Image Preprocessing: Geometric Correction Image Preprocessing: Geometric Correction Jensen, 2003 John R. Jensen Department of Geography University of South Carolina Columbia, South Carolina John R. Jensen Department of Geography University of South Carolina Columbia, South Carolina 29208

Geometric Correction There are two basic types of geometric correction: * Image-to-image registration - useful for registering two or more images together when it is not necessary to have the interpreted output in a formal map projection. Image-to-image registration may also be used to perform simple non-quantitative change detection. * Image-to-map rectification - Useful when preparing images and interpreted output for presentation in a rigorous map projection using a known geoid and datum. Especially valuable when performing digital change detection. There are two basic types of geometric correction: * Image-to-image registration - useful for registering two or more images together when it is not necessary to have the interpreted output in a formal map projection. Image-to-image registration may also be used to perform simple non-quantitative change detection. * Image-to-map rectification - Useful when preparing images and interpreted output for presentation in a rigorous map projection using a known geoid and datum. Especially valuable when performing digital change detection. Jensen, 2003

Image-to-Map Geometric Rectification Image-to-map rectification requires two basic operations: * Spatial Interpolation Using Coordinate Transformation * Intensity Interpolation Image-to-map rectification requires two basic operations: * Spatial Interpolation Using Coordinate Transformation * Intensity Interpolation Jensen, 2003 We will focus our attention on image-to-map rectification because: it is the most widely adopted geometric correction methodology, and it is the most widely adopted geometric correction methodology, and the image-to-image registration process is very similar. the image-to-image registration process is very similar. We will focus our attention on image-to-map rectification because: it is the most widely adopted geometric correction methodology, and it is the most widely adopted geometric correction methodology, and the image-to-image registration process is very similar. the image-to-image registration process is very similar.

Spatial Interpolation Using Coordinate Transformations where: x and y are positions in the output-rectified image or map, and x’ and y’ represent corresponding positions in the original input image. where: x and y are positions in the output-rectified image or map, and x’ and y’ represent corresponding positions in the original input image. For moderate distortions in a relatively small area of an image, a 1 st order, six-parameter, affine transformation is sufficient to rectify the imagery to a geographic frame of reference: Jensen, 2003

Spatial Interpolation Using Coordinate Transformations This first order transformation can model six kinds of distortion in the remote sensor data, including: translation in x and y, translation in x and y, scale changes in x and y, scale changes in x and y, skew, and skew, and rotation. rotation. This first order transformation can model six kinds of distortion in the remote sensor data, including: translation in x and y, translation in x and y, scale changes in x and y, scale changes in x and y, skew, and skew, and rotation. rotation. Jensen, 2003

How Different Affine Transformations Fit a Hypothetical Surface How Different Affine Transformations Fit a Hypothetical Surface Original surface Original surface 1 st order 2 nd order 3 rd order Jensen, 2003

Spatial Interpolation Logic Jensen, 2003 The goal is to fill a matrix that is in a standard map projection with the appropriate values from a non- planimetric image.

Spatial Interpolation Using Coordinate Transformation All of the original GCPs selected are usually not used to compute the final six-parameter coefficients and constants used to rectify the input image. There is an iterative process that takes place. First, all of the original GCPs (e.g., 20 GCPs) are used to compute an initial set of six coefficients and constants. The root mean squared error (RMSE) associated with each of these initial 20 GCPs is computed and summed. Then, the individual GCPs that contributed the greatest amount of error are determined and deleted. After the first iteration, this might only leave 16 of 20 GCPs. A new set of coefficients is then computed using the16 GCPs. The process continues until the RMSE reaches a user-specified threshold (e.g., <1 pixel error in the x - direction and <1 pixel error in the y-direction). The goal is to remove the GCPs that introduce the most error into the multiple-regression coefficient computation. When the acceptable threshold is reached, the final coefficients and constants are used to rectify the input image to an output image in a standard map projection as previously discussed. Jensen, 2003

Spatial Interpolation Using Coordinate Transformation where: x orig and y orig 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 when we utilize the six coefficients. 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 RMS error for all GCPs, it is possible to (1) see which GCPs contribute the greatest error, and 2) sum all the RMS error. where: A way to measure the accuracy of a geometric rectification algorithm (actually, its coefficients) is to compute the Root Mean Squared Error (RMS error ) for each ground control point using the equation: Jensen, 2003

Characteristics of Ground Control Points Point Number Order of Points Deleted Easting on Map X1 Northing on Map Y1 X’ pixel Y’ Pixel Total RMS error after this point deleted ,627, ,6803,627, … ,7003,632, Total RMS error with all 20 GCPs used: If we delete GCP #20, the RMSE will be 8.452

Image-to-Map Geometric Rectification Intensity Interpolation: Unfortunately, geometric correction algorithms rarely direct us to go to an integer row and column in the original imagery (e.g., row 2, column 2) to get a brightness value to fill a location in the rectified output image. Rather, the location is usually a floating point number (e.g., column 2.4, row 2.7). Therefore, intensity interpolation algorithms (often referred to as resampling) are used to obtain a brightness value from the desired location in the original image and then place this value in the output matrix. Unfortunately, geometric correction algorithms rarely direct us to go to an integer row and column in the original imagery (e.g., row 2, column 2) to get a brightness value to fill a location in the rectified output image. Rather, the location is usually a floating point number (e.g., column 2.4, row 2.7). Therefore, intensity interpolation algorithms (often referred to as resampling) are used to obtain a brightness value from the desired location in the original image and then place this value in the output matrix. Intensity Interpolation: Unfortunately, geometric correction algorithms rarely direct us to go to an integer row and column in the original imagery (e.g., row 2, column 2) to get a brightness value to fill a location in the rectified output image. Rather, the location is usually a floating point number (e.g., column 2.4, row 2.7). Therefore, intensity interpolation algorithms (often referred to as resampling) are used to obtain a brightness value from the desired location in the original image and then place this value in the output matrix. Unfortunately, geometric correction algorithms rarely direct us to go to an integer row and column in the original imagery (e.g., row 2, column 2) to get a brightness value to fill a location in the rectified output image. Rather, the location is usually a floating point number (e.g., column 2.4, row 2.7). Therefore, intensity interpolation algorithms (often referred to as resampling) are used to obtain a brightness value from the desired location in the original image and then place this value in the output matrix. Jensen, 2003

Image-to-Map Geometric Rectification There are several intensity interpolation (resampling) algorithms, including: * Nearest Neighbor * Nearest Neighbor * Bilinear Interpolation * Bilinear Interpolation * Cubic Convolution * Cubic Convolution There are several intensity interpolation (resampling) algorithms, including: * Nearest Neighbor * Nearest Neighbor * Bilinear Interpolation * Bilinear Interpolation * Cubic Convolution * Cubic Convolution Jensen, 2003

Nearest-Neighbor Resampling The brightness value closest to the predicted x’, y’ coordinate is assigned to the output x,y coordinate. Jensen, 2003

Bilinear Interpolation Assigns output pixel values by interpolating brightness values in two orthogonal direction in the input image. It basically fits a plane to the 4 pixel values nearest to the desired position ( x’, y’ ) and then computes a new brightness value based on the weighted distances to these points. For example, the distances from the requested ( x’, y’ ) position at 2.4, 2.7 in the input image to the closest four input pixel coordinates (2,2; 3,2; 2,3;3,3) are computed. Also, the closer a pixel is to the desired x’,y’ location, the more weight it will have in the final computation of the average. where Z k are the surrounding four data point values, and D 2 k are the distances squared from the point in question ( x’, y’ ) to the these data points. Jensen, 2003

Bilinear Interpolation Jensen, 2003

Cubic Convolution Assigns values to output pixels in much the same manner as bilinear interpolation, except that the weighted values of 16 pixels surrounding the location of the desired x’, y’ pixel are used to determine the value of the output pixel. where Z k are the surrounding four data point values, and D 2 k are the distances squared from the point in question ( x’, y’ ) to the these data points. Jensen, 2003

Cubic Convolution Jensen, 2003