Module 5: The Perfect Map: Ironing Out the Bumps

By Gopi Krishna Tummala

The Problem of Relief Displacement

Look at a raw satellite image of a hilly area. The side of a mountain facing the satellite looks stretched out, and a tall building appears to lean away from the image center. This is called Relief Displacement, and it means the object’s position on the map is wrong. You can’t trust the coordinates.

To create a true, measurable map—an orthophoto or orthomosaic—we must remove this displacement. Orthorectification is the process of using the Digital Elevation Model (DEM) we created in Module 3 and the Collinearity Equations from Module 2 to mathematically correct the position of every single pixel. It’s like ironing the topography flat onto a perfect 2D surface.

💡 The Math Hook: The Geometric Correction

The math behind this process is a geometric shift calculation. For a given pixel, the algorithm checks the DEM to find its true height, $Z$. It then calculates exactly how far that pixel needs to be moved horizontally on the map plane to remove the lean caused by its height.

This movement is not arbitrary; it’s a precise vector calculated by incorporating the known position of the satellite, the camera angle, and the ground elevation. When done correctly, the final orthomosaic is planimetrically correct, meaning distances and coordinates are accurate everywhere.

The Orthorectification Process:

1. Forward Transformation:

   • For each pixel in the output orthoimage
   • Calculate its ground coordinates $(X, Y)$ using the DEM height $(Z)$
   • Use sensor model to find corresponding pixel in raw image

2. Resampling:

   • Raw image pixels rarely align perfectly with output grid
   • Interpolate pixel values (nearest neighbor, bilinear, cubic)

3. Output:

   • Georeferenced orthoimage
   • Every pixel has accurate ground coordinates
   • Uniform scale across the image

Key Topics

The Necessity of Orthorectification

Raw satellite images have geometric distortions, especially relief displacement—where tall objects (buildings, mountains) appear to lean away from the image center.

Types of Geometric Errors:

1. Relief Displacement:

   • Objects at different elevations appear shifted
   • Most significant in hilly/mountainous terrain
   • Increases with distance from nadir (image center)

2. Sensor Distortions:

   • Lens distortion
   • Detector misalignment
   • Scan line timing errors

3. Earth Curvature:

   • Earth’s curvature causes scale variations
   • More significant for wide-swath sensors

4. Atmospheric Refraction:

   • Light bends as it passes through atmosphere
   • Small but measurable effect

Why Orthorectification?

• Creates a planimetrically correct map
• Every pixel represents a true ground position
• Enables accurate distance and area measurements
• Essential for overlaying with other geospatial data

The Orthorectification Process

Combining Raw Image, DEM, and Sensor Model:

Orthorectification requires three inputs:

1. Raw Image: The distorted satellite image
2. DEM (Digital Elevation Model): Height information for every pixel
3. Sensor Model: Camera geometry and orientation

Mathematical Foundation: Uses the collinearity equations from Module 2, but now solving for image coordinates given ground coordinates and elevation.

Creating Seamless Orthomosaics

Image Mosaicking:

A single satellite image covers a limited area. Mosaicking combines multiple orthorectified images into one seamless map.

Key Steps:

1. Seam-line Generation:

   • Find optimal boundaries between overlapping images
   • Avoid cutting through important features (buildings, roads)
   • Minimize color differences along seams

2. Radiometric Balancing:

   • Images captured at different times have different brightness/color
   • Adjust histogram matching or gain/offset correction
   • Create visually seamless appearance

3. Blending:

   • Feathering: Gradual transition along seam lines
   • Cutline: Sharp boundary (requires careful balancing)
   • Multi-image blending: Weighted average of overlapping pixels

Challenges:

• Color differences between images
• Cloud shadows and artifacts
• Temporal changes (crops, construction)
• Maintaining geometric accuracy across seams

Accuracy Assessment

Evaluating the Final Map’s Accuracy:

Root Mean Square Error (RMSE):

The standard measure of geometric accuracy:

$\text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \hat{x}_i)^2 + (y_i - \hat{y}_i)^2}$

Where:

• $(x_i, y_i)$: Measured coordinates from the map
• $(\hat{x}_i, \hat{y}_i)$: True coordinates from ground control
• $n$: Number of check points

Accuracy Metrics:

1. Absolute Accuracy: Error relative to true ground coordinates

   • Typically reported as RMSE in meters
   • Example: “RMSE = 2.5 meters”

2. Relative Accuracy: Error between features within the map

   • Important for feature extraction
   • Usually better than absolute accuracy

3. CE90/LE90: Circular/Linear Error at 90% confidence

   • 90% of points have error less than this value
   • Common military/mapping standard

Validation Process:

• Use independent check points (not used in georeferencing)
• Measure errors in X, Y, and Z (if DEM available)
• Report statistics: mean, RMSE, maximum error

Orthorectification creates accurate, usable maps. In the next module, we’ll explore how AI automates feature extraction from these maps.