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         Chapter 7
     Geometric Correction

   It is usually necessary to preprocess remotely sensed data and
    remove geometric distortion so that individual picture elements
    (pixels) are in their proper planimetric (x, y) map locations.

 Geometrically corrected imagery can be used to extract
  accurate distance, polygon area, and direction (bearing)
       Internal and External Geometric Error

   Internal geometric errors are introduced by the remote sensing system itself
    or in combination with Earth rotation or curvature characteristics

   These distortions are often systematic (predictable) and may be identified and
    corrected using pre-launch or in-flight platform ephemeris (i.e., information
    about the geometric characteristics of the sensor system and the Earth at the
    time of data acquisition).

 Geometric distortions in imagery that can sometimes be corrected through
  analysis of sensor characteristics and ephemeris data include:
   • skew caused by Earth rotation effects,
   • scanning system–induced variation in ground resolution cell size,
   • scanning system one-dimensional relief displacement, and
   • scanning system tangential scale distortion.
             Image Skew
a) Landsat satellites 4, 5, and 7 are in a Sun-
synchronous orbit with an angle of
inclination of 98.2. The Earth rotates on its
axis from west to east as imagery is
b) Pixels in three hypothetical scans
(consisting of 16 lines each) of Landsat TM
data. While the matrix (raster) may look
correct, it actually contains systematic
geometric distortion caused by the angular
velocity of the satellite in its descending
orbital path in conjunction with the surface
velocity of the Earth as it rotates on its axis
while collecting a frame of imagery.
c) The result of adjusting (deskewing) the
original Landsat TM data to the west to
compensate for Earth rotation effects.
Landsats 4, 5, and 7 use a bidirectional
cross-track scanning mirror.
Scanning System-induced Variation in Ground Resolution Cell Size

                                     The ground resolution cell size along a single
                                          across-track scan is a function of

                                     a)   the distance from the aircraft to the
                                          observation where H is the altitude of the
                                          aircraft above ground level (AGL) at
                                          nadir and H sec f off-nadir;
                                     b)    the instantaneous-field-of-view of the
                                          sensor, b, measured in radians; and
                                     c)   the scan angle off-nadir, f. Pixels off-
                                          nadir have semi-major and semi-minor
                                          axes (diameters) that define the
                                          resolution cell size. The total field of
                                          view of one scan line is q.
                   Scanning System One-Dimensional Relief Displacement

a)   Hypothetical perspective geometry of a vertical aerial photograph obtained over level terrain. Four 50-ft-tall
     water tanks are distributed throughout the landscape and experience varying degrees of radial relief
     displacement the farther they are from the principal point (PP).
b)   Across-track scanning system introduces one-dimensional relief displacement perpendicular to the line of
     flight and tangential scale distortion and compression the farther the object is from nadir. Linear features
     trending across the terrain are often recorded with s-shaped or sigmoid curvature characteristics due to
     tangential scale distortion and image compression.
         External Geometric Error

   External geometric errors are usually introduced by phenomena that vary in
    nature through space and time.

 The most important external variables that can cause geometric error in remote
  sensor data are random movements by the aircraft (or spacecraft) at the exact
  time of data collection, which usually involve:
    • altitude changes, and/or
       • The diameter of the spot size on the ground (D; the nominal spatial
          resolution) is a function of the instantaneous-field-of-view (b) and the
          altitude above ground level (H) of the sensor system, i.e.,

                            D  b H
a) Geometric modification in imagery may be
introduced by changes in the aircraft or satellite
platform altitude above ground level (AGL) at the
time of data collection. Increasing altitude results in
smaller-scale imagery while decreasing altitude
results in larger-scale imagery.
b) Geometric modification may also be introduced
by aircraft or spacecraft changes in attitude,
including roll, pitch, and yaw. An aircraft flies in the
x-direction. Roll occurs when the aircraft or
spacecraft fuselage maintains directional stability
but the wings move up or down, i.e. they rotate
about the x-axis angle (omega: w). Pitch occurs
when the wings are stable but the fuselage nose or
tail moves up or down, i.e., they rotate about the y-
axis angle (phi: f). Yaw occurs when the wings
remain parallel but the fuselage is forced by wind to
be oriented some angle to the left or right of the
intended line of flight, i.e., it rotates about the z-axis
angle (kappa: k). Thus, the plane flies straight but all
remote sensor data are displaced by k. Remote
sensing data often are distorted due to a combination
of changes in altitude and attitude (roll, pitch, and
        Ground Control Points

 A ground control point (GCP) is a location on the surface of the Earth (e.g., a
  road intersection) that can be identified on the imagery and located accurately
  on a map. The image analyst must be able to obtain two distinct sets of
  coordinates associated with each GCP:

        • image coordinates specified in i rows and j columns, and
        • map coordinates (e.g., x, y measured in degrees of latitude and
          longitude, feet in a state plane coordinate system, or meters in a
          Universal Transverse Mercator projection).

 The paired coordinates (i, j and x, y) from many GCPs (e.g., 20) can be
  modeled to derive geometric transformation coefficients. These coefficients
  may be used to geometrically rectify the remote sensor data to a standard datum
  and map projection.
      Ground Control Points

 Several alternatives to obtaining accurate ground control point (GCP) map
  coordinate information for image-to-map rectification include:
       • hard-copy planimetric maps (e.g., U.S.G.S. 7.5-minute 1:24,000-
          scale topographic maps) where GCP coordinates are extracted using
          simple ruler measurements or a coordinate digitizer;
       • digital planimetric maps (e.g., the U.S.G.S. digital 7.5-minute
          topographic map series) where GCP coordinates are extracted
          directly from the digital map on the screen;
       • digital orthophotoquads that are already geometrically rectified (e.g.,
          U.S.G.S. digital orthophoto quarter quadrangles —DOQQ); and/or
       • global positioning system (GPS) instruments that may be taken into
          the field to obtain the coordinates of objects to within +20 cm if the
          GPS data are differentially corrected.
      Types of Geometric Correction

 Two common geometric correction procedures are often used by scientists to
  make the digital remote sensor data of value:
      • image-to-map rectification, and
      • image-to-image registration.

 The general rule of thumb is to rectify remotely sensed data to a standard map
  projection whereby it may be used in conjunction with other spatial
  information in a GIS to solve problems.
                                Image to Map Rectification

a) U.S. Geological Survey 7.5-minute 1:24,000-scale topographic map of Charleston, SC, with three
ground control points identified (13, 14, and 16). The GCP map coordinates are measured in meters
easting (x) and northing (y) in a Universal Transverse Mercator projection. b) Unrectified 11/09/82
Landsat TM band 4 image with the three ground control points identified. The image GCP coordinates are
measured in rows and columns.
                                  Image to Image Hybrid Rectification

a) Previously rectified Landsat TM band 4 data obtained on November 9, 1982, resampled to 30  30 m pixels using
nearest-neighbor resampling logic and a UTM map projection.

b) Unrectified October 14, 1987, Landsat TM band 4 data to be registered to the rectified 1982 Landsat scene.
                           Spatial Interpolation Using Coordinate Transformation

         A way to measure the accuracy of a geometric rectification algorithm
         (actually, its coefficients) is to compute the Root Mean Squared Error
         (RMSerror) for each ground control point using the equation:

                  RMS error                x  x   y  y 

xorig and yorig are are the original row and column coordinates of the GCP in the image and x’ and y’ are the computed
or estimated coordinates in the original image. Basically, the closer these paired values are to one another, the more
accurate the algorithm (and its coefficients). The square root of the squared deviations represents a measure of the
accuracy of each GCP. By computing RMSerror for all GCPs, it is possible to (1) see which GCPs contribute the
greatest error, and 2) sum all the RMSerror.
       Intensity Interpolation

 Intensity interpolation involves the extraction of a brightness value from an
  x, y location in the original (distorted) input image and its relocation to the
  appropriate x, y coordinate location in the rectified output image.
 There are several methods of brightness value (BV) intensity interpolation that
  can be applied, including:

         • nearest neighbor,
         • bilinear interpolation, and
         • cubic convolution.

 The practice is commonly referred to as resampling.
       Nearest-Neighbor Resampling

The brightness value closest to the predicted x’, y’ coordinate is assigned to the output x, y
Bilinear Interpolation
Cubic Convolution
Image Mosaicking

    Individual images should be rectified to the same map projection and datum. (e.g., multiple Landsat
     TM scenes to be mosaicked are often resampled to 30  30 m).
    One of the images to be mosaicked is designated as the base image. The base image and image 2 will
     normally overlap a certain amount (e.g., 20% to 30%).
    A representative geographic area in the overlap region is identified.