5. Estimating spatial resolution of astronaut photographs
In order to use astronaut
photographs for digital remote sensing, it is important to be able to
calculate the equivalent to an IFOV—the ground area represented
by a single pixel in a digitised orbital photograph. The obliquity of
most photographs means that pixel 'sizes' vary at different places in
an image. Given a ground distance (D) represented by a photograph in each
direction (horizontal, vertical) an approximate average pixel width (P,
the equivalent of IFOV) for the entire image can be calculated as follows:
(2)
where D is the projected distance on the ground covered
by the image in the same direction as the pixel is measured, d is the
actual width of the image on the original film (table 1), and S is the
digitising spatial resolution.
Here, we present three
mathematical formulations for estimating the size of the footprint, or
area on the ground covered by the image. Example results from the application
of all three formulations are in table 5. The first and simplest calculation
(formulation 1) gives an idea of the maximum spatial resolution attainable
at a given altitude of orbit with a given film format and a perfectly
vertical (nadir) view downward. Formulation 2 takes into account obliquity
by calculating look angle from the difference between the location on
the ground represented at the centre of the photograph, and the nadir
location of the spacecraft at the time the photograph was taken (figure
1). Formulation 3 describes an alternate solution to the oblique look
angle problem using coordinatesystem transformations. This formulation
has been implemented in a documented spreadsheet and is available for
download (http://eol.jsc.nasa.gov/sseop/Low_Oblique_301_Locked.xls)
and has been implemented on our Webbased user interface
to the Astronaut Photography Database.
Although formulations 2
and 3 account for obliquity, for purposes of calculation they treat the
position of the spacecraft and position of the camera as one. In actuality,
astronauts are generally holding the cameras by hand (although cameras
in bracketed in the window are also used), and the selection of window,
position of the astronaut in the window, and rotation of the camera relative
to the movement of the spacecraft are not known. Thus, calculations using
only the photo centre point (PC) and spacecraft nadir point (SN) give
a locator ellipse and not the locations of the corners of the photograph.
A locator ellipse describes an estimated area on the ground that is likely
to be included in a specific photograph regardless of the rotation of
the film plane about the camera's optical axis (figure 6).
Estimating
the corner positions of the photo requires additional user input of a
single auxiliary pointa location on the image that has a know location
on the ground. Addition of this auxiliary point is an option available
to users of the spreadsheet. An example of the results of adding an auxiliary
point is shown in figure 7 with comparisons of the various calculations
in table 5.
5.1. Formulation 1. Footprint for a nadir view.
The simplest way to estimate footprint size
is to use the geometry of camera lens and spacecraft altitude to calculate
the scaling relationship between the image in the film and the area covered
on the ground. For a perfect nadir view, the scale relationship from the
geometry of similar triangles is
(3)
where d = original image size, D = distance of footprint
on the ground, f = focal length of lens, and H = altitude. Once D is known,
pixel size (length = width) can be calculated from equation 2. These calculations
represent the minimum footprint and minimum pixel size possible for a
given camera system, altitude and digitising spatial resolution (table
2). Formulation 1 was used for calculating minimum pixel sizes shown in
tables 2 and 4.
By
assuming digitising at 2400 ppi (10.6 mm/pixel), currently a spatial resolution
commonly attainable from multipurpose colour transparency scanners (see
section 4.4.1, above), we used this formulation to convert area covered
to an IFOV equivalent for missions of different altitudes (table 2). Table
4 provides a comparison of IFOV of images from various satellites, including
the equivalent for astronaut photography. Values in this table were derived
by using formulation 1 to estimate the area covered because a perfect
nadir view represents the best possible spatial resolution and smallest
field of view that could be obtained.
5.2. Formulation 2. Footprint for oblique views using simplified
geometry and the great circle distance
A more realistic approach to determining
the footprint of the photograph accounts for the fact that the camera
is not usually pointing perfectly down at the nadir point. The look angle
(the angle off nadir that the camera is pointing) can be calculated trigonometrically
by assuming a spherical earth and calculating the distance between the
coordinates of SN and PC (figure 1) using the Great Circle distance, haversine
solution (Sinnott 1984, Snyder 1987:3032, Chamberlain 1996). The difference
between the spacecraft centre and nadir point latitudes, Dlat = lat2 
lat1, and the difference between the spacecraft centre and nadir point
longitudes, Dlon = lon2  lon1, enter the following equations:
Assuming that the camera was positioned so that the imaginary
line between the centre and nadir points (the principal line) runs vertically
through the centre of the photograph, the distance between the geometric
center of the photograph (principal point, PP) and the top of the photograph
is d/2 (figure 8, A). The scale at any point in the photograph varies
as a function of the distance, y, along the principal line between the
isocentre and the point according to the relationship
(7)
(Wong 1980, equation 2.14, H >> the ground elevation, h ). Using
figure 8, A, at the top of the photo,
(8)
and at the bottom of the photo,
(9)
Thus, for given PC and SN coordinates and assuming a
photo orientation like figure 8A and not figure 8B, we can estimate a
minimum D (using equations 7 and 8) and a maximum D (using equations 7
and 9), and then average the two to determine the pixel size (P) via equation
2.
5.3. Formulation 3. The Low Oblique Space Photo
Footprint Calculator.
The Low Oblique Space Photo Footprint Calculator was developed to provide
a more accurate estimation of the geographic coordinates for the footprint
of a low oblique photo of the Earth's surface taken from a humanoccupied
spacecraft in orbit. The calculator performs a series of 3dimensional
coordinate transformations to compute the location and orientation of
the centre of the photo exposure plane relative to an earth referenced
coordinate system. The nominal camera focal length is then used to create
a vector from the photo's perspective point, through each of eight points
around the parameter of the image, as defined by the format size. The
geographical coordinates for the photo footprint are then computed by
intersecting these photo vectors with a spherical earth model. Although
more sophisticated projection algorithms are available, no significant
increase in the accuracy of the results would be produced by these algorithms
due to inherent uncertainties in the available input data (i.e. the spacecraft
altitude, photo centre location, etc.)
The calculations were initially
implemented within a Microsoft Excel workbook, which allowed us to embed
the mathematical and graphical documentation next to the actual calculations.
Thus, interested users can inspect the mathematical processing. A set
of error traps was also built into the calculations to detect erroneous
results. A summary of any errors generated is reported to the user with
the results of the calculations. Interested individuals are invited to
download the Excel workbook from http://eol.jsc.nasa.gov/sseop/Low_Oblique_301_Locked.xls.
The calculations are currently being encoded in a high level programming
language and should soon be available alongside other background data
provided for each photograph at Office of Earth Sciences (2000).
For the purposes of these
calculations, we defined a low oblique photograph as one with the centre
within 10 degrees of latitude and longitude of the spacecraft nadir point.
For the typical range of spacecraft altitudes to date, this restricted
the calculations to photographs in which Earth’s horizon does not
appear (the general definition of a low oblique photograph, e.g. Campbell
1996:71, and figure 4).
5.3.1 Input data and results
Upon opening the Excel workbook, the user
is presented with a program introduction providing instructions for using
the calculator. The second worksheet tab ("HowToUse"), provides
detailed stepbystep instructions for preparing the baseline data for
the program. The third tab ("InputOutput") contains the user
input fields and displays the results of the calculations. The additional
worksheets contain the actual calculations and program documentation.
Although users are welcome to review these sheets, an experienced user
need only access the "InputOutput" spreadsheet.
To begin a calculation
the user enters the following information which is available for each
photo in the NASA Astronaut Photography Database: (1) SN, geographical
coordinates of spacecraft nadir position at the time of photo; (2) H,
spacecraft altitude; (3) PC, the geographical coordinates of the centre
of the photo; (4) f, nominal focal length; and (5) d, image format size.
The automatic implementation of the workbook on the web will automatically
enter these values and complete calculations.
For more accurate results
the user may optionally enter the geographic coordinates and orientation
for an auxiliary point on the photo, which resolves the camera's rotation
uncertainty about the optical axis. The auxiliary point data must be computed
by the user following the instruction contained in the "HowToUse"
tab of the spreadsheet.
After entering the input
data the geographic coordinates of the photo footprint (i.e. four photo
corner points, four points at the bisector of each edge and the centre
of the photo) are immediately displayed below the input fields along with
any error messages generated by the user input or by the calculations
(figure 7). Although results are computed and displayed, they should not
be used when error messages are produced by the program. The program also
computes the tilt angle for each of the photo vectors relative to the
spacecraft nadir vector. To the right of the photo footprint coordinates
is displayed the arc distance along the surface of the sphere between
adjacent computed points.
5.3.2 Calculation Assumptions
The mathematical calculations implemented
in the Low Oblique Space Photo Footprint Calculator use the following
assumptions:
 The SN location is used as exact, even though the true value may
vary by up to ± 0.5 from the location provided with the photo.
 The spacecraft altitude is used as exact. Although our determination
of the nadir point at the instant of a known spacecraft vector is relatively
precise (± 1.15 ´ 104°), degrees), the propagator
interpolates between sets of approximately 10  40 known vectors per
day, and the time code recorded on the film can drift. Thus, the true
value for SN may vary by up to ± 0.1 ° degree from the value
provided with the photo.
 The perspective centre of the camera is assumed to be at the given
altitude over the specified spacecraft nadir location at the time of
photo exposure.
 The PC location is used as exact, even though the true value may
vary by up to ±0.5 ° latitude and ±0.5 ° longitude
from the location provided with the photo.
 A spherical earth model is used with a nominal radius of 6,372,161.54
m (a common first order approximation for a spherical earth used in
geodetic computations).
 The nominal lens focal length of the camera lens is used in the computations
(calibrated focal length values are not available).
 The photo projection is based on the classic pinhole camera model.
 No correction for lens distortion or atmospheric reflection is made.
 If no auxiliary point data is provided, the "Top of the Image"
is oriented perpendicular to the vector from SN towards PC.
5.3.3 Transformation from Earth to photo coordinate systems
The calculations begin by converting the
geographic coordinates (latitude and longitude) of the SN and PC to a
Rectangular EarthCentred Coordinate System (REarth), defined as shown
in figure 9 (with the centre of the Earth at [0, 0, 0]). Using the vector
from the Earth's centre through SN and the spacecraft altitude, the spacecraft
location (SC) is also computed in REarth.
For ease of computation, we next define
a Rectangular SpacecraftCentred Coordinate System (RSpacecraft), as
shown in figure 9. The origin of RSpacecraft is located at SC, with its
+Z axis aligned with vector from the centre of the Earth through SN and
its +X axis aligned with the vector from SN to PC (figure 9). The specific
rotations and translation used to convert from the REarth to the RSpacecraft
are computed and documented in the spreadsheet.
With the mathematical positions of the
SC, SN, PC, and the centre of the Earth computed in RSpacecraft, the
program next computes the location of the camera's principal point (PP).
The principle point is the point of intersection of the optical axis of
the lens with the image plane (i.e. the film). It is nominally positioned
at a distance equal to the focal length from the perspective centre of
the camera (which is assumed to be at SC) along the vector from PC through
SC, as shown in figure 9.
We next create a third coordinate system,
the Rectangular Photo Coordinate System (RPhoto) with its origin at PP,
its XY axial plane normal to the vector from PC through SC and its +X
axis aligned with the +X axis of RSpacecraft, as shown in figure 9. The
XY plane of this coordinate system represents the image plane of the
photograph.
5.3.4 Auxiliary Point Calculations
The calculations above employ a critical
assumption that all photos are taken with the "top of the image"
oriented perpendicular to the vector from the SN towards PC, as shown
in figure 9. To avoid nonuniform solar heating of the external surface,
most orbiting spacecraft are slowly and continually rotated about one
or more axes. In this condition a flight crew member taking a photo while
floating in microgravity could orient the photo with practically any orientation
relative to the horizon (see also figure 8, B). Unfortunately, since these
photos are taken with conventional hand held cameras there is no other
information available which can be used to resolve the photo's rotational
ambiguity about the optical axis, other then the photo itself. This is
why the above assumption is used and the footprint computed by this calculator
is actually a "locator ellipse", which estimates the area on
the ground what is likely to be included in a specific photograph (see
figure 6). This locator ellipse is most accurate for square image formats
and subject to additional distortion as the photograph format becomes
more rectangular.
If the user wants a more precise calculation
of the image footprint, the photo's rotational ambiguity about the optical
axis must be resolved. This can be done in the calculator by adding data
for an auxiliary point. Detailed instructions regarding how to prepare
and use auxiliary point data in the computations are included in the "HowToUse"
tab of the spreadsheet. Basically, the user determines which side of the
photograph is top, and then measures the angle between the line from PP
to the top of the photo and from PP to the auxiliary point on the photo
(figure 9).
If the user includes data for an auxiliary
point, a series of computations are completed to resolve the photo rotation
ambiguity about the optical axis (i.e. the +Z axis in Rphoto). A vector
from the Auxiliary Point on the Earth (AE) through the photograph perspective
centre (located at SC) is intersected with the photo image plane (XY
plane of RPhoto) to compute the coordinates of the Auxiliary Point on
the Photo (AP) in RPhoto. A twodimensional angle in the XY plane of
RPhoto, from the –Xaxis to a line from PP to AP is calculated,
as shown in figure 9. The –Xaxis is used as the origin of the angle
since it represents the top of the photo once it passes through the perspective
centre. The difference between the computed angle and the angle measured
by the user on the photo resolves the ambiguity in the rotation of the
photo relative to the principal line (figures 7 and 9). The transformations
from RSpacecraft and the RPhoto are then modified to include an additional
rotation angle about the +Zaxis in RPhoto.
5.3.5 "Footprint" Calculations
The program next computes the coordinates
of eight points about the perimeter of the image format (i.e. located
at the four photo corners, plus a bisector point along each edge of the
image) . These points are identified in RPhoto based upon the photograph
format size and then converted to RSpacecraft. Since all computations
are done in orthogonal coordinate systems, the RSpacecraft to RPhoto
rotation matrix is transposed to produce an RPhoto to RSpacecraft rotation
matrix. Once in RSpacecraft, a unit vector from each of the eight perimeter
points, through the photo perspective centre (the same point as SC), is
computed. This provides the coordinates for points about the perimeter
of the image format, with their direction vectors in a common coordinate
system with other key points needed to compute the photo footprint.
The next step is to compute the point of
intersection between the spherical earth model and each of the eight perimeter
point vectors. The scalar value for each perimeter point unit vector is
computed using twodimensional planar trigonometry. An angle g is computed
using the formula for the cosine of an angle between two included vectors
(the perimeter point unit vector and the vector from SC to the centre
of the Earth). Angle y is computed using the Law of Sines. Angle e = 180
degrees  g y. The scalar value of the perimeter point vector is computed
using e and the Law of Cosines. The scalar value is then multiplied by
the perimeter point unit vector to produce the threedimensional point
of intersection of the vector with Earth's surface in RSpacecraft. The
process is repeated independently for each of the eight perimeter point
vectors. Aside from its mathematical simplicity, the value of arcsine
(computed in step 2) will exceed the normal range for the sin of an angle
when the perimeter point vectors fail to intersect with the surface of
the earth. A simple test, based on this principle, allows the program
to correctly handle oblique photos which image a portion of the horizon
(see results for high oblique photographs in Table 5).
The final step in the process converts
the eight earth intersection points from the RSpacecraft to REarth.
The results are then converted to the standard geographic coordinate system
and displayed on the "InputOutput" page of the spreadsheet.
5.4. Examples
We applied all three formulations to the
photographs included in this paper and the results are compared in table
5. Formulation 1 gives too small of a value for distance across the photograph
(D) for all but the most nadir shots, and thus serves as an indicator
of the best theoretical case, but is not a good measure for a specific
photograph. For example, the photograph of Lake Eyre taken from 276 km
altitude (figure 2, A) and the photograph of Limmen Bight (figure 7),
were closest to being nadir views (offsets < 68 km or t < 15°,
table 5). For these photographs, D calculated using Formulation 1 was
similar to the minimum D calculated using Formulations 2 and 3. For almost
all other more oblique photographs, Formulation 1 gave a significant underestimate
of the distance covered in the photograph. For figure 5, A (the picture
of Houston taken with a 40 mm lens), Formulation 1 did not give an underestimate
for D. This is because Formulation 1 does not account for curvature of
the Earth in any way. With this large field of view, assuming a flat Earth
inflated the value of D above the minimum from calculations that included
Earth curvature.
A
major difference between Formulations 2 and 3 is the ability to estimate
pixel sizes (P) in both directions (along the principal line and perpendicular
to the principal line). For the more oblique photographs, the vertical
estimate of D and pixel sizes is much larger than in the horizontal direction
(e.g. the low oblique and high oblique photographs of Hawaii, figure 4,
table 5).
For the area of Limmen Bight (figure 7),
table 5 illustrates the improvement in the estimate of distance and pixel
size that can be obtained by reestimating the location of the PC with
greater accuracy. Centre points in the catalogued data are ± 0.5°
of latitude and longitude. When the centre point was reestimated to ±
0.02°, we determined that the photograph was not taken as obliquely
as first thought (change in the estimate of the look angle t from 16.9
to 12.8°, table 5). When the auxiliary point was added to the calculations
of Formula 3, the calculated look angle shrank further to 11.0°, indicating
that this photograph was taken at very close to a nadir view. Of course,
this improvement in accuracy could have also led to estimates of greater
obliquity, and corresponding larger pixel sizes.
We also tested the performance of the scale
calculator with auxiliary point by estimating the corner and point locations
on the photograph using a 1:1,000,000 Operational Navigational Chart.
For this test, we estimated our ability to read coordinates from the map
as ± 0.02°and our error in finding the locations of the corner
points as ± 0.15° (this error varies among photographs depending
on the detail that can be matched between photo and map). For Limmen Bight
(figure 7), the mean difference between map estimates and calculator estimates
for 4 points was 0.31° (SD = 0.18, n = 8). For a photograph of San
Francisco Bay (STS062151291) the mean difference between map estimates
and calculator estimates for 4 points was 0.064° (SD = 0.18, n = 8).
For a photograph of San Francisco Bay (STS062151291) the mean difference
between map estimates and calculator estimates for 8 points was 0.196°
(SD = .146, n = 16). Thus in one case, the calculator estimates were better
than our estimate of the error in locating corner points on the map. It
is reasonable to expect that for nadirviewing photographs, the calculator
used with an auxiliary point can estimate locations of the edges of a
photograph to within ± 0.3°.
5.5. Empirical Confirmation of Spatial Resolution Estimates
As stated previously, a challenge to estimating
systemAWAR for astronaut photography of Earth is the lack of suitable
targets. Small features in an image can sometimes be used as a check on
the size of objects that can be successfully resolved, giving an approximate
value for GRD. Similarly, the number of pixels that make up those features
in the digitised image can be used to make an independent calculation
of pixel size. We have successfully used features such as roads and airport
runways to make estimates of spatial scale and resolution (e.g. Robinson
et al. 2000c). While recognizing that the use of linear detail in an image
is a poor approximation to a bar target, and that linear objects smaller
than the resolving power can often be detected (Charman 1965), we could
find few objects other than roads to make any direct estimates of GRD.
Thus, we used roads and runways in the images of Houston (where we can
readily conduct ground verifications, and where a number of highercontrast
concrete roadways were available), to obtain empirical estimates of GRD
and pixel size for comparison with table 5.
In the alldigital ESC image of Houston
(figure 3, D) we examined Ellington Field runway 422 (centre left of
the image) which is 2438.4 ´ 45.7 m. This runway is approximately
67 pixels in width and 304309.4 pixels in length, so pixels represent
an distance on the ground 7 – 8 m. Using a lower contrast measure
of a street length between two intersections (212.5 m = 21.1 pixels),
we estimate a pixel width of 10.1 m. These results compare favourably
with the minimum estimate of 8.1 m pixels using Formulation 1 (table 5).
For an estimate of GRD that would be more comparable to aerial photography
of a line target, the smallest street without tree cover that we could
clearly distinguish on the photograph was 7.92 m wide. The smallest nonstreet
object (a gap between stages of the Saturn rocket on display in a park
at Johnson Space Center) we could clearly distinguish on the photograph
was 8.53 m wide.
For the photograph of Houston taken with
250mm lens (figure 3, C), and digitised from 2nd generation film at 2400
ppi (10.6 mm/pixel), Ellington Field runway 422 is 3 pixels in width
and 161.3 pixels in length, so pixels represent 15.1 – 15.2 m on
the ground. These results compare favourably with the minimum estimate
of 15.2 m using Formulation 2, and 15.4  18.5 m pixels using Formulation
3 (table 5). For an estimate of GRD using an 8 ´ 8inch print (1:3.69
enlargement) and 4 ´ magnification, the smallest street we could
clearly distinguish was 8.22 m wide, the same feature could barely be
distinguished on the digitised image.
We
also made an empirical estimate of spatial resolution for lower contrast
vegetation boundaries. By clearing forest so that a pattern would be visible
to landing aircraft, a landowner outside Austin, Texas (see also aerial
photo in Lisheron 2000), created a target that is also useful for evaluating
spatial resolution of astronaut photographs. The forest was selectively
cleared in order to spell the landowner's name 'LUECKE' with the remaining
trees (figure 10). According to local surveyors who planned the clearing,
the plan was to create letters that were 3100 ´ 1700 ft (944.9 ´
518.2 m). Photographed at a high altitude relative to most Shuttle missions
(543 km) with a 250mm lens, Formula 3 predicts that each pixel would
represent an area 28.6 ´ 36.0 m on the ground (table 5). When original
film was digitised at 2400 ppi (10.6 mm/pixel), letters correspond to
29.4 ´ 18.8 pixels for a comparable pixel size of 27 – 32
m.
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