GEOG 4110/5100 Advanced Remote Sensing Lecture 4

Transcription

1 GEOG 4110/5100 Advanced Remote Sensing Lecture 4 Geometric Distortion Relevant Reading: Richards, Sections

2 Geometric Distortion Geometric Distortion: Errors in image geometry, (location, dimensions, etc.) Sources Earth rotation during image acquisition Sensor scan characteristics Wide field of view of some sensors Curvature of the Earth Sensor realities (not perfect) Variations in platform altitude, attitude and velocity Panoramic effects related to image geometry

3 Image Edge Compression (S-Bend Distortion) θ θ β/2 For constant angle increments (β), pixels are offset more near the edge of a scan than near nadir (S-bend distortion) Pixel spacing = P/cos 2 θ (P/2) θ P/2 Resulting effect when pixels are placed on a uniform grid is cross-track compression

4 Image Edge Compression (S-Bend Distortion) Cross-track distortion is SN/TN = q/tanq For edge of Landsat swath: (q/tanq) max = At scan edge (92.5 km from nadir point) pixel will be 314 m out of position, compared with pixel at scan point.

5 Correction of Geometric Distortion Mapping Polynomials for Image Projection Assumes geometrically correct map of image region is available Need to map known locations (map) to corresponding locations in image u = f(x,y) v = g(x,y) 5

6 Resampling Change in grid spacing (cell-size) in an image as compared to those in the reference map or a reference image In the case below, we show a one-to-one correspondence between control point location and the corresponding image location. This need not be the case. 6

7 Resampling Often we wish to represent data on a different grid than was acquired or than corresponds to a map Shifted Grid Interpolation Higher apparent resolution interpolation Lower apparent resolution Averaging/Drop-inbucket Interpolation/weighted averaging 7

8 Resampling to Lower Resolution Averaging/Drop-in-Bucket Interpolation/Weighted Averaging Average of all points within a grid cell Weight of points assigned based on distance from grid cell center 8

9 Interpolation Nearest Neighbor Resampling Assigns the value of the actual pixel that is closest to the grid point in the image Preferred method for classification, since it leaves the actual pixel values intact, and just rearranges them in position match desired image geometry Bilinear Interpolation Uses 3 linear interpolations over four pixels surrounding the image grid point to display grid position Cubic Convolution Cubic Polynomials are fit along four lines of four pixels and a fifth is fit to the four interpolants 9

10 Interpolation: Nearest Neighbor Resampling 10

11 Interpolation: Nearest Neighbor Resampling f(i,j) = Pixel Brightness Value at grid point i,j 11

12 Interpolation: Nearest Neighbor Resampling Distortion Rotation 12

13 Interpolation: Bilinear Uses 3 linear interpolations over four pixels surrounding the image grid point to display grid position 13

14 Interpolation: Bilinear f(i,j+j )=j f(i,j+1)+(1-j ) f(i,j) f(i+1,j+j )=j f(i+1,j+1)+(1-j ) f(i+1,j) f(i+i',j+j )= (1-i )[j f(i,j+1)+(1-j ) f(i,j)] + i [j f(i+1,j+1)+(1-j )f(i+1,j) 14

15 Interpolation: Cubic Convolution Cubic Polynomials are fit along four lines of four pixels and a fifth is fit to the four interpolants From Map Cubic Polynomial Interpolation 15

16 Correction of Geometric Distortion Mapping Polynomials for Image Projection Assumes geometrically correct map of image region is available Need to map known locations (map) to corresponding locations in image u = f(x,y) v = g(x,y) 16

17 Choice of Control Points Sufficient number of well-defined control points needed to ensure generation of accurate mapping polynomials Must be sufficiently distributed so as to represent the full image and avoid large extrapolation errors. Example of curve fitting to reinforce the potentially poor behavior of high order mathematical functions when used to extrapolate (Richards and Jia, 2006, Fig. 2.15) 17

18 Good point distribution Bad point distribution 18

19 Mathematical Modeling: Example Aspect Ratio Distortion Correction Aspect Ratio: relative vertical and horizontal scales (width/heigh) x ½ x x AR=1 x AR=0.5 Samples are sometimes acquired too quickly across a scan line compared to the instrument IFOV e.g. Landsat MSS acquires pixels at 56 m intervals with an IFOV of 79 m Landsat effective pixel size is 79 m x 56 m (along-track across-track) Image displayed on a square grid will be too wide for its height Have to compress in width by a factor of 79/56 or Another example is aircraft moving too slow or too fast compared to cross-track scan 19

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21 Mathematical Modeling Example Aspect Ratio Distortion Correction Samples are sometimes acquired too quickly across a scan line compared to the instrument IFOV E.g. Landsat MSS acquires pixels at 56 m intervals with an IFOV of 79 m Landsat effective pixel size is 79 m x 56 m (along-track across-track) Image displayed on a square grid will be too wide for its height Have to compress in width by a factor of 79/56 or x y = u v x = 0.709u y = v 80 m 56 m 80 m 80 m u v = x y u= x v = y 21

22 Mathematical Modeling Example Aspect Ratio Distortion Correction Samples are sometimes acquired too quickly across a scan line compared to the instrument IFOV E.g. Landsat MSS acquires pixels at 56 m intervals with an IFOV of 79 m Landsat effective pixel size is 79 m x 56 m (along-track across-track) Image displayed on a square grid will be too wide for its height Have to compress in width by a factor of 79/56 or or expand height by the same factor Compress Width Expand Height! # "# x y $ & %& =! # " 0 1 $ & %! # " u v $ & % x = 0.709u y = v! # "# x y $ & %& =! # 1 0 " $ & %! # " u v $ & % x = u y = 1.411v! # " u v $! & = # % " 0 1 $ & %! # "# x y $ & %& u= x v = y! # " u v $! & = # 1 0 % " $ & %! # "# x y $ & %& u= x v =.709 y 22

23 Earth Rotation Effect at 40 Latitude In the ~27.4 s it takes to acquire a Landsat scene, a point on the Earth s surface moves (Dx e ) 9.73 km, or 5.4% of the scene width Dx e = v e DT = 355 m/s 27.4 s = 9727 m Accounting for ~13 angle N/S offset at 40 latitude associated with inclination: D x = Dx e cos13 = 9.48 km (5.3% of scene width)

24 Mathematical Corrections Earth Rotation Skew Correction a For Landsat at 40 latitude associated with inclination: Dx = Dx e cos13 = 9.48 km (5.3% of scene width) (Eqs. from Richards, p. 49) x y α = u v u v α = x y Combined corrections for aspect ratio and Earth rotation skew x y = α 0 1 u v 24

25 Mathematical Corrections North/South Orientation for an angle offset from the vertical For Landsat, z is approximately 9 o at equator z x = y cos z sin z - sin z cosz u u = v v cos z sinz sin z cos z (In image space, North is oriented clockwise by 9 degrees) - x y 25

26 Image Registration Georeferencing: registration of an image to a known map coordinate system Pixels addressable in map coordinates such as lats and lons or Eastings or Northings Enables comparisons to other data types registered similarly Image to image Registration Multiple images registered to a single map Image pairs registered to one another Requires fewer steps Relationships to only between image pairs or sets One image chosen as reference (master) image that remains unchanged; other (slave) images are corrected to match features to the reference image 26

27 Image Registration Features are clearly located at different pixel positions in the different images In order to compare, we need to co-register them to the same coordinate system/reference-frame Image to image, or each to a known reference 27

28 Identification of Control Points By Correlation (Image to Image co-registration) Feature identified in subordinate image We search for same feature in reference image to determine its offset Define a small rectangle centered on the feature Define a search region in reference image Incrementally move the rectangle pixel by pixel in the reference image to find maximum correlation Point where sum of the magnitude of the differences is least Quality of point depends on strength of maximum correlation in relation to next highest correlation values à greatest when features are local and distinct Repeat for multiple features at different locations in image 28

29 Subordinate Image Reference Image GEOG 4110/

30 Subordinate Image Reference Image GEOG 4110/

31 Subordinate Image Reference Image GEOG 4110/

32 Subordinate Image Reference Image Need to translate the subordinate image 1 pixel north and 1 pixel west to coregister GEOG 4110/