Error Surfaces

Georegistration is usually a tradeoff between a desire for high accuracy and the operator's patience or willingness to enter many control points or to tolerate long georegistration processing times required by using higher orders. When fewer control points are used the accuracy of georegistration declines. If desired, Manifold can compute a measure of accuracy of transformation when using the numeric method. The accuracy report is presented in the form of a surface where the value at each location in the surface provides a measure of georegistration accuracy at that location.

Checking the Save error surface as dialog box will create a surface in the same coordinate system and the same size as the component being registered. The value of the surface at each location reports the georegistration accuracy by reporting the RMS (root mean square) error at that location. The RMS values are computed by applying an inverse georegistration to the registered surface and then comparing the inverse result to the original component.

Beginning with an original component and then georegistering the component introduces some errors and then reversing the georegistration through an inverse transformation introduces additional errors. The component resulting from georegistration and then an inverse georegistration is therefore different from the original. It can be compared to the original using a root mean square comparison of original and transformed / re-transformed values for X and Y coordinates at each location.

The root mean square computation is reported in units of the source coordinate system. Values range from zero in regions where the transformation results in exact matches to some non-zero value in regions where the transformation and inverse transformation results in imperfect matches. The RMS error values are assembled into a surface called an error surface for convenient display of error values. The RMS error value in each pixel of the error surface is treated as the "height" of the surface at that pixel.

Error surfaces have their View - Display Options set to no shading and no palette. When opened in a surface window the error surface will have black tones in regions of low error transformations and lighter tones in regions of higher error transformations. High accuracy regions will be near control points and lower accuracy regions will be further away from control points.

If greater accuracy is desired we can then add more control points in regions of lighter tones and repeat the georegistration (save a copy of the original component so that the georegistration may be repeated using the original). Error surfaces created during georegistration will inherit control points.

Important: Computing an error surface requires not only the original georegistration transformation but also calculation of an inverse transformation. This requires additional computation and results in a longer georegistration process, approximately doubling the time required to georegister images and surfaces. Do not check the Save error surface as box unless you are willing to wait twice as long for the georegistration process.

Examples

Caution: This example uses some options that are not available in production Manifold releases. It is an artificial example cobbled up using a special version of Manifold created by manifold.net engineering. The special version was used to create illustrations of the effects of the Order parameter that will fit within the very small screen space available for Help illustrations.

The difference between this example and production Manifold releases is that this example uses the Numeric method with an Order of 1. That's not possible in production releases, where the minimum Order usable with the Numeric method is 2. However, using an Order of 1 provides a dramatic, obvious example when compared to an Order of 2 even though a relatively small number of control points (seven) has been used.

These examples use a portion of the SanFran.jpg sample image. The image was cropped and resized to a smaller number of pixels.

We have placed seven control points in the image that match seven control points in the Bay_hydro example drawing.

Clicking the Register button in the control points pane launches the georegistration dialog. We choose the Numeric method, check Save error surface as and press OK. The Order is left at 1. [Note: Remember, this is possible only in the special version of Manifold used for this illustration. Production Manifold releases require a minimum Order of 2].

The image is georegistered to the bay_hydro drawing…

…and a new surface reporting errors is created.

If we open the error surface we see that by default it is seen in grayscale with no shading applied. Dark regions show lower errors and lighter regions show higher errors.

We can choose View - Display Options and assign a palette to the error surface. This also has the side effect of showing the minimum and maximum error values in the surface within the Display Options dialog.

Seen using the Spectrum palette the error surface shows finer visual detail of the distribution of errors. The surface is shown next to the georegistered image.

The error pattern seen in the above example is a very simple pattern that results from our use of a low order, 1, for the numeric georegistration. It's fairly obvious why the Numeric method in production versions of Manifold requires an order of 2 or higher in order to avoid error surfaces like the above. If we were to repeat the georegistration using a higher order a more sophisticated georegistration would occur showing a more detailed error pattern.

To see this effect we can begin with a copy of the original image and use an Order of 2 in the Register dialog, as seen above.

The result is a slightly different appearance in the georegistered image as it is warped into registration with the latitude / longitude projection of the Bay_hydro drawing.

Opening the created error surface we see that it has a different pattern of grayscale tones.

Applying the Spectrum palette we can see that there is a region of high accuracy that correlates well to the placement of control points. The least accuracy occurs in the upper right corner of the image which is farthest from any control points.

Error surfaces are created using the same projection as the georegistered component. This allows us to overlay the error surface in a map with the georegistered component.

The illustration above shows the error surface in a map overlaid upon the georegistered SanFran image together with the Bay_hydro drawing. The opacity of the error surface has been reduced using Layer Opacity so the SanFran image may be partially seen through the error surface. This allows direct comparison between the accuracy implied by the error surface and the georegistered image.

Technical Discussion

As mentioned earlier, the above illustrations were created using a specially engineered version of Manifold that allows use of an Order of 1. Production versions do not allow this. Why was a special version created? There are two reasons why:

First, although production Manifold versions could use the Affine (scale, shift, rotate) method that is algorithmically equivalent to using Numeric with an Order of 1, only the Numeric method allows creation of an error surface since the Affine method is not exactly the same as the Numeric method.

Second, although it would have been possible to show a comparison between using an Order of 2 with an Order of 3. However, in order to use an Order of 3 there would have had to be many more control points in the image, so many that the relatively small sizes of images used for illustrations would have appeared very cluttered. By using a small number of control points any relationships between the pattern of control points and any resulting pattern in the error surface is more easily visible.