FIG. 1. Vertical panoramic photograph of Washington, D.C. with the Capitol on the left. DONALD A. KAWACHI* Fairchild Space & Defense Systems Syosset, L. I., N. Y. Image Geometry of Vertical & Oblique Panoramic Photography A transparent grid overlay, like the Canadian Grid, provides a practical means for overcoming the effects of image motion and geometric distortion. (Abstract on page 300) INTRODUCTION the fil m. The radius of the cylinder is the y FAR THE MOST COMMON TYPE of camera focal length of the lens. The fil m is exposed B used in aerial photography is the frame by rotating both lens and slit on the axis of camera, but another type, the panoramic the cylinder, as shown in Figure 3. camera, has found increasing usage in recent I t is easy to visualize how this camera years. This increased usage stems from the ex­ achieves a much larger field of view. By tensive enlargement of the continuous field of rotating the lens the viewing angle is con­ view for each photograph, a factor which tinuously varied for the same piece of film. comes at the cost of distorting the view of the In one direction the coverage extends to ground space. nearly 180°, limited only by the film getting in its own way. For some of the more sophisti­ DESCRIPTION OF PANORAMIC PHOTOGRAPHY cated configurations even this limitation is In its simplest form the panoramic camera circumvented, with the result that more than consists of a film curved in the shape of a 180° of coverage is achieved. This is an im­ cylinder with the lens on the axis of this portant factor, for it allows horizon-to­ cylinder and an exposure slit just in front of horizon coverage in vertical aerial photog­ raphy. * Two previous articles on the same general subject are cited as Rderellces 5 and 6 at the end A typical panoramic photograph wi th of this artide.. horizon-to-horizon coverage is shown in Fig- 298 iMAGE GEOMETRY OF VERTICAL & OBLIQUE PANORA.fIC PHOTOGRAPHY 299 FIG. 2. Top view of the ground coverage for a vertical panoramic photograph. The fractions refer to the ratio of the scale to the scale of a vertical frame photograph. ure 1. This is a view of Washington, D. c., is achieved by rotating a mirror or prism in with the Capitol building appearing on the front of the lens in synchronization with the left. Note that the scale diminishes toward movemen t of the ftl m on the rollers. The rota­ the horizon with a corresponding increase in tion of the reflector accomplishes the lateral the area coverage. The actual coverage of a coverage by bending the light rays. Actually panoramic photograph is drawn in Figure 2. a mirror is impractical because it fails to re­ The borders of this coverage are hyperbolas. flect the light rays when vie\\'ing directly Included among the more sophisticated below. This obstacle is overcome by replacing panoramic configurationsl is a camera with a the mirror with a double dove prism shown in non-rotating lens and the fil m stretched be­ Figure 3, which reflects the light rays in all tween two rollers behind an exposure slit as positions. shown in Figure 3. The panoramic coverage Al though a tri metrogon array of frame FIG. 3. Simplified diagrams of two types of panoramic cameras. 300 PHOTOGRAMMETRIC ENGI EERING cameras conSlstlllg of a vertical camera and motion compensation are fully accurate, the two side oblique cameras is also able to only sources of image motion from the for­ achieve horizon-to-horizon lateral coverage, ward movement of the aircraft are the vari­ its imagery is not on a single strip of film and able scale within the slit (parabolic image 2 the system requires three cameras. If longer motion ) and the ground topography. focal length lenses are included in the oblique For the forward oblique position, however, cameras to achieve higher ground resolutions, an additional source exists, caused by the abrupt changes in scale result between the variation in magnitude and direction of the \'ertical and oblique cameras. image velocity over the film. Complete com­ pensation of this velocity is impossible. For­ IMAGE Mono, AND DrSTORTlON tunately the image velocity is smaller for the Two sources of image motion in aerial oblique orientation since the scale is less, so photography are the forward veloci ty of the that the effect of the residual velocity may be aircraft and the random movements due to negligible. varying air densities and atmospheric turbu- The forward movemen t of the aircraft also ABSTRACT: This is the last of three articles on image motion in aerial photog­ raphy. Thejint one, in the January 1965 issue, was a mathematical analysis of the image motion compensation and the image motion from the aircraft's for­ ward movemen t for the oblique frame camera. The second article, in the Sep­ temberissue, analyzedthe imagemotionfromrandom rotational motionsofthe air­ craft for the frame and panoramic cameras. To complete the series of studies, the image movement from the aircraft's velocity for vertical and oblique panoramic cameras is investigated here. A brief description of the panoramic camera includes the distortions associated with it. Several equations of interest pertain to the panoramic cameras, such as the coordinate transformation and image velocity equations, from which the proper image motion compensation velocity, residual image motion, and the image displacements are derived. A n easy method of rapid location of ground points from the panoramic photograph can be applied. lence. The random motions of the aircraft causes distortion in the photograph because generally are not a problem except at high of the focal plane shutter. The primary dis­ altitudes where high resolution is necessary. tortion in panoramic photography from the A stabilized mount may be included to damp true pictlu'e of the ground is, of course, the out the motions. At very low altitudes fast panoramic distortion caused by the curvature shutter speeds minimize the effect of the of the film. The shutter introduces two addi­ larger pitch and roll motions. The effect of tional image displacements, These displace­ random or rotational motions was examined ments arise from the movement of the air­ in Reference 6, but the equations for the im­ craft as the shutter sweeps across the ftlm age motion on the vertical panoramic photo­ and the changing position of the lens relative graph are repeated later in this article. to the film for image motion compensation. For vertical panoramic photography the The first is known as the sweep posi tional effect of the forward velocity is greatly re­ displacement and the second, the IMC dis­ duced by imparting a movement to the lens placement. Both are depicted in Figure 4. or film (image motion compensation). The A detailed description of the sweep posi­ proper velocity of this movement is deter­ tional and IMC displacements was included mined by a measurement of the angular in Reference 5 as the displacements apply to velocity of the object space, obtained from oblique frame photography. It is sufficient to a velocity-to-height sensor. The required state here that the two displacements do not velocity varies \\·ith the cosine of the lateral cancel each other, and there is a residual dis­ viewing angle. If the sensor and the image placement. However, these displacemen ts are IMAGE GEOMETRY OF VERTICAL & OBLIQUE PANORAMIC PHOTOGRAPHY 301 FIG. 4. Displacement of center line due to the sweep positional 'l.nd 1Me distortions. known and are, therefore, recoverable. For velocity of the ground point into three mu­ the oblique orientations the tilt of the camera tually perpendicular components, one along produces another distortion. the line-of-sight which does not contribute These distortions prevent rapid photointer­ to the image velocity and the others in direc­ pretation without special equipment. How­ tions parallel to the x and y-directions of the ever, there are methods of overcoming this photograph, and mul tiplying these com­ obstacle. One such technique involves a ponents by the appropriate scale factors. This transparent grid overlay which allows the procedure is valid because one component rapid location of image points appearing on contributes only to the velocity in the x­ the panoramic photograph. This technique is direction and the other to the y-direction. described in more detail later. The formulas for the sweep positional and IMC displacement are derived by integrating DESCRIPTION OF THE EQUATIONS the image velocity with respect to time, where AND METHOD OF DERIVATION time is measured from the moment at which Equa tions for several parameters of interest the nadir point is exposed. The difference in are given in this article, as they apply to the integrations for the two displacements vertical and forward oblique panoramic arises in the treatment of the sweep angle; for photography. These are: the sweep positional displacement this angle Transformation equations from photo to is treated as a constant and for the IMC dis­ ground coordinates. placement it is made a function of time. The The velocity of the image from the forward reason is that the velocity of an image point movement of the aircraft. depends only upon the sweep angle of the The velocity of image motion compensation. image point, which is a constant, but the The residual image velocity. velocity of the lens varies wi th the sweep The image velocity from rotational motions angle of the lens. of the aircraft. The transformation equations for the ob­ The sweep positional displacement. lique camera are derived by treating the The image motion compensation displace­ panoramic photograph as a series of infini­ ment.