An Analysis of the Size of the Protest Sign on North Table Mountain
29 August 2008
By Edmond W. Holroyd, III, Ph.D.
[Address & phone withheld]
eholroyd@juno.com
In the mid-morning of 26 August, 2008, I photographed the protest sign on North Table Mountain from a position somewhat north of my home. The location was east of Indiana Street on the northeast side bank of the Croke Canal. The position was N 39 deg 47' 45.7", E 105 deg 09' 51.7" (GPS measurement). The distance from there to the center of the sign was about 2875 meters (9432 feet, 1.79 miles). The viewing direction was approximately perpendicular to the general slope of the mountain where the sign was located. That geometry increases the accuracy of the measurements compared to a view looking at a great angle to the slope.
Qualifications related to the measurement:
In late September 2005 I retired from the U.S. Bureau of Reclamation after 31 years of service as a research scientist. Since 1988 I was the leading scientist in Reclamation's remote sensing group. Since January 1999 I have been teaching graduate courses in remote sensing and image processing at the University of Denver, University College. Since June 2005 I have been teaching similar courses at the Denver (and Buckley AFB) campus of Webster University. At both universities I am an adjunct professor.
The analysis was done with the TNTmips, version 7.3, software from MicroImages, Inc. (http://www.microimages.com) I have used their software since about 1989. I provide the free TNTlite version to my graduate students for their homework assignments.
Data sources:
My photograph was automatically named P8261036.jpg by the Olympus Evolt E-500 8 Megapixel digital camera using a 40-150 mm zoom lens at the 150 mm setting. The photo is 2448 pixels tall and 3264 pixels wide.
An orthophotograph of the region was obtained from USGS at 1 meter pixel resolution. A digital elevation model (DEM) was obtained at 1/3 arc second (latitude / longitude) resolution from USGS. These were used for absolute calibration of the other sources. The DEM was analyzed to create elevation contours at 10 meter intervals.
Screen captures from Google Earth were used for a finer resolution and more modern image (2007) of the mountain region. Upon georeferencing using the USGS sources the Google Earth imagery had a resolution of 0.535 meters/pixel.
Analysis procedures:
My photo was georeferenced to the Google Earth image using the standard techniques in the TNTmips 7.3 software. That involves identifying landmarks (light and dark patches that serve as control points) visible in both images. The file Georef2Poly.gif is a screen capture showing my photo on the left and the Google Earth image on the right. The numbered red crosses show the locations of the landmarks used.
Two different "models" were used to resample the photo. A second degree polynomial model has a slight curvature in a parabolic sense, which allows for the surface of the mountain to differ from being purely flat though tilted. A plane projective model assumes that a flat tilted surface is being viewed. Both models are not perfect for the mountain slope but are close to ideal.
In the middle of the Georef2poly.gif image are two columns of numbers showing the residual errors between actual and predicted positions. They are ordered according to descending total residual error. Upon examination, the direction of the residual errors was mostly along the line of sight (x-axis) and therefore is a result of small changes in the slope of the mountainside. Differences right and left (y-axis) were much smaller. An initial resampling was done with a set of control points having y-axis residuals less than 2 meters. A subsequent resampling was done after some control points were removed and others adjusted to less than 1 meter y-axis residuals. The second set of control points was then used for the plane projective model. The different geometry of the plane projective resampling increased the residual errors because the mountainside is not perfectly flat. (See GeorefPP.gif)
The resampling of the photo, as seen in the left side of Georef2poly.gif, orients the image so that north is at the top and features are in correct map positions, as if looking at the sign from a vertical position. Distance measurements along the elevation contours (along the sequence of word letters) are then accurate. Distance measurements from the base of the sign to its top must take into account the tilted mountainside surface. There were 3 such resamplings: two for the second degree polynomial model and one for the plane projective model. All resamplings resulted in pixel sizes of 0.1 meters. For an unknown reason, the polynomial resamplings have fuzzy patches in places, but fortunately not where they affect measurements. The plane projective resampling does not have such artifacts.
Next the orientations of the 10-meter elevation contours at the location of the sign were measured. The sign spanned elevations 1845 to 1920 meters, for a total rise of 75 meters.
Elevation: 1920 1910 1900 1890 1880 1870 1860 1850 1840 meters
Heading: 340.61 341.67 342.54 343.49 344.21 342.18 339.97 340.19 339.72 deg.
The average heading is 341.62 degrees. Opposite and perpendicular directions are 71.62, 161.62, and 251.62 degrees. These four directions were then used to construct rectangles tightly around the sign letterings.
For each resampled image the line and column coordinates of a pixel on the edge of 4 letters were selected: the left edge of the "u" in the middle line; the top edge of the "D" in the top line; the bottom edge of the "n" in the bottom line; and the right edge of the "s" in the top line. The three resamplings have different coordinates for such positions. From each of those coordinates lines were drawn on the image in opposite pairs of headings so that straight lines would result in the sides of rectangles enclosing the sign letters and just touching the letter edges. The lines were intentionally drawn longer than needed so that they crossed each other at the corners of the rectangles. The following table shows the line and column coordinates of the resulting rectangle corners. (In image processing, lines are numbered from the image top to the bottom. Columns are numbered from left to right.) The corners are labeled as if viewing the sign as in the photo.
Rectangle corner coordinates (line, column)
Corner Initial polynomial Refined polynomial Plane projective
upper right (1528, 1829) (1518, 1830) (1502, 1898)
lower right (1085, 3163) (1076, 3162) (1062, 3224)
lower left (3060, 3819) (3048, 3816) (3030, 3878)
upper left (3503, 2486) (3491, 2482) (3470, 2551)
image Boxed2p.gif Boxed2P2.gif BoxedPP.gif screen captures
To determine the width of the sign, right to left, the long lengths of the rectangles were calculated as the hypotenuse of right angle triangles. The differences in corner line numbers and in column numbers were squared and added. The square root was then taken. Such lengths are in pixel units. Dividing by 10 gives them in meters because of the resampling scale of 0.1 meters/pixel. (Note: the numbers below should be rounded to no more significance than 0.1 meters.)
Sign widths (long edges of rectangles)
side Initial polynomial Refined polynomial Plane projective Average
top 208.141 m 207.791 m 207.351 m
bottom 208.110 m 207.762 m 207.382 m
average 208.126 m 207.778 m 207.366 m 207.757 m
682.83 ft 681.69 ft 680.33 ft 681.62 ft
This is slightly longer than the target 666 feet of the design.
The sign height as projected on a horizontal surface was calculated by the same hypotenuse formula applied to the corner coordinates. To get the sign height on a tilted surface tangential to the mountainside, the "horizontal" height must be squared and added to the square of the 75 meter rise in elevation. Then the square root provides the sign height on the tilted surface.
Sign heights (short edges of rectangles)
side Initial polynomial Refined polynomial Plane projective Average
left 140.468 m 140.563 m 139.804 m
right 140.563 m 140.342 m 139.710 m
average 140.516 m 140.453 m 139.757 m
Sign heights tilted with 75 meter rise
159.28 m 159.22 m 158.61 m 159.04 m
522.57 ft 522.38 ft 520.37 ft 521.77 ft
This is slightly less than the target 530 feet of the design.
RESULT: (average of three calculations, appropriate to a flat surface tangential to the mountainside.)
The sign had a width of 207.8 meters (681.6 feet) and a height of 159.0 meters (521.8 feet).