1790 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20
Terrestrial Photogrammetry of Weather Images Acquired in Uncontrolled Circumstances
ERIK N. RASMUSSEN Cooperative Institute for Mesoscale Meteorological Studies, and NOAA/National Severe Storms Laboratory, Norman, Oklahoma
ROBERT DAVIES-JONES NOAA/National Severe Storms Laboratory, Norman, Oklahoma
RONALD L. HOLLE Tucson, Arizona
(Manuscript received 12 August 2002, in ®nal form 14 April 2003)
ABSTRACT This paper describes an accurate automated technique of terrestrial photogrammetry that is applied to weather images obtained in uncontrolled circumstances such as unknown focal length and 3D camera orientation (azimuth and tilt of the optical axis, and roll about this axis), principal point unmarked on the image, and undetermined lens horizon. With the possible exception of the principal point, these quantities are deduced rapidly by a computer algorithm, with input consisting of accurate azimuth and elevation angles of landmarks that appear in the image. The algorithm works for wide-angle as well as for telephoto images and is more accurate than previous methods, which are based on assumptions of small angles and zero roll. Results are insensitive to the exact position of the principal point for telephoto images. For wide-angle photography, the principal point can be determined only if there is a suf®cient number of accurately measured landmarks with diverse azimuth and elevation angles. If all the landmarks have low elevation angles, the principal point is impossible to determine and must be assumed to lie at the intersection of the diagonals of the uncropped image. The algorithm also provides the azimuth and elevation angle of any object, given the position of its image in the photograph. A photogrammetric search technique is described for ®nding an entity, which is visible in one camera's photography, in the simultaneous image obtained from a different direction by a second camera. Once the same object has been identi®ed in both images, its 3D position is determined by triangulation.
1. Introduction metrically. Wakimoto and Bringi (1988) utilized radar data overlaid on still photographs to document the de- This paper describes an accurate automated technique velopment and descent of precipitation in deep cumulus of terrestrial1 photogrammetry that applies to weather clouds during the Microburst and Severe Thunderstorm images obtained in uncontrolled circumstances. Tradi- tionally, photogrammetry has been performed on care- (MIST) project. Colorado microbursts were similarly fully obtained images from special airborne cameras analyzed using still photographs (Wakimoto et al. 1994). with carefully calibrated focal length and orientation A set of images of a Colorado tornado was overlaid (Slama 1980). In meteorological terrestrial photogram- with Doppler radar data in the analysis of Wakimoto metry, images have provided much useful information. and Martner (1992). These studies were performed with For example, Hoecker (1960), Golden and Purcell telephoto images obtained under relatively controlled (1978), and others cited in Bluestein and Golden (1993) circumstances: that is, known focal length lenses and have measured wind speeds in tornadoes photogram- negligible difference between the lens and visible ho- rizons (for de®nitions of terms, see Table 1). This is not always the case with terrestrial weather photographs. 1 Photogrammetry in general is much more often concerned with These investigators used established photogrammetry measurements in airborne photographs obtained at near-vertical inci- dence. Terrestrial photogrammetry is concerned with measurements in procedures (Holle 1982, hereafter H82) that apply only photographs obtained from the ground at incidence close to horizontal. to objects with small angular displacements from the principal axis of the lens and assume that the camera is held perfectly horizontally. In other words, the roll angle Corresponding author address: Erik Rasmussen, 50742 Bear Run Dr., P.O. Box 267, Mesa, CO 81643. of the camera, or the angle at the principal point between E-mail: [email protected] the ``vertical'' side on the photograph and true vertical
᭧ 2003 American Meteorological Society
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TABLE 1. De®nitions of common photogrammetry terms. Focal length The distance from the rear nodal point to the plane of best focus for distant objects Lens horizon, horizon line The intersection of the photograph (or image) with the horizontal plane through the rear nodal point of the lens Magni®cation Ratio of size of projected or enlarged image to the size on ®lm (or electronic imager) Nodal points Front and rear nodal points are points on the optical axis of a lens such that when all object distances O are measured along the optical axis from the front nodal point (FNP) and all image distances I are measured similarly from the rear nodal point (RNP), they satisfy the thin lens relation 1/ f ϭ 1/O ϩ 1/I. The ray that emerges from the RNP is parallel to the ray that is incident at the FNP Optical axis, principal axis A straight line connecting the center of curvature of the lens elements and passing through the principal point of the image Principal azimuth Azimuth of the camera principal axis relative to true north Principal horizontal line Line through principal point in image, parallel to the bottom edge Principal point Point near the center of the image through which the principal axis passes; at center of image for correctly positioned ®lm or image sensor Principal vertical line Line through principal point in image, parallel to the side Roll Angle at which the camera is rotated about its own optical axis Tilt Angle of the camera optical axis above the true horizon Visible horizon Where the earth meets the sky in object space, is assumed to be identically zero. Saun- Very little information is present in the formal mete- ders (1963) outlined a more general technique that in- orological literature concerning techniques for single- volves ®nding the lens horizon by a manual trial-and- camera or multicamera photogrammetry of images ob- error method. The fast computer algorithm presented in tained in these uncontrolled circumstances. For that mat- this paper is essentially an automated and more precise ter, papers in the formal literature rarely address issues version of Saunders' method with the labor-intensive such as the actual location of the horizon line, camera manual iterations replaced by convergent iterative solu- roll angle about the optical axis, or other crucial details tions of the photogrammetric equations. of photogrammetric analysis. This paper describes a new The methods in this paper apply to movie or video algorithm for analyzing terrestrial weather images ob- images as well as still images. Historically, movies have tained with any type of lens (telephoto, normal, or wide been used to assess the motion of cloud or debris by angle). The input for this algorithm consists of measure- comparing positions of a feature between frames ex- ments obtained in prephotogrammetry surveys from the posed at a known time interval [e.g., Golden and Purcell camera site (section 2). The information that must be (1978), Hoecker (1960), and references cited by Blue- obtained in these surveys consists of locating the exact stein and Golden (1993)]. Much of the literature con- camera position and from this point measuring precisely cerning motion picture weather photogrammetry is in- the azimuth and elevation angles of landmarks that appear formal and will not be cited here. In most cases the in the images. The mathematical method for retrieving small-angle (linear scaling) approximation described in focal length, principal azimuth, and camera tilt and roll section 3 was appropriately utilized for image scaling, is developed in section 3. Once these parameters are and prephotogrammetry determination of landmark az- found, the azimuth and elevation angle of any feature in imuth and elevation was used for image orientation. the image can be determined. The scale distortion in- In recent ®eld programs such as the Veri®cation of herent in photographs obtained with wide-angle lenses is Rotation in Tornadoes Experiment (VORTEX; Rasmus- accommodated by the nonlinear equations developed sen et al. 1994) numerous photographs and video images herein. The algorithm is tested in section 4 using a tele- were obtained and proved very useful for deducing cloud photo image obtained during VORTEX and also simu- and tornado locations. These photographs were typically lated wide-angle photography. The range of a visible fea- obtained with unknown focal length and camera orien- ture from a camera is unknown unless further information tation, unmarked principal point, and poorly calibrated is available such as the map of the damage path in the image placement with respect to the camera optical axis. case of a tornado or a simultaneous image from a second Often, cameras were hand-held with little attention to camera with a different viewing angle. Section 5 de- careful orientation with respect to the horizon. The cam- scribes a search method used in analyses of VORTEX eras used lenses with a variety of focal lengths that often data for locating the same feature in photographs from changed between exposures. Exact camera position was different directions and then deducing its 3D position. seldom recorded. Photographic parameters are not re- corded at the time either by the general public or by 2. The prephotogrammetry survey scientists in the stressful, rapidly changing circumstances of a severe storm intercept. Thus the parameters have to A prephotogrammetric ®eld survey is unnecessary in be deduced a posteriori using information gained from the ideal situation when full images are obtained with revisiting the camera site with surveying equipment. a special camera and the camera's exact Cartesian and
Unauthenticated | Downloaded 09/30/21 08:41 PM UTC 1792 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 angular coordinates are measured and recorded. The ment is placed as close as possible to the actual camera special camera would have a ®xed and calibrated focal location, and it is leveled very precisely because both length and would mark the position of the principal azimuth and elevation of landmarks will be measured. In point on the image (Slama 1980). Such controlled pro- practice, a sketch is ®rst made of the scene showing and cedures are impractical when pursuing rare, short-lived numbering the approximate location of the landmarks in phenomena such as tornadoes because of time con- the image, and a numbered list is prepared that describes straints, the expense of special cameras, and the need the landmark (e.g., ``top of leaning power pole''). The for zoom lenses to obtain an optimum ®eld of view for actual location of the survey point must be described a given situation. The meteorologist has even less con- because both azimuth and elevation will be measured. trol of the data gathering if photography is obtained Increasing the number of landmarks that are utilized in- from the public, rather than as part of a scienti®c project. creases the con®dence with which both subjective and Thus focal length, camera orientation, and sometimes objective scaling and orientation can be performed. In camera location are unknowns, which can be deduced practical experience, it has proven valuable to have two accurately only through information acquired in a pre- people involved in the survey, each making independent photogrammetric survey. Lacking a survey, the analyst measurements in order to catch gross errors. typically assumes that the visible horizon line coincides The azimuths obtained in the ®eld are measured rel- with the lens horizon and that this line can then be used ative to the survey instrument because it is not generally to determine camera orientation. However, this only possible to know the orientation of true north during works with a ¯at horizon at the same elevation as the the survey. In order to determine earth-relative azimuths camera (a condition that is hard to verify). in later processing, it is possible to locate two or more We now present guidelines for conducting a prepho- ``reference landmarks'' and measure the azimuth of togrammetry survey that are suf®cient to obtain the data these during the ®eld survey. Reference landmarks needed for scaling and orienting a weather image. The should be tall objects such as transmission towers that goals of the survey are to determine the camera location, are visible from many or all of the camera sites that are as well as azimuths and elevations of landmarks visible being surveyed. It is imperative that the actual location in the image(s). Further, information must be obtained of the reference landmarks be ascertained (through to- so that ®eld-measured azimuths can be made earth rel- pographic maps or GPS). Then, using the known camera ative in later analysis. Certain equipment is essential for and reference landmark locations, the actual azimuth of the survey, including a global positioning system (GPS) the reference landmark can be computed to within a receiver and measuring wheels to determine camera lo- fraction of a degree, and by comparing this value with cation, and a survey transit capable of measuring azi- the measured azimuth, the azimuth bias of the survey muth and elevation with an accuracy of 1Ј of arc instrument at the camera site can be determined. (0.0167Њ). Also essential are prints or traces of the orig- The preceding is suf®cient for scaling images in terms inal photography with marked identi®able landmarks, of angular separations. Scaling in terms of linear dis- and a camera of ®xed focal length. tances requires range information. For example, the hor- First, the exact camera location must be determined. izontal range of tornado debris is determined approxi- This can be facilitated in a ®eld project if the location mately by the intersection of the object's azimuth with is marked at the time of the original photography by the centerline of the tornado's damage track. Hence a spray painting the ground or recording the location from careful survey of the tornado track is also required. The GPS (the former is presently more accurate and may be range of cloud features often can be obtained by tri- essential if nearby, tall landmarks are used for image angulation (section 5) if there is simultaneous photog- scaling and orientation). During a survey, the camera raphy from different viewing directions. site is located through simple comparison of perspective between foreground and background objects in the pho- 3. A computer algorithm for ground-based tograph. In practice, a camera site can usually be de- photogrammetry termined to within 1 m unless the image is lacking in landmarks (which occurs most often in telephoto im- We now present a mathematical technique for retriev- ages). After the site is found, it is necessary to determine ing the parameters describing image orientation and the geographical location of the site, preferably with scaling from information gathered during the prepho- GPS. For gross error checking, distance to nearby land- togrammetry survey, and measurements made in the im- marks can be measured with a measuring wheel. Further, age. The image orientation (␣, , ; principal azimuth, by measuring the azimuths (as described below) of dis- elevation angle, roll angle) and scaling ( f ; focal length) tant landmarks of a known location, triangulation can are based on the azimuths and elevation angles (, ) be used to validate the camera location information. of two or more landmarks that are visible in the image After establishing the camera location and leveling the (see Table 2 for a summary of variable de®nitions). instrument, it is necessary to measure the azimuth and We begin the mathematical formulation by de®ning elevation angles (, ) of as many landmarks as practical relevant coordinate systems. Let (X*, Y*, Z*) be a Car- in order to minimize errors (section 3). A survey instru- tesian coordinate system in the object space where the
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TABLE 2. De®nitions of symbols used in text.
␣ Principal azimuth (of optical axis) Azimuth angle of an object measured clockwise from true north Roll angle of camera (ϭangle between principal horizontal line and the lens horizon) Elevation angle of an object Elevation angle of the optical axis f Focal length of the camera F ``Effective'' focal length; F ϭ fM M Magni®cation; ratio of size of projected or enlarged image to the size on ®lm (or electronic im- ager). R Slant range of an object (X*, Y*, Z*) Eastward, northward, and vertical coordinates from convenient origin (e.g., road intersection)
(X000***, Y , Z ) (X*, Y*, Z*) coordinates of the camera
(X, Y, Z) Rotation of (X* Ϫ X000***, Y* Ϫ Y , Z* Ϫ Z ) coordinate system about Z* axis through angle ␣
such that Y axis is in direction of principal azimuth (Z ϭ Z* Ϫ Z*0 ) (XЈ, YЈ, ZЈ) Rotation of (X, Y, Z) coordinate system about X axis through angle such that YЈ axis is along optical axis (XЈϭX) (xЈ, zЈ) Coordinates in the image plane with origin at the principal point and coordinate axes in direc- tions of horizon line and upward normal to this line, respectively yЈ Distance of ®lm plane from rear nodal point; yЈϭ f if focus is on in®nity (xЉ, zЉ) Coordinates relative to center of photograph parallel to the horizontal and vertical sides of the photograph [rotation of (xЈ, zЈ) coordinates through angle followed by a translation to center of photograph]
(xpЉ, zpЉ) Location of principal point in (xЉ, zЉ) coordinates
X*, Y*, and Z* axes are eastward, northward, and up- XЈ R sin( Ϫ ␣) cos ward, respectively, and let the front nodal point of the YЈϭ R cos( Ϫ ␣) cos cos ϩ R sin sin . camera lens be at (X*000 ,Y* ,Z* ). Translation of the origin to the camera location, followed by rotation of the X* ZЈ R sin cos Ϫ R cos( Ϫ ␣) cos sin and Y* axes clockwise about the Z* axis so that the new (4) Y axis is along the principal azimuth ␣, results in a new coordinate system [X, Y, Z] (Fig. 1a, where square Let (xЈ, yЈ, zЈ) be a coordinate system in the image space brackets are used to distinguish it from the prime system behind the lens with origin at the rear nodal point of introduced below) and the lens and axes in the opposite directions to the (XЈ, YЈ, ZЈ) system. The image of a distant object at (R, , X cos␣ Ϫsin␣ 0 X* Ϫ X 0* ) is in focus on the ®lm at a distance f behind the rear Y ϭ sin␣ cos␣ 0 Y* Ϫ Y 0*, nodal point where f is the focal length of the lens. By similar triangles in Fig. 1d, the image is at Z 001 Z* Ϫ Z 0* (x , y , z ) ( fX /Y , f, fZ /Y ), (5) Ј Ј Ј ϭ Ј Ј Ј Ј X* Ϫ X 0* cos␣ sin␣ 0 X where the principal point is at xЈϭ0, zЈϭ0. In practice, Y* Ϫ Y* ϭϪsin␣ cos␣ 0 Y . (1) 0 the image is magni®ed for more accurate measurements. Z* Ϫ Z 0*001 Z Henceforth we make the image-space coordinates apply to the enlarged image instead of the actual ®lm by re- In terms of spherical coordinates (R, , ), R is slant range placing f everywhere by F Mf, where M is the mag- from the camera, is the azimuth angle measured clock- ϵ ni®cation. From (4) and (5) the image of an object that wise from true north, and is the elevation angle: is effectively at (ϱ, , ) appears on the ®lm at X ϭ R sin( Ϫ ␣) cos, Y ϭ R cos( Ϫ ␣) cos, F (xЈ, zЈ) ϭ Z ϭ R sin. (2) 1 ϩ sec( Ϫ ␣) tan tan The relationships between [X, Y, Z] and (R, , ) are ϫ [tan( Ϫ ␣) sec, shown in Fig. 1b. The geometry of a camera tilted up- ward at an elevation or pitch angle is shown in Fig. sec( Ϫ ␣) tan Ϫ tan]. (6) 1c. Rotation of the Y and Z axes about the X axis so Conversely, the azimuth and elevation angles of an im- that the new Y axis (now YЈ axis) is along the optical age in the ®lm plane at (xЈ, zЈ) are given by axis of the camera lens results in the primed system xЈ sec XЈ 10 0 X ϭ ␣ ϩ tanϪ1 , and YЈϭ 0 cos sin Y . (3) F Ϫ zЈ tan ZЈ 0 Ϫsin cos Z cos( Ϫ ␣)(zЈϩF tan) ϭ tanϪ1 , (7) From (2) and (3), []F Ϫ zЈ tan
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FIG. 1. (a) The transformation of horizontal coordinates. (b) The relationship between [X, Y, Z] and (R, , ) coordinates. Here T is the object or target and T1 is the projection of T onto the vertical plane that contains the optical axis. (c) The geometry in the principal plane of image formation for a camera with a thin lens focused on in®nity, pointing along azimuth ␣, and tilted upward at an elevation angle . (d) The geometry of the image formation projected onto the plane that contains the optical axis and is normal to the principal plane. Symbols as in text. from (6). Note that (7) is equivalent to Eqs. (4) and (5) Often the camera is held askew so that the lens ho- in H82. When ϭ 0, zЈϭϪF tan, which is the rizon is not horizontal, and the location of the principal equation of the lens horizon. point on a print may not be known owing to cropping. We can construct a grid of constant azimuth and el- These complexities can be taken into account with a evation on the photograph by drawing the following rotation of the xЈ and zЈ axis about the yЈ axis counter- curves. A contour of constant azimuth, say c,onthe clockwise through an angle of roll or bank , and a photograph is given by translation so that the origin of the new coordinate sys-
zЈϭϪxЈ cosec cot(c Ϫ ␣) ϩ F cot (8a) tem (xЉ, zЉ) is at the center of the print and its xЉ and from (7). Thus the contours are straight lines that zЉ axes are parallel to the horizontal and vertical sides intersect at the image of the zenith point, which is at of the photograph, respectively (Fig. 2). The new pho- (xЈ, zЈ) ϭ (0, F cot) (usually outside the picture). It tographic coordinates are related to the previous ones can be shown that a contour of constant elevation angle, by say c, satis®es the equation {zЈϩ0.5F[tan( ϩ ) Ϫ tan( Ϫ )]}2 xЉϪxЉ cos sin xЈ cc P ϭ and 0.25F 22[tan( ϩ ) ϩ tan( Ϫ )] zЉϪzЉϪsin cos zЈ cc [][P ][] xЈ2 Ϫϭ1. xЈ cos Ϫsin xЉϪxPЉ 2 ϭ , (9) 0.5F cotcc[tan( ϩ ) ϩ tan( cϪ )] zЈ sin cos zЉϪzЉ [] [ ][P ] (8b)
This is a hyperbola with center at (xЈ, zЈ) ϭ {0, where (xЉPP ,zЉ ) is the location of the principal point
Ϫ0.5F[tan(c ϩ ) Ϫ tan(c Ϫ )]} and asymptotic on the photograph in the rotated coordinates. From 2 2 Ϫ0.5 slopes of Ϯtanc sec(1 Ϫ tan c tan ) . The upper (6) and (9) we obtain the following equations for the (lower) branch of the hyperbola is the contour ϭ c( image location of an object with angular coordinates ϭϪc). (, ):
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xЉϪxЉ tan( Ϫ ␣) sec cos ϩ [sec( Ϫ ␣) tan Ϫ tan] sin P ϭ , F 1 ϩ sec( Ϫ ␣) tan tan zЉϪzЉϪtan( Ϫ ␣) sec sin ϩ [sec( Ϫ ␣) tan Ϫ tan] cos P ϭ . (10) F 1 ϩ sec( Ϫ ␣) tan tan Conversely, the azimuth and elevation angle of an object, whose image in the photograph is at the point (xЉ, zЉ), is determined by
(xЉϪxЉ) cos Ϫ (zЉϪzЉ) sin ϭ ␣ ϩ tanϪ1 PP and F cos Ϫ (xЉϪxЉ) sin sin Ϫ (zЉϪzЉ) sin cos []PP (xЉϪxЉ) cos sin ϩ (zЉϪzЉ) cos cos ϩ F sin ϭ tanϪ1 cos( Ϫ ␣)PP , (11a) F cos Ϫ (xЉϪxЉ) sin sin Ϫ (zЉϪzЉ) sin cos []PP
from (9) and (7). When ϭ 0, these formulas reduce Gn(F, ␣, , ) to Eqs. (4) and (5) in H82. Although Holle's equations are correct for large angles when ϭ 0, his method ϵ (xnpЉϪxЉ)[1 ϩ sec( nϪ ␣) tan ntan] utilizes small-angle approximations in obtaining focal Ϫ F tan(n Ϫ ␣) sec cos length, principal azimuth, and camera elevation angle. Saunders (1963) measured distances on the print from Ϫ F[sec(nnϪ ␣) tan Ϫ tan] sin ϭ 0, the principal plane and the lens horizon. His coordinates n ϭ 1, 2, are given by Gnϩ2(F, ␣, , ) xSPϭ (xЉϪxЉ) cos Ϫ (zЉϪz PЉ) sin, ϵ (znpЉϪzЉ)[1 ϩ sec( nϪ ␣) tan ntan] zSPϭ (xЉϪxЉ) sin ϩ (zЉϪz PЉ) cos ϩ F tan, (11b) ϩ F tan(n Ϫ ␣) sec sin and (11a) becomes Ϫ F[sec(nnϪ ␣) tan Ϫ tan] cos ϭ 0,
xS n ϭ 1, 2, (12) ϭ ␣ ϩ tanϪ1 , F sec Ϫ z sin []S z cos( Ϫ ␣) cos ϭ tanϪ1 S , (11c) F sec Ϫ z sin []S in agreement with Saunders' Eqs. (6) and (7). In uncontrolled circumstances, the parameters F, ␣, , and are unknown. Consider a grid, consisting of labeled contours of and at 1Њ intervals, overlaid on the photograph. The parameters F, ␣, , and control the mesh size, horizontal and vertical displacement of the grid, and orientation of the grid, respectively. Since lines of constant intersect in the image at the zenith point, which varies with cot , the tilt also affects the shape of the grid. Assume for now that the principal point is at the center of the photograph, as is generally the case. Then F, ␣, , and can be determined a posteriori by measuring the azimuth and elevation an- gles, (n, n), and the corresponding locations on the photograph (xЉnn ,zЉ ) of N landmarks in the photograph where N Ն 2. Since the solution for F, ␣, , and is obtained iteratively, a good initial approximate solution FIG. 2. Schematic of a photograph taken with a camera at a roll is desirable. This is obtained by selecting two landmarks angle . The intersection C of the diagonals (dashed) approximately locates the principal point P (the separation between C and P is (labeled n ϭ 1 and n ϭ 2) and using the small-angle exaggerated for clarity). The xЈ and zЈ axes are horizontal and vertical; approximation. From (10), F, ␣, , and are the roots the xЉ and zЉ axes are parallel to the long and short edges of the of the nonlinear system of four equations: photograph.
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where xЉPPϭϭzЉ 0 here. Assuming that | n Ϫ ␣ |, can set xЉPPϭϭzЉ 0 without losing precision. Variations | n |, | |, | |, |(xЉnPϪ xЉ)/F | and | (zЉ nPϪ zЉ)/F | are inxЉpp /F andzЉ /F are compensated by equal changes in all much less than one results in the second-order equa- ␣ ϩ and Ϫ ␣, respectively. tions, Dif®culties in determining the principal point in a wide-angle photograph may be anticipated since the grid (xnpЉϪxЉ) ϭ (nnϪ ␣) ϩ ( Ϫ ) and is locally Cartesian in the neighborhood of the principal F point. Our fears were con®rmed by the results of sim- ulated tests (section 4) in which the above method for (znpЉϪzЉ) ϭϪ(nnϪ ␣) ϩ ( Ϫ ), (13a) ®nding the principal point generally converged to the F wrong point. Therefore, in scaling wide-angle photo- which have the solutions (see the appendix) graphs, the ``crossed-diagonals technique'' (H82) should be used on the original negative or slide in order F Ϫ1 ( )/( x z ), ϭ ⌬⌬ ϩ⌬⌬ ⌬⌬ Љϩ⌬⌬ Љ to obtain a good estimate of the location of the principal ϭ (⌬⌬xЉϪ⌬⌬zЉ)/(⌬⌬xЉϩ⌬⌬zЉ), point in the image. For example, the principal point of the Canon 16 MS 16-mm movie camera used during ϭ Ϫ f Ϫ12[(xЉϪxЉ) ϩ zЉϪzЉ]/(1 ϩ ), 11P 1 P VORTEX is within (Ϯ0.2 mm, Ϯ0.3 mm) of the center and of the 10.37 mm ϫ 7.52 mm image frame according to speci®cations obtained from the manufacturer. Ϫ12 ␣ ϭ 11Ϫ F [xЉϪxPЉϪ(z1ЉϪzPЉ)]/(1 ϩ ), (13b) 4. Tests of the algorithm where ⌬ ϵ ( 2 Ϫ 1), etc. If either | (xЉnPϪ xЉ)/F | With telephoto images or in situations in which only K 1or|(zЉnPϪ zЉ)/F | K 1 is false, the solutions are a feature in the central part of the image is of interest, good only to ®rst order. the H82 method is found to be acceptable. In other More accurate solutions are found by using the Polak± words, a scaled Cartesian grid can be placed on the Ribiere conjugate±gradient method exactly as detailed image, and rotated so that its horizontal axis is parallel by Press et al. (1986, 303±306) to minimize the cost to the visible horizon, which is assumed to be at 0Њ function: elevation. This has been done in Fig. 3 with an image 1 N of the Dimmitt, Texas, tornado of 2 June 1995 recorded E 2(F, ␣, , ) ϵ [( Ϫ ˆ )22ϩ ( Ϫ ˆ ) ], (14) in Super-VHS video format during a VORTEX inter- N nn nn nϭ1 cept. The overlay was created using drawing software, where (n, n) is the measured angular position of the and scaled, translated, and rotated to obtain a subjective nth landmark, (ˆ n, ˆ n) is the location of the nth landmark best ®t to the four survey landmarks. This ϳ10.5Њ image predicted by (11) as a function of F, ␣, , and , and is typical of the sort of telephoto image in which the (13b) is used as a ®rst guess for F, ␣, , and . The small-angle approximation does not lead to signi®cant squared angular distance between the actual and pre- errors (i.e., a Cartesian grid can be used for scaling). 2 2 dicted position of the nth landmark is cos n(n Ϫ ˜ n) This photograph (video image capture) also has been 2 ϩ (n Ϫ ˜ n) . Therefore, E is the root-mean-square error analyzed by the algorithm (Fig. 4). The resulting (slight- (rmse) in angular distance in the usual case when all ly non-Cartesian) grid is generated in a matter of a few 2 the landmarks have low elevation angles (cos n ഠ 1 seconds at most, and is similar to that obtained by the for all n). The method requires formulas for the deriv- more labor-intensive subjective method (Fig. 3). The .. These are grid ®ts the landmark positions with an rmse of 0.045Њץ/E 2ץ , andץ/E 2ץ ,␣ץ/E 2ץ ,Fץ/E 2ץ atives obtained by differentiating (14) and (11a). The small-angle solution based on the two outer land- What happens if the principal point is unknown, ow- marks has an rmse that is larger by 20% [or 40% if ing to the photograph being cropped or other causes? is set to zero in (13b)]. Although the H82 method may Do three or more landmarks provide enough information be accurate enough for telephoto images, the algorithm to determine the six unknowns: F, ␣, , , xЉpp,?IfzЉ still has a considerable speed advantage. Once the com- a telephoto lens was used, the grid is approximately puter code has been written (a one-time effort), the work Cartesian (constant spacing in xЉ and zЉ) across the in generating the grid consists merely of inputting the photograph and locating the principal point is virtually azimuths and elevation angles (n, n) into the program impossible for the following reason. It is clear from the and running it. Moreover, the algorithm computes the appendix that (13a) for n ϭ 1, 2, . . . , N is a linear magni®ed focal length F (2242 pixels where the pixel system in only the four independent variables, 1/F, , dimensions of the image are 416 ϫ 238), principal az-
␣ ϩ Ϫ xЉpp/F, and Ϫ ␣ Ϫ zЉ/F, regardless of the imuth ␣(262.67Њ), camera elevation angle (1.33Њ), and number of landmarks N, and hence the best we can do roll angle (0.74Њ). These parameters are then available is to determine F, ␣, , and in terms ofxЉPP andzЉ . for immediately computing via (11a) the azimuth and Fortunately, the solutions for F, ␣, , and are highly elevation angles (, ) of any object, given the Cartesian insensitive to the location of the principal point and we coordinates (xЉ, zЉ) of its image.
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FIG. 3. Image of the Dimmitt, TX, tornado of 2 Jun 1995 obtained by the ``CAM-1'' VORTEX intercept team using Super-VHS video. The four survey landmarks are marked with arrowheads near the survey point, and azimuth and elevation notated in decimal degrees. From left to right, the landmarks are a faint power pole, the left roof peak of the barn, a power pole to the right of the barn, and the top of the second power pole from the image edge. The Cartesian grid overlay was generated in graphics drawing software and magni®ed, translated, and rotated to ®nd a subjective best ®t with the survey data.
FIG. 4. As in Fig. 3 but the principal vertical and horizontal lines are solid white with a gap at the principal point (at image center), and the overlay is the objectively computed image scaling and orientation. Lines of constant azimuth and elevation are dash-dotted at 5Њ intervals, and dotted at 1Њ intervals. This and subsequent images are the output of an IDL image analysis program.
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A more challenging test is provided by the following 5. Application example using triangulation of simulation of a 35-mm photograph (image dimensions VORTEX data 36 mm ϫ 24 mm) taken with a 28-mm wide-angle lens. It is assumed that the camera is pointed at 270Њ azimuth, One of the uses of photogrammetry in VORTEX data is tilted upward at 15Њ, and has a roll of 7.5Њ, and that analysis has been to locate tornadoes, wall clouds, and other accessory clouds by triangulation. In many events, there are four landmarks at angular positions (1, 1) ϭ (240Њ, 0.5Њ), ( , ) ϭ (290Њ, 1.0Њ), ( , ) ϭ (260Њ, numerous photographers, working with VORTEX and 2 2 3 3 independently, obtained still and video images from dif- 2.0Њ), and ( 4, 4) ϭ (275Њ, 0.7Њ). The principal point is assumed to be at the center of the slide. The corre- ferent viewing directions. The video images were reg- sponding positions of the images of the landmarks on istered in time to the nearest ϳ5 s (time registration the slide, computed from (10) and then rounded to the techniques are beyond the scope of this paper). Time nearest 0.1 mm to allow for observational imprecision, registration of still photos is more problematic, but in the case of tornado cyclones, the cloud features tend to are (xЉnn ,zЉ ) ϭ (Ϫ17.5, Ϫ5.0), (9.5, Ϫ8.3), (Ϫ5.9, Ϫ 5.7), and (1.6, Ϫ7.4) mm for n ϭ 1, . . . , 4. Given the rotate and evolve quickly enough that comparisons with above angular positions and coarsened image locations time-stamped video provides time estimates that have of the landmarks, the algorithm was used to retrieve the an uncertainty of about 15 s in practice. Using two or camera parameters F, ␣, , and , and to compute the more simultaneous images from different angles (stereo rmse of the predicted angular positions of the landmarks photogrammetry), the locations of common features in compared to the actual positions. The small-angle so- the photos can be triangulated. (In ®eld research pro- lution based on the two outer landmarks gives (F, ␣, grams, whenever practical, camera teams should syn- , ) ϭ (30.9 mm, 270.7Њ, 13.8Њ, 7.6Њ) and has an rmse chronize their photographs via radio communication.) of 1.7Њ. Using the two inner landmarks instead of the The technique of graphical intersection simply in- two outer ones provides the more accurate result: (F, volves plotting, on an isometric map projection, straight ␣, , ) ϭ (29.0 mm, 269.9Њ, 14.6Њ, 7.9Њ) with an rmse lines from the camera locations oriented along the az- of 0.54Њ. Ignoring the terms in this case degrades the imuths of the feature. The intersection of these lines rmse to 2.4Њ. The corresponding values for the conju- gives the horizontal coordinates of the common feature. gate-gradient solution, (F, ␣, , ) ϭ (28.03 mm, The height of the feature can be computed directly from 270.01Њ, 15.00Њ, 7.50Њ) and rmse ϭ 0.07Њ, illustrate the (16) below using the range and elevation angle of the superiority of the conjugate±gradient solution. feature from one of the cameras and the camera's height We did not succeed in ®nding a reliable method for above mean sea level. The graphical intersection tech- locating the principal point (PP) in wide-angle photo- nique is illustrated in Fig. 5 for the Dimmitt, Texas, graphs. Our attempts consisted of running the algorithm tornado photographed at ϳ0106:30 UTC 2 June 1995. with different assumed positions of the PP to determine (In this illustration, the two images were not obtained simulataneously, so there is a small error in tornado rmse as a function ofxЉPP andzЉ . The assumed positions position compared to accurate trackings that have been formed a regular grid inxЉPP ±zЉ space with a grid spacing of 0.5 mm. It was hoped that the actual principal point, made using stereo photogrammetry and mobile Doppler (0, 0), would be the locus of a single or global minimum data for a formal study in progress.) The center of the visible tornado near the ground, and the left and right of rmse(xЉPP ,zЉ ). However, several minima were found and the one closest to (0, 0) usually was not the global extent of the completely opaque portion of the debris minimum when the input data were rounded to obser- cloud were mapped. Examination of the scaled images vational accuracy. In cases with ϭ 0, the variation in as well as the output from the objective scaling and zЉP/F between minima seemed to be compensated by a orientation technique indicates that uncertainty in azi- nearly equal change in . This indicated that the in- muth from orientation and scaling errors is ϳ0.1Њ, while ability to locate the PP might be associated with the uncertainty owing to the ``nebulous'' appearance and low elevation angles of all the landmarks. To test this de®nition of the features is probably closer to 0.2Њ.It hypothesis, the experiment was repeated with 4 is typically the case that more uncertainty accrues from changed from 0.7Њ to 20Њ. In this case, there was a global the identi®cation of a cloud feature than from image minimum near the PP at (0, 0.5) mm with an rmse of measurement, scaling, and orientation errors. At the 0.068Њ. There was also a comparable minimum at ranges of 6±10 km, an uncertainty in azimuth of 0.3Њ (0.5,2.5) mm with an rmse of 0.086Њ, which introduces equates to an uncertainty in position of about 30±50 m. uncertainty into the PP determination. When 3 also was This process can be automated easily to provide es- changed, from 2Њ to 15Њ, there was a single minimum timates of the 3D location of a certain object (the ``tar- at the PP with an rmse of 0.062Њ. Recall that accurate get''), given camera locations, and the azimuth and el- surveying of high-elevation-angle landmarks is very evation angles of the object from each site. Let a target sensitive to the precision with which the camera position T be identi®ed at angular position (PT, PT) in the image is determined. Since the likelihood of having two land- of camera P. We wish ®rst to ®nd the angular position marks with precise high-elevation angles is small, de- (CT, CT) of the same feature in a simulataneous image termination of the PP generally is impossible. taken by another camera C at an azimuth  and a hor-
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FIG. 5. Example of graphical intersection technique. The top image is from a 35-mm still photograph obtained by the ``PROBE-4'' VORTEX team located at ``P4'' on Texas Highway 194 southeast of Dimmitt. The bottom image is a digitized frame of Super-VHS video from the CAM-1 team located on Texas Highway 86 east of Dimmitt. The images were objectively scaled and oriented using the method described in the text. The map shows lines plotted from the camera sites along the measured azimuths of the center of the tornado near the ground, as well as the left and right extent of the opaque portion of the debris cloud. The circle represents the ϳ350 m diameter location of this opaque debris cloud.
Unauthenticated | Downloaded 09/30/21 08:41 PM UTC 1800 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 izontal distance S from P, and then ®nd the location in space of T. By triangulation and the sine rule (Fig. 6), the horizontal ranges from P and C to T are
S sin(PT Ϫ ) RCTcos CT ϭ and sin(PTϪ CT)
S sin(CT Ϫ ) RPTcos PT ϭ , (15) sin(PTϪ CT) where subscripts T, P, and C identify the target and the camera. The height of the target is simply Z* ϭϩZ C* RCT sinCT ϭϩZ P* RPT sinPT. If there are several fea- tures in C's image that might correspond to the target in P's image, then the automated technique is used as follows. The ray commencing at camera P and passing through the target is de®ned parametrically by
X* ϭ X 0P* ϩ R PTsin PTcos PT,
Y* ϭ Y 0P* ϩ R PTcos PTcos PT, and
Z* ϭ Z 0P* ϩ R PTsin PT, (16) from (1) and (2). Points along this ray appear in C's image at
xЉ cos sin FXЈ /YЈ CCCCCCϭ (17) zЉϪsin cos FZЈ /YЈ []CCCCCC [ ][ ] from (5) and (9) (assuming that the PP is at the origin), where FIG. 6. The geometrical relationship in two vertical planes and in projection onto a horizontal plane between a target T and two cameras X CCCЈ 1 0 0 cos␣ Ϫsin␣ 0 C and P. The numerical values represent the solution of the problem YЈϭ 0 cos sin sin␣ cos␣ 0 depicted in Fig. 7. CCCCC Z CCCЈ 0 Ϫsin cos 001 target image. In this case, the lowered portion of the X 0P* Ϫ X 0C* ϩ R PTsin PTcos PT distant cloud base to the right of the tornado at roughly ϫ Y* Ϫ Y* ϩ R cos cos (18) 15 750-m range is chosen as the solution. This technique 0P 0C PT PT PT Z* Ϫ Z* ϩ R sin has the attractive feature of allowing estimates of un- 0P 0C PT PT certainty based on how close the solution ray passes to from (1), (3), and (16). This givesx CCЉ andzЉ as a function target features. In the example, it appears that the un- of RPT and allows us to project the ray from P onto C's certainty is perhaps around 0.5Њ. Hence, this cloud fea- image with points along the ray labeled by values of ture is near azimuth 250.7Њ, elevation 0.8Њ, and range
RPT. Narrowing the search to a narrow strip (allowing 15 750 m from camera C. A correction term for earth for experimental error) is usually suf®cient to identify curvature and atmospheric refraction, ϩ6.76 ϫ 10Ϫ8 R 2 2 T in C's image and hence to deduce RPT. Then the an- cos (H82), is used in the height computations. This gular position (CT, CT) of T in the image of C can be correction is ϩ3 m at P and ϩ16 m at C. The target obtained from (11a) and the now-knownx CCЉ andzЉ , and height is roughly 1400 m MSL from Fig. 6. [The close RCT can be determined from (15). agreement between the calculations using P's and C's This procedure is illustrated by another example from data is probably fortuitous.] VORTEX. Figure 6 depicts the camera positions and This pair of images was chosen because of their image geometry for this example. A target cloud feature in a clarity in publication. In reality, they were not obtained photograph from camera P is chosen (i.e., the diamond at the same instant (as evidenced by the differing mor- near 267Њ,2Њ in Fig. 7a). The coordinates of points along phology of the tornado funnel), although the trailing the ray are computed from (16), and the ray is then low cloud feature was evolving slowly enough that this projected into the image of a second camera, C, using measurement might not be too much in error. Another (17) and (18). The projected points in C's image are pathology of this image pair is that the two cameras labeled with the pertinent values of RCT (see Fig. 7b). were quite close to collinear in orientation near the time The analyst then chooses the location along the ray of these images were obtained, which can lead to serious points that coincides with the same cloud feature of the errors in practice: stereo photogrammetry, like dual-
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FIG. 7. Example of automated graphical intersection for a full 3D target location solution. In the top image (obtained by I. Wittmeyer of the VORTEX PROBE-4 team) a target has been identi®ed on the lower edge of the cloud base trailing the tornado (black box containing white diamond). In the lower image, obtained by the VORTEX CAM-1 team, the ray from the PROBE-4 camera through the target is traced. Symbols are marked with range from the CAM-1 camera in tens of meters. The likely location of the target along the ray is between the 14 990- and 15 940-m range marks.
Doppler analysis, relies on adequate separation in the amorphous target, such as the edge of a cloud base. The angle of view. analyst simply chooses the point in the solution image It is not strictly necessary to identify distinctive tar- where the ray passes through the cloud base (being cog- gets. For example, it is often suf®cient to choose a more nizant of perspective), irrespective of whether a dis-
Unauthenticated | Downloaded 09/30/21 08:41 PM UTC 1802 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 tinctive feature can be noted on the cloud base. This APPENDIX approach is the one utilized in the foregoing discussion of the actual photographs. Small-Angle Solution for a Known or Assumed Principal Point 6. Conclusions The rearrangement of (13a),
Ϫ1 After deducing the camera site accurately by lining (x1ЉϪxPЉ)F Ϫ 11ϩ ϩ ␣ ϭ , (A1) up foreground and background landmarks in the im- (xЉϪxЉ)F Ϫ1 Ϫ ϩ ϩ ␣ ϭ , (A2) agery with those seen in person and then carefully mea- 2 P 22 Ϫ1 suring the azimuth and elevation angles of the land- (z1ЉϪzPЉ)F ϩ 11ϩ Ϫ ␣ ϭ , and (A3) marks, it is possible to perform accurate photogram- (z z )F Ϫ1 , (A4) metry on images with unrecorded focal length and cam- 2ЉϪ PЉ ϩ 22ϩ Ϫ ␣ ϭ era orientation angles. We have developed an algorithm reveals that (13a) is really a linear system in the four (section 3) that rapidly deduces the focal length, azi- variables F Ϫ1, , ϩ ␣, and Ϫ ␣ (or F Ϫ1, , muth, and tilt of the optical axis, and the roll angle of Ϫ1 Ϫ1 ϩ ␣ Ϫ xЉPPPPF , and Ϫ ␣ Ϫ zЉF ifxЉ and if zЉ the camera, and then computes the azimuth and ele- also are regarded as unknowns). We de®ne the differ- vation angles of any feature in the image. Assumptions ence operator ⌬ by ⌬()ϵ ()1 Ϫ ()2. Subtracting (A2) that the roll is negligible, that the visible horizon is the from (A1) and (A4) from (A3) results in a linear system lens horizon, and that angles are small are unnecessary in two of the unknowns, namely and potential sources of error. Although the small-angle solution derived in section 3 and the appendix might be ⌬xЉF Ϫ1 Ϫ⌬ ϭ⌬ and (A5) suf®ciently accurate for telephoto images, there is no ⌬zЉF Ϫ1 ϩ⌬ ϭ⌬, (A6) advantage to using it over the more general and accurate algorithm if the time to develop the computer code has which, by Cramer's rule has the solution been invested previously. The method accommodates F Ϫ1 ϭ (⌬⌬ ϩ⌬⌬)/(⌬⌬xЉϩ⌬⌬zЉ) and the scale distortion inherent in wide-angle photographs (e.g., the distortion visible in the wide-angle photograph ϭ (⌬⌬xЉϪ⌬⌬zЉ)/(⌬⌬xЉϩ⌬⌬zЉ), (A7) of Fig. 5). Optical effects, such as pincushion and barrel provided that ⌬⌬xЉϩ⌬⌬zЉ 0. With F Ϫ1 and distortion, are not accommodated here, but are treatable ± now known, we rewrite (A3) and (A1) as a linear system using techniques readily found in Web searches. Results in the unknowns and ␣, are insensitive to the exact position of the principal point Ϫ1 for telephoto images. For wide-angle photography, the Ϫ ␣ ϭ 11Ϫ Ϫ (z 1ЉϪzPЉ)F and (A8) principal point can be determined only if there is a suf- Ϫ1 ®cient number of precisely measured landmarks with ϩ ␣ ϭ 11ϩ Ϫ (x 1ЉϪxPЉ)F , (A9) diverse azimuth and elevation angles. If all the land- and thus obtain marks have low elevation angles, the PP is impossible Ϫ12 to determine and must be assumed to lie at the inter- ϭ 11Ϫ F [(xЉϪxPЉ) ϩ z1ЉϪzPЉ]/(1 ϩ ) (A10) section of the diagonals of the uncropped image. A photogrammetric search technique is described for and ®nding an entity, which is visible in one camera's pho- Ϫ12 ␣ ϭ 11Ϫ F [xЉϪxPЉϪ(z1ЉϪzPЉ)]/(1 ϩ ). (A11) tography, in a simultaneous image obtained from a dif- ferent direction by a second camera. Once the same object has been identi®ed in both images, its 3D position REFERENCES is determined by triangulation. This method could be used to superpose visible cloud features onto ®elds of Bluestein, H. B., and J. H. Golden, 1993: A review of tornado ob- re¯ectivity and Doppler velocity observed by mobile servations. The Tornado: Its Structure, Dynamics, Prediction, Doppler radar. and Hazards, Geophys. Monogr., No. 79, Amer. Geophys. Union, 319±352. Golden, J. H., and D. Purcell, 1978: Air¯ow characteristics around Acknowledgments. This work was supported under the Union City tornado. Mon. Wea. Rev., 106, 22±28. Grants ATM-9617318 and ATM-0003869 from the Na- Hoecker, W. H., 1960: Wind speed and air¯ow patterns in the Dallas tornado of April 2, 1957. Mon. Wea. Rev., 88, 167±180. tional Science Foundation. Additional support was pro- Holle, R. L., 1982: Photogrammetry of thunderstorms. Thunder- vided by the National Severe Storms Laboratory. We storms: A Social and Technological Documentary, E. Kessler, gratefully acknowledge very thorough and helpful re- Ed., University of Oklahoma Press, 77±98. views of the anonymous reviewers. Computer routines Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, 1986: Numerical Recipes: The Art of Scienti®c Computing. Cam- in the IDL programming language can be obtained from bridge University Press, 818 pp. the corresponding author to perform some of the pho- Rasmussen, E. N., J. M. Straka, R. Davies-Jones, C. A. Doswell III, togrammetric functions described in this paper. F. H. Carr, M. D. Eilts, and D. R. MacGorman, 1994: Veri®cation
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