Image Processing: Geometric Correction

• Image processing includes image pre-processing which requires operations to be done before main data analysis.
• Image pre-processing removes errors caused by image acquisition, using image restoration and rectification.
• Image processing includes image enhancement, image correction and transformation, and image classification.

Geometric Correction

• Geometric correction preprocesses remotely sensed data to remove geometric distortion.

• It ensures individual pixels have proper planimetric (x, y) map locations.

• Remote sensing-derived information is related to other thematic information in GIS or SDSS by geometric corrections.

• Geometrically corrected imagery helps extract accurate distance, polygon area, and direction/bearing information.

• Geometric correction of remotely sensed images is needed when using them, or a product derived from them to:

  • Transform an image to match a map projection.
  • Locate points of interest on a map and image.
  • Overlay temporal sequences of images of the same area, acquired by different sensors.
  • Overlay images and maps for GIS data analysis.

Internal and External Geometric Errors

• Remotely sensed imagery may have internal and external geometric errors.
• Identifying if errors are systematic (predictable) or nonsystematic (random) is important.
• Systematic geometric error is easier to identify and correct than random geometric error.
• Internal geometric errors (systematic) are introduced by the remote sensing system itself or in combination with Earth's rotation/curvature.
• External geometric errors (unsystematic) are introduced by phenomena varying in nature through space and time.
• Random movements by aircraft/spacecraft during data collection are the most important external variables causing geometric error.

Sources of Geometric Errors

• Systematic errors include:

  • Earth's rotation effect that skews the image.
  • Relief distortion.
  • Panoramic distortion which is a scale distortion.
  • Earth Curvature.

• Unsystematic (random) errors include variations in altitude and attitude of the sensor (roll, yaw, pitch).

• These errors contribute unequally to the overall geometric distortion in an image.

Error Correction

• Systematic errors can be corrected using:

  • Data from platform ephemeris.
  • Knowledge of internal sensor distortions.
  • Systematic errors are mostly removed after data acquisition by satellite data providers.
  • Easily corrected by formulas that model distortions.

• Ephemeris is the calculated positions of a celestial object at regular intervals throughout a period.

• Unsystematic errors cannot be corrected with acceptable accuracy without enough GCPs (Ground Control Points).

• Unsystematic errors may remain in the image, making it non-planimetric and not in proper x,y locations.

Geometric Rectification Levels

• Level 1 involves mostly systematic errors corrected through original rectification.
• Level 2 uses GCPs (X, Y) and transformation for geometric rectification on less and flat area.
• Level 3 uses 3D GCPs, DEM (Digital Elevation Model) and transformation for ortho rectification.

Image to Map Geometric Rectification Logic

• Two basic operations must be performed.

• Geometric rectification of remotely sensed image to a map coordinate system:

  • Spatial interpolation
    • Geometric relationship that relates/relocates x,y to X,Y coordinates is required.
  • Intensity interpolation
    • Proper input BV (Brightness Value) must be assigned to the output pixel location.

Geometric Rectification via GCPs

• Corresponding points are selected (usually manually) from map/image or reference images.

• Transformation is computed to transform distorted image to map/reference image:

  • x = f1(X,Y)
  • y = f2(X,Y)
  • (x,y) = distorted-image coordinates (col, row)
  • (X,Y) = correct (map) coordinates
  • (f1, f2) = transformation functions created using ground control points (GCPs)

• Low order polynomials commonly accomplish transformation.

Polynomial Models

• They relate two-dimensional ground coordinates to image coordinates.

• The equation for 1st order polynomial needs two equations for each GCP:

  • x = a0+ a1X + a2Y
  • y = b0+ b1X + b2Y

• Transformation accounts for six parameters: shift (translation) in x and y, scale in x and y, shear (skew), and rotation.

• The minimum number of GCPs needed is 3.

• Six equations can be written for three GCPs, and solved simultaneously for unknowns.

Affine/Polynomial Model

• Calculates the equation for minimum number of GCPs
  • No of GCPs = ((T+1)*(T+2)) / 2 , where T=order of transformation (polynomial)
• Simple transformation works for moderate distortions and small areas.
• Complex transformations are involved for large areas and more distortion.

Transformations

• Projective transformations can transform a square into any more general quadrilateral, maps lines to lines, and does not preserve parallelism.
• Affine transformations preserve lines and parallelism and maps parallel lines to parallel lines.
• Similarity transformations are shape-preserving and are proportional scaling transformations.
• Euclidean transformations allow only rotations and translations.

Important Notes - GCPs

• Using more than the minimum GCPs is desirable.
• Results from a small number of GCPs should be viewed cautiously.
• Spatial distribution of points is critical to avoid biased fit, which may be a problem in sea, water areas, or featureless regions like deserts.
• Avoid GCPs from maps not revised for a long period of time.

Intensity (Spectral) Interpolation

• It is the transfer of brightness values from original image and their relocation to the rectified image.
• The mechanism is for determining brightness value (BV) to assign to the output rectified pixel.
• There is no direct one-to-one relationship between the movement of input pixel values to output pixel locations.

Intensity Interpolation/Resampling

• Includes extraction of brightness values from original image and their relocation to the rectified image.

• Methods include:

  • Nearest Neighbor (Zero order interpolation)
  • Bilinear interpolation (1st order interpolation)
  • Cubic convolution (Bicubic interpolation or 2nd order interpolation)

• The optimal resampling method depends upon the data application and the advantages/disadvantages of each.

• Nearest Neighbor selects the actual pixel whose center is closest to a point located on the image, maintaining actual pixel values, but boundaries are not smooth, and have Jagged edges.

• Nearest Neighbor is recommended if data are to be classified later.

• Bilinear interpolation calculates a value for each corrected pixel based on the weighted average of the four nearest input pixels, resulting in a more natural look, and is geometrically accurate

• Result is interpolated pixel values (altered values), not used prior to classification, rather post classification.

• Cubic convolution uses cubic polynomials and 16 surrounding pixels, resulting in interpolated pixel values with data values altered more than other methods.

Error and Accuracy

• Errors remaining after transformation are residual errors.

• The magnitude of residual errors indicates the quality of transformation.

• Overall accuracy of transformation is expressed by the Root Mean Square Error.

• RMS error calculates a mean value from the individual residuals.

• It helps to check which GCP contributes more error and find overall RMS error of the process.

• If it does not meet the user defined threshold:

  • Delete the GCP contributing higher error.
  • Re-compute Transformation parameters.
  • Re-compute Error.

• RMS error is calculated using the equation:

  • RMS error = √(xr - xi)² + (yr - yi)², where xi and yi are the source coordinates in the reference image/map and xr and yr are the transformed coordinates.

• RMS error test sets a window of pixels that is considered as correct.

• If the RMS error is set at 1.0, the output location is considered correct if it falls within 1 pixel of the source.

• the output location can be off by one pixel in either the x or y direction.