41 Global functions:interpolation n i d n i i d i i v v n i d n i i d p i p i v v Distance-based IDW

42 Global functions: interpolation IDW x y f(x,y)=zi Dist. à obs. 8 = di weights wi= 1/ di Normalised weights (1/di) / (S1/di) ? S1/di = f ^(65,137) =

43 Global functions : Interpolation x y z ? (x,y) = z = a0 + a1x + a2y Sz = a 0 n + a 1 S x + a 2 S y Sz x = a 0 S x + a 1 S x 2 + a 2 S x y S z y = a 0 S y + a 1 S x y + a 2 S y = a 0 (6) + a 1 (15) + a 2 (15) 420 = a 0 (15) + a 1 (47)+ a 2 (36) 451 = a 0 (15) + a 1 (36)+ a 2 (47) f^(2,3) = 29.9 f(x,y) = x -0.3y 43

44 Global functions : Interpolation IDW weights normalized weights x y f(x,y)=zi Distance to observation 8 = di wi= 1/ di (1/di) / (S1/di) ? S1/di = f ^(65,137) =

45 Global functions: interpolation IDW: The larger the power, the larger the difference between neighbouring cells Exact interpolator (estimate on sample points gives the point value) As one moves away from the sample points, values will tend to the mean value Needs a good sample distribution 45

46 Global functions : Interpolation Triangulation f(x,y) = z = ax + by + c J z 1 = ax 1 + by 1 + c z 2 = ax 2 + by 2 + c z 3 = ax 3 + by 3 + c A OIJ O A OJK I K A OIK 46

47 Global functions : Interpolation Quantification of error margins e = f^(x,y) - f(x,y) e = 1/n S f^(x,y) - f(x,y) for n points estimation error mean error MAE = e = 1/n S f^(x,y) - f(x,y) mean abs. error MSE = 1/n S [ f^(x,y) - f(x,y) ] 2 mean squared error 47

48 Converting vector raster 48

49 Raster advantages Simple data structure Easy analysis Low-tech platforms Remote sensing data Modeling is simple 49

50 Spatial inaccuracy Generalization Implicit data. Raster disadvantages Each cell must be classified. Large data sets 50

51 Vector advantages Closest to maps mental model Higher resolution Accuracy in positioning Node-vertex storage Understandable Topology 51

52 Raster disadvantages Complex data structure Demands geometric processing More complex editing 52