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Image Rectification

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Rectification Introduction Registration Rectification Rectification Raw, remotely sensed image data gathered by a satellite or aircraft are representations of the irregular surface of the Earth. Even images of seemingly flat areas are distorted by both the curvature of the Earth and the sensor being used. This chapter covers the processes of geometrically correcting an image so that it can be represented on a planar surface, conform to other images, and have the integrity of a map. A map projection system is any system designed to represent the surface of a sphere or spheroid (such as the Earth) on a plane. There are a number of different map projection methods. Since flattening a sphere to a plane causes distortions to the surface, each map projection system compromises accuracy between certain properties, such as conservation of distance, angle, or area. For example, in equal area map projections, a circle of a specified diameter drawn at any location on the map represents the same total area. This is useful for comparing land use area, density, and many other applications. However, to maintain equal area, the shapes, angles, and scale in parts of the map may be distorted (Jensen, 1996). There are a number of map coordinate systems for determining location on an image. These coordinate systems conform to a grid, and are expressed as X,Y (column, row) pairs of numbers. Each map projection system is associated with a map coordinate system. Rectification is the process of transforming the data from one grid system into another grid system using a geometric transformation. While polynomial transformation and triangle-based methods are described in this chapter, discussion about various rectification techniques can be found in Yang (Yang, 1997). Since the pixels of the new grid may not align with the pixels of the original grid, the pixels must be resampled. Resampling is the process of extrapolating data values for the pixels on the new grid from the values of the source pixels. In many cases, images of one area that are collected from different sources must be used together. To be able to compare separate images pixel by pixel, the pixel grids of each image must conform
Transcript
Page 1: Image Rectification

Rectification

Introduction

Registration

Rectification

Rectification

Raw, remotely sensed image data gathered by a satellite or aircraft arerepresentations of the irregular surface of the Earth. Even images ofseemingly flat areas are distorted by both the curvature of the Earth andthe sensor being used. This chapter covers the processes ofgeometrically correcting an image so that it can be represented on aplanar surface, conform to other images, and have the integrity of amap.

A map projection system is any system designed to represent thesurface of a sphere or spheroid (such as the Earth) on a plane. Thereare a number of different map projection methods. Since flattening asphere to a plane causes distortions to the surface, each mapprojection system compromises accuracy between certain properties,such as conservation of distance, angle, or area. For example, in equalarea map projections, a circle of a specified diameter drawn at anylocation on the map represents the same total area. This is useful forcomparing land use area, density, and many other applications.However, to maintain equal area, the shapes, angles, and scale in partsof the map may be distorted (Jensen, 1996).

There are a number of map coordinate systems for determining locationon an image. These coordinate systems conform to a grid, and areexpressed as X,Y (column, row) pairs of numbers. Each map projectionsystem is associated with a map coordinate system.

Rectification is the process of transforming the data from one gridsystem into another grid system using a geometric transformation.While polynomial transformation and triangle-based methods aredescribed in this chapter, discussion about various rectificationtechniques can be found in Yang (Yang, 1997). Since the pixels of thenew grid may not align with the pixels of the original grid, the pixels mustbe resampled. Resampling is the process of extrapolating data valuesfor the pixels on the new grid from the values of the source pixels.

In many cases, images of one area that are collected from differentsources must be used together. To be able to compare separateimages pixel by pixel, the pixel grids of each image must conform to theother images in the data base. The tools for rectifying image data areused to transform disparate images to the same coordinate system.

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Registration is the process of making an image conform to anotherimage. A map coordinate system is not necessarily involved. Forexample, if image A is not rectified and it is being used with image B,then image B must be registered to image A so that they conform toeach other. In this example, image A is not rectified to a particular mapprojection, so there is no need to rectify image B to a map projection.

Georeferencing

Latitude/Longitude

Orthorectification

258

Georeferencing refers to the process of assigning map coordinates toimage data. The image data may already be projected onto the desiredplane, but not yet referenced to the proper coordinate system.Rectification, by definition, involves georeferencing, since all mapprojection systems are associated with map coordinates. Image-to-image registration involves georeferencing only if the reference imageis already georeferenced. Georeferencing, by itself, involves changingonly the map coordinate information in the image file. The grid of theimage does not change.

Geocoded data are images that have been rectified to a particular mapprojection and pixel size, and usually have had radiometric correctionsapplied. It is possible to purchase image data that is already geocoded.Geocoded data should be rectified only if they must conform to adifferent projection system or be registered to other rectified data.

Lat/Lon is a spherical coordinate system that is not associated with amap projection. Lat/Lon expresses locations in the terms of a spheroid,not a plane. Therefore, an image is not usually rectified to Lat/Lon,although it is possible to convert images to Lat/Lon, and some tips fordoing so are included in this chapter.

You can view map projection information for a particular file usingthe Image Information utility. Image Information allows you to

modify map information that is incorrect. However, you cannot

rectify data using Image Information. You must use the

Rectification tools described in this chapter.

The properties of map projections and of particular map projectionsystems are discussed in "Cartography" on page 217 and "Map

Projections" on page 303.

Orthorectification is a form of rectification that corrects for terraindisplacement and can be used if there is a DEM of the study area. It isbased on collinearity equations, which can be derived by using 3DGCPs. In relatively flat areas, orthorectification is not necessary, but inmountainous areas (or on aerial photographs of buildings), where ahigh degree of accuracy is required, orthorectification is recommended.

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See "Photogrammetric Concepts" on page 609 for moreinformation on orthocorrection.

When to Rectify Rectification is necessary in cases where the pixel grid of the imagemust be changed to fit a map projection system or a reference image.There are several reasons for rectifying image data:

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·

·

·

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comparing pixels scene to scene in applications, such as changedetection or thermal inertia mapping (day and night comparison)

developing GIS data bases for GIS modeling

identifying training samples according to map coordinates prior toclassification

creating accurate scaled photomaps

overlaying an image with vector data, such as ArcInfo

comparing images that are originally at different scales

extracting accurate distance and area measurements

mosaicking images

performing any other analyses requiring precise geographiclocations

Before rectifying the data, you must determine the appropriatecoordinate system for the data base. To select the optimum mapprojection and coordinate system, the primary use for the data basemust be considered.

If you are doing a government project, the projection may bepredetermined. A commonly used projection in the United Statesgovernment is State Plane. Use an equal area projection for thematicor distribution maps and conformal or equal area projections forpresentation maps. Before selecting a map projection, consider thefollowing:

Rectification

·

·

How large or small an area is mapped? Different projections areintended for different size areas.

Where on the globe is the study area? Polar regions and equatorialregions require different projections for maximum accuracy.

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· What is the extent of the study area? Circular, north-south, east-west, and oblique areas may all require different projection systems(Environmental Systems Research Institute, 1992).

When to GeoreferenceOnly

Rectification is not necessary if there is no distortion in the image. Forexample, if an image file is produced by scanning or digitizing a papermap that is in the desired projection system, then that image is alreadyplanar and does not require rectification unless there is some skew orrotation of the image. Scanning and digitizing produce images that areplanar, but do not contain any map coordinate information. Theseimages need only to be georeferenced, which is a much simplerprocess than rectification. In many cases, the image header can simplybe updated with new map coordinate information. This involvesredefining:

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·

the map coordinate of the upper left corner of the image

the cell size (the area represented by each pixel)

This information is usually the same for each layer of an image file,although it could be different. For example, the cell size of band 6 ofLandsat TM data is different than the cell size of the other bands.

Use the Image Information utility to modify image file headerinformation that is incorrect.

Disadvantages ofRectification

During rectification, the data file values of rectified pixels must beresampled to fit into a new grid of pixel rows and columns. Althoughsome of the algorithms for calculating these values are highly reliable,some spectral integrity of the data can be lost during rectification. If mapcoordinates or map units are not needed in the application, then it maybe wiser not to rectify the image. An unrectified image is more spectrallycorrect than a rectified image.

Classification

Some analysts recommend classification before rectification, since theclassification is then based on the original data values. Another benefitis that a thematic file has only one band to rectify instead of the multiplebands of a continuous file. On the other hand, it may be beneficial torectify the data first, especially when using GPS data for the GCPs.Since these data are very accurate, the classification may be moreaccurate if the new coordinates help to locate better training samples.

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Thematic Files

Nearest neighbor is the only appropriate resampling method forthematic files, which may be a drawback in some applications. Theavailable resampling methods are discussed in detail later in thischapter.

Rectification Steps NOTE: Registration and rectification involve similar sets of procedures.Throughout this documentation, many references to rectification alsoapply to image-to-image registration.

Usually, rectification is the conversion of data file coordinates to someother grid and coordinate system, called a reference system. Rectifyingor registering image data on disk involves the following general steps,regardless of the application:

1. Locate GCPs.

2. Compute and test a transformation.

3. Create an output image file with the new coordinate information in theheader. The pixels must be resampled to conform to the new grid.

Images can be rectified on the display (in a Viewer) or on the disk.Display rectification is temporary, but disk rectification is permanent,because a new file is created. Disk rectification involves:

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·

rearranging the pixels of the image onto a new grid, which conformsto a plane in the new map projection and coordinate system

inserting new information to the header of the file, such as the upperleft corner map coordinates and the area represented by each pixel

ResamplingMethods

The next step in the rectification/registration process is to create theoutput file. Since the grid of pixels in the source image rarely matchesthe grid for the reference image, the pixels are resampled so that newdata file values for the output file can be calculated.

Figure 83: Resampling

GCP

GCP

1. The input image withsource GCPs.

GCP

GCP

2. The output grid, withreference GCPs shown.

3. To compare the two grids, theinput image is laid over theoutput grid, so that the GCPsof the two grids fit together.

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4. Using a resampling method,the pixel values of the inputimage are assigned to pixelsin the output grid.

The following resampling methods are supported in ERDAS IMAGINE:

282

· Nearest Neighbor on page 283—uses the value of the closest pixelto assign to the output pixel value.

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·

·

·

Bilinear Interpolation on page 284—uses the data file values offour pixels in a 2 × 2 window to calculate an output value with abilinear function.

Cubic Convolution on page 288—uses the data file values ofsixteen pixels in a 4 × 4 window to calculate an output value with acubic function.

Bicubic Spline Interpolation on page 290—fits a cubic splinesurface through the current block of points.

In all methods, the number of rows and columns of pixels in the outputis calculated from the dimensions of the output map, which isdetermined by the geometric transformation and the cell size. Theoutput corners (upper left and lower right) of the output file can bespecified. The default values are calculated so that the entire source fileis resampled to the destination file.

If an image to image rectification is being performed, it may bebeneficial to specify the output corners relative to the reference filesystem, so that the images are coregistered. In this case, the upper leftX and upper left Y coordinate are 0,0 and not the defaults.

If the output units are pixels, then the origin of the image is theupper left corner. Otherwise, the origin is the lower left corner.

Rectifying to Lat/Lon

Nearest Neighbor

Rectification

You can specify the nominal cell size if the output coordinate system isLat/Lon. The output cell size for a geographic projection (i.e., Lat/Lon)is always in angular units of decimal degrees. However, if you want thecell to be a specific size in meters, you can enter meters and calculatethe equivalent size in decimal degrees. For example, if you want theoutput file cell size to be 30 × 30 meters, then the program wouldcalculate what this size would be in decimal degrees and automaticallyupdate the output cell size. Since the transformation between angular(decimal degrees) and nominal (meters) measurements varies acrossthe image, the transformation is based on the center of the output file.

Enter the nominal cell size in the Nominal Cell Size dialog.

To determine an output pixel’s nearest neighbor, the rectified

coordinates (xo, yo) of the pixel are retransformed back to the sourcecoordinate system using the inverse of the transformation. The

retransformed coordinates (xr, yr) are used in bilinear interpolation andcubic convolution as well. The pixel that is closest to the retransformed

coordinates (xr, yr) is the nearest neighbor. The data file value(s) forthat pixel become the data file value(s) of the pixel in the output image.

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Advantages Disadvantages

Transfers original data valueswithout averaging them as the othermethods do; therefore, theextremes and subtleties of the datavalues are not lost. This is animportant consideration whendiscriminating between vegetationtypes, locating an edge associatedwith a lineament, or determiningdifferent levels of turbidity ortemperatures in a lake(Jensen, 1996).

When this method is used toresample from a larger to a smallergrid size, there is usually a stairstepped effect around diagonal linesand curves.

Suitable for use beforeclassification.

Data values may be dropped, whileother values may be duplicated.

The easiest of the three methods tocompute and the fastest to use.

Using on linear thematic data (e.g.,roads, streams) may result in breaksor gaps in a network of linear data.

Appropriate for thematic files, whichcan have data file values based ona qualitative (nominal or ordinal)system or a quantitative (interval orratio) system. The averaging that isperformed with bilinear interpolationand cubic convolution is not suitedto a qualitative class value system.

Figure 84: Nearest Neighbor

(xr,yr)

nearest to

(xr,yr)

Bilinear Interpolation

284

In bilinear interpolation, the data file value of the rectified pixel is based

upon the distances between the retransformed coordinate location (xr,

yr) and the four closest pixels in the input (source) image (see Figure85). In this example, the neighbor pixels are numbered 1, 2, 3, and 4.Given the data file values of these four pixels on a grid, the task is to

calculate a data file value for r (Vr).

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data

file

val

ues

Vm = ------------------ dy + V1

Figure 85: Bilinear Interpolation

1

dy

2

mdx

r n

(xr,yr)

3D

4

r is the location of the retransformed coordinate

To calculate Vr, first Vm and Vn are considered. By interpolating Vm and

Vn, you can perform linear interpolation, which is a simple process toillustrate. If the data file values are plotted in a graph relative to theirdistances from one another, then a visual linear interpolation is

apparent. The data file value of m (Vm) is a function of the change in the

data file value between pixels 3 and 1 (that is, V3 - V1).

Figure 86: Linear InterpolationCalculating a data file value as a function

of spatial distance between two pixels

V3

Vm

V1

(V3 - V1) / D

Y1 Ym Y3

Ddata file coordinates

(Y)

The equation for calculating Vm from V1 and V3 is:

V3 – V1

D

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Vn = ------------------ dy + V2

Vr = ------------------- dx + Vm

------------------ dy + V2 – -----------------1- dy + V1

Vr = ------------------------------------------------------------------------------------------------------- dx + ------------------ dy + V1

V1 D – dx D – dy + V2 dx D – dy + V3 D – dx dy + V4 dx dyVr = --------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Where:

Yi

Vi

dy

D

= the Y coordinate for pixel i

= the data file value for pixel i

= the distance between Y1 and Ym in the source coordinatesystem

= the distance between Y1 and Y3 in the source coordinatesystem

If one considers that (V3 - V1 / D) is the slope of the line in the graph

above, then this equation translates to the equation of a line in y = mx +b form.

Similarly, the equation for calculating the data file value for n (Vn) in thepixel grid is:

V4 – V2

D

From Vn and Vm, the data file value for r, which is at the retransformed

coordinate location (xr,yr),can be calculated in the same manner:

Vn – Vm

D

The following is attained by plugging in the equations for Vm and Vn to

this final equation for Vr :

V4 – V2 V3 – VD D V3 – V1

D D

D2

In most cases D = 1, since data file coordinates are used as the sourcecoordinates and data file coordinates increment by 1.

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Advantages Disadvantages

Results in output images that aresmoother, without the stair-steppedeffect that is possible with nearestneighbor.

Since pixels are averaged, bilinearinterpolation has the effect of a low-frequency convolution. Edges aresmoothed, and some extremes of thedata file values are lost.

More spatially accurate than nearestneighbor.

This method is often used whenchanging the cell size of the data,such as in SPOT/TM merges withinthe 2 × 2 resampling matrix limit.

D –∆xi D –∆yi∑ ---------------------------------------------- Vi

D

Some equations for bilinear interpolation express the output data filevalue as:

Vr = ∑ wiVi

Where:

wi is a weighting factor

The equation above could be expressed in a similar format, in which the

calculation of wi is apparent:

Where:

Vr =

4

i = 1

∆xi

∆yi

Vi

D

= the change in the X direction between (xr,yr) and the datafile coordinate of pixel i

= the change in the Y direction between (xr,yr) and the datafile coordinate of pixel i

= the data file value for pixel i

= the distance between pixels (in X or Y) in the sourcecoordinate system

For each of the four pixels, the data file value is weighted more if the

pixel is closer to (xr, yr).

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See "Enhancement" on page 459 for more about convolutionfiltering.

Cubic Convolution Cubic convolution is similar to bilinear interpolation, except that:

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·

a set of 16 pixels, in a 4 × 4 array, are averaged to determine theoutput data file value, and

an approximation of a cubic function, rather than a linear function,is applied to those 16 input values.

To identify the 16 pixels in relation to the retransformed coordinate

(xr,yr), the pixel (i,j) is used, such that:

i = int (xr)j = int (yr)

This assumes that (xr,yr) is expressed in data file coordinates (pixels).

The pixels around (i,j) make up a 4 × 4 grid of input pixels, as illustratedin Figure 87.

Figure 87: Cubic Convolution

(i,j)

(Xr,Yr)

Since a cubic, rather than a linear, function is used to weight the 16

input pixels, the pixels farther from (xr, yr) have exponentially less

weight than those closer to (xr, yr).

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a + 2 x 3 – a + 3 x 2 + 1

f x = a x 3 – 5a x 2 + 8a x – 4a

Several versions of the cubic convolution equation are used in the field.Different equations have different effects upon the output data filevalues. Some convolutions may have more of the effect of a low-frequency filter (like bilinear interpolation), serving to average andsmooth the values. Others may tend to sharpen the image, like a high-frequency filter. The cubic convolution used in ERDAS IMAGINE is acompromise between low-frequency and high-frequency. The generaleffect of the cubic convolution depends upon the data.

The formula used in ERDAS IMAGINE is:

4

Vr = ∑ V i – 1 j + n – 2 f d i – 1 j + n – 2 + 1 +n = 1

V i j + n – 2 f d i j + n – 2 +

V i + 1 j + n – 2 f d i + 1 j + n – 2 – 1 +

Where:

i

j= int (xr)

= int (yr)

d(i,j) = the distance between a pixel with coordinates (i,j) and

(xr,yr)

V(i,j)

Vr

a

f(x)

= the data file value of pixel (i,j)

= the output data file value

= -1 (a constant)

= the following function:

Rectification

0

Source: Atkinson, 1985

if x 1

if 1 x 2otherwise

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Advantages Disadvantages

Uses 4 × 4 resampling. In mostcases, the mean and standarddeviation of the output pixels matchthe mean and standard deviation ofthe input pixels more closely thanany other resampling method.

Data values may be altered.

The effect of the cubic curveweighting can both sharpen theimage and smooth out noise(Atkinson, 1985). The actualeffects depend upon the data beingused.

This method is extremely slow.

This method is recommended whenyou are dramatically changing thecell size of the data, such as inTM/aerial photo merges (i.e.,matches the 4 × 4 window moreclosely than the 2 × 2 window).

Bicubic SplineInterpolation

Bicubic Spline Interpolation is based on fitting a cubic spline surfacethrough the current block of points. The output value is derived from thefitting surface that will retain the values of the known points. Thisalgorithm is much slower than other methods of interpolation, but it hasthe advantage of giving a more exact fit to the curve without theoscillations that other interpolation methods can create. Bicubic SplineInterpolation is so similar to Bilinear Interpolation that unless you havethe need to maximize surface smoothness, you should use BilinearInterpolation.

Data Points

The known data points are an array of raster of m × n,

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