system errors of differential rectifiers with optical …...horst scholer lenoptiklena g.m.b.h....

HORST SCHOLERlenoptik lena G.m.b.H.

German Democratic Republic

System Errors of DifferentialRectifiers with Optical ProjectionErrors in on-line optical projection orthophoto systems, andthe correction of those errors, are discussed.

IMAGING ERRORS IN "ON-LINE" INSTRUMENTSYSTEMS

I N A NUMBER of articles on the orthophototechnique published in recent years the

term "sytem errors" has been used repeatedly. These papers were usually published with the view of propagating "offline" systems. "Off-line" systems in thssense are understood to be combinations ofstereoplotters and differential rectifiers withwhich profile plotting and differential rectification are done in successive processesor in which the differential rectification ismade with the aid of a stored digital terrainmodel from previous profile scannings.

taneously with the exposure of an imagestrip along this profile. For economicreasons it is necessary to make this strip asbroad as possible so as to reduce the scanning time of a model. Since, however, thegained height data for the control of the differential magnification of the image detailslimited by the scanning diaphragm are correctly valid only for the profile scanned withthe floating mark, projection errors must resuIt at the edges of the image strips. If a terrain slope exists across the scanning strip,discrepancies in the scale must occur in theorthophoto along the strip edges due to thestaircase arrangement of the parallel strips.Consequences of this are, on the one hand,

ABSTRACT: The paper investigates the imaging errors in "on-line"instrument systems. Errors present themselves as double images,missing image detail, and image point displacements along the strip.A discussion of the errors is followed by pointing out instrumentalcorrection capabilities. Considerations regarding image motion, resolving power, and the flatness of photographic material concludethe paper.

On the other hand, in a true "on-line" system the stereoplotter is directly connectedwith the differential rectifier without interposing other functional elements (whichmay also include digital process computers).The differential rectifier may be an orthophoto equipment with "oblique projection" (orthoprojector) or with "frontal projection." In all hitherto built instrument combinations of this type information on the elevation pattern in the profile is gained simul-

* Presented Paper at the Orthophoto Symposium of the International Society for Photogrammetry (I.S.P.) in Sao Paulo/Brazil in July1975.

the double imaging or the omission of imagedetails along such edges and, on the otherhand, displacements of image points in thestrip direction on either side of the scannedprofile line.

DOUBLE IMAGING AND MISSING IMAGEDETAILS

After these general explanations it will beplausible that the size of such image defectsmust be dependent on the dimensions of theslit diaphragm, the terrain slope across thestrip direction, and the field angle of thecamera used. Figure 1 may be used for amathematical consideration. Assuming the

PHOTOGRAMMETRIC ENGINEERING AND REMOTE SENSING,

Vol. 42, No. 12, December 1976, pp. 1505-1509.

1505

1506 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976

r ]I

plane ofrectlficotlon 1 [mplane ofrecflficotion2

I(errain

I

- -- region ofdoubt" imagery

FIG. 1. Image formation in optical differential rectifiers.

c = b . tan ax . tan Tx , (3)

DISPLACEMENTS OF IMAGE POINTS IN THE

STRIP DIRECTION

the width of double imaging in strip I becomes

If Figure 1 is now considered as a sectionof the projection process in the y-z plane(Le., parallel to the strip direction), it be-

comes clear that due to the staircase formation of the horizontally lying strips a displacement of image points in the strip direction (here the y direction) must also takeplace at the separating edge. This is demonstrated by a sawtoothlike formation ofpatterns crossing the strip edges as a result ofa terrain slope normal to the strip.

Analogous to the preceding considerationson double imaging, a y displacement is produced for terrain point 3, which is mathematically formulated as

v = 1/2 b . tan ax . tan T y , (4)

where T y is the component of the fieldangle of the camera used, which lies in they-z plane.

DISCUSSION OF THE IMAGING ERRORS

Equations 3 and 4 describe the defects occurring at the strip edges in the orthophoto,which was produced in differential rectifierswithout devices for performing a "slope correction" (i.e., consideration of the terrainslope across the strip direction). Since thetechnical realization of such a correction ofthe image details limited by 'the slit diaphragm is only possible by a considerableequipment outlay and, on the other hand,requires a departure from the highly productive and simple "on-line" technique, an ideaof the magnitude of the errors involvedshould first of all be conveyed.

Equations 3 and 4 show that the size oftheimaging errors is a direct function of thecomponents Tx and Ty of the field angle T ofthe camera used and of the terrain slope ax.There are possibilities, on the one hand, to

(1)

(2)

c = li.h . tan Tx .

Since li.h = b . tan ax,

scanning strips to be situated in the y direction, a section through the z-x plane is shown(perpendicular to the strips produced by theslit width b). The terrain surface 1-2-3 inclined by the angle ax is imaged in the orthophoto within the section 1-4 in strip I andwithin the section 2-5 in strip II. One willeasily see in our assumed case that doubleimaging takes place in the strips I and II.The area ofdouble imaging is determined bythe projection of the height difference 4-5between the profile axes A and B onto thehorizontal slit planes in the strips I and II bymeans of the rays passing through the projection center O. Thus, the width of the overlaparea c in strip I is produced by the ray passing through edge 5 of the image strip II.

According to Figure 1, the width of theoverlap area is

where ax is the slope of the terrain plane and'7x the component of the field angle of thecamera used, which lies in the x-z plane.

SYSTEM ERRORS OF DIFFERENTIAL RECTIFIERS 1507

influence the angle T during the taking process by choosing an aerial camera with suitable focal length with due consideration ofthe existing terrain shape. On the otherhand, it is equally of importance which partsof the aerial image are to be used for theproduction of the orthophoto.

In a picture series with a normal imageoverlap of 66 percent each stereogram generally has a width of two thirds of the useable format side, which corresponds to twobase lengths at the image scale. Hence, successive models overlap by 50 percent in thestrip. Thus, for the compilation of an orthophoto map one requires orthophotos witha width of one third of the useable imageformat side. When according to Figure 2 weuse the right image of a stereogram for theproduction of the orthophoto, we have thechoice between the image area I (with theimage coordinates y' = +y'n, x' = 0; y' =-y'n, x' = 0; y' = -y'n, x' = -b; y' = +y'n, x'= -b) and the image area II (with the imagecoordinates y' = +y'n, x' = + b12; y' = -y'n,x' = + b12· y' = -y' x' = -b12 . y' = +y' x'= -bI2).' In combinations of severaln~r_thophotos there is no doubt that preferenceshould be given to the image area II as central part of photograph 2 in the interest of anoptimum image quality. Also in our considerations of the imaging errors it is of advantage that the component Tx of the field angleof the camera (in Equation 3) becomes considerably smaller than when image area I isused.

The mean errors appearing as double im-

FIG. 2. Use of different photo areas for the production of orthophotos.

aging and omission of image details at theseparating edge of the extreme marginalstrips of the model or as y displacements inthe extreme model corner are listed in Tables 1 to 3 with the argument of a maximumterrain slope (these are the maximum inclinations of the terrain normal to the contourlines) for various strip widths.

INSTRUMENTAL POSSIBILITIES FOR

CORRECTING IMAGING ERRORS

As is shown by the discussion of the imaging errors to be expected, it will certainly bepossible to dispense with slope correctionfor a very large variety of tasks. If, however,these imaging errors are to be compensatedwith instrumental devices, then the imagedetail lying in the scanning diaphragm mustbe additionally magnified as a function ofthe terrain slope ax and the component Tx of

TABLE 1. MEAN PROJECTION ERROR AT THE MODEL EDGE WITH OMITTED SLOPE CORRECTION FOR AMODEL FORMAT OF 76 X 200 mm2 WHEN USING ACAMERA WITH A FOCAL LENGTH OF 305 mm AND

AN IMAGE FORMAT OF 230 X 230 mm2•

Slit Imaging errors (mm)

width Model area II Model area I

(mm) 5° 10° W 25° 35° 45° 5° 10° 15° 20° 25° 35° 45° Terrain slope (0) max.

2 0.01 0.02 0.02 0.04 0.06 0.01 0.02 0.03 0.04 0.05 0.08 0.11 Overlap (mm)0.02 0.02 0.03 0.05 0.08 same as model area II y displacement (mm)

in the model corner

4 0.01 0.02 0.03 0.05 0.08 0.12 0.02 0.04 0.06 0.08 0.11 0.16 0.23 Overlap (mm)0.01 0.03 0.04 0.07 0.11 0.16 same as model area II y displacement (mm)

in the model corner


in the model corner


in the model corner

1508 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976

TABLE 2. MEAN PROJECTION ERROR AT THE MODEL EDGE WITH OMITTED SLOPE CORRECTION FOR AMODEL FORMAT OF 76 X 200 mm2 WHEN USING A CAMERA WITH FOCAL LENGTH OF 153 mm AND

AN IMAGE FORMAT OF 230 X 230 mm2 •

Imaging errors (mm)

Slit Model area II Model area Iwidth -------------.-------------(mm) 5° . 10° 15° 20° 25° 35° 45° 5° 10° 15° 20° 25° 35° 45° Terrain slope (0) max.

2 0.01 0.02 0.03 0.04 0.05 0.98 0.12 0.02 0.03 0.06 0.08 0.11 0.16 0.24 Overlap (mm)0.01 0.04 0.06 0.08 0.11 0.16 0.24 same as model area II y displacement (mm)

in the model corner


in the model corner


in the model corner


in the model corner

the field angle of the camera used and itmust be rotated in azimuth about the slitcenter as a function of the terrain slope ax

and the component Ty of the field angle ofthe camera used. In differential rectifierswith frontal projection the center of the profile (in Figure 1, point 2 for strip II and point1 for strip I) lies always in the axis of theprojection equipment.

From Figure 1 it follows that the image ofthe terrain surface 2-3 must cover the entirehalf width b/2 of the scanning diaphragm inorder to avoid double imaging. If e is the

terrain section 2-3 imaged on slit II by thecentral perspective ray 3-0, then the additional magnification Zv for e with the designations used in Figure 1 is

b 1Zv =- = (5)

2e 1 - tan Tx . tan ax

The image displacement in the y directionoccurring at the strip edge had been expressed by Equation 4. Since the center of thescanning diaphragm is free from errors, one

TABLE 3. MEAN PROJECTION ERROR AT THE MODEL EDGE WITH OMITTED SLOPE CORRECTION FOR AMODEL FORMAT OF 76 x 200 mm2 WHEN USING A CAMERA WITH FOCAL LENGTH OF 88 mm AND

AN IMAGE FORMAT OF 230 x 230 mm2 •

Slit

width Model area II

Imaging errors (mm)

Model area I

(mm) 5° 10° 15° 20° 25° 35° 45° 5° 10° 15° 20° 25° 35° 45° Terrain slope (0) max.


in the model corner


in the model corner


in the model comer


in the model corner

SYSTEM ERRORS OF DIFFERENTIAL RECTIFIERS 1509

obtains corrected image point positions outside the center by the azimuthal rotation ofthe image details lying in the slit diaphragm.With Equation 4 this angle of rotation {3 is

{3 = arc (tan ax . tan Ty ) (6)

In these derivations for the effect of correction devices it is assumed that the terrainrelief in the scanning strip can be approximated exactly enough by a flat plane lyingobliquely in space. Only in this case it ispossible to increase the slit width by the application of slope correction and thus shorten the scanning time. When this conditiondoes not apply, one can actually eliminatethe disturbances at the strip edges, but instead "unvisible errors" are produced withinthe strip.

IMAGE MOTION AND RESOLVING POWER

In most differential rectifiers with opticalprojection the scanning slit is continuouslymoved along the scanning profile. Here theeffective exposure time (the same as for afocal-plane shutter) results from the scanning speed and the dimension of the slit inthe direction of motion (slit depth). Considering the geometric imaging conditions at aninfinitely short moment, we see that thesame viewpoints are of relevance as in Figure 1 transverse to the direction of motionwith regard to the double imaging. Since,however, the slit depth is considerably smaller than the slit width. these effects are reduced considerably. If a stepwise motion ofthe scanning slit and an exposure understationary conditions were carried out, theseeffects would, however, become noticeablefor a terrain slope au lying in the profile di-

rection. With continuous exposure, however,the image detail lying in the diaphragm is,due to the z tracking, subjected to a permanent change of magnification. Owing to thefinite extension of the diaphragm in the direction of motion, an image motion is produced during the exposure time which affects the resolving power of the resultingimage in exactly the same way as in aerialphotography. This effect becomes less significant as the slit length and the height differences of the terrain become smaller. Thisis the reason why the narrowest possible slitdiaphragms are recommended for use.

FLATNESS OF PHOTOGRAPHIC MATERIAL

The geometry of the image can in additionbe influenced to a noticeable order of magnitude by an insufficient flatness of thephotographic material to be exposed. It iseasily understood that image point displacements caused by this are a function ofthe field angle during imaging. In the orthophoto projectors with reconstruction ofthe camera's path of rays it is therefore expedient to take special measures to ensure asufficient flatness of the photographic material. In consideration of the abovementioned model ranges recommended forthe orthophoto production, a flatrress tolerance of 0.1 mm must be maintained forwide-angle photographs and of 0.07 mm forsuper-wide-angle photographs, when theimage point position at the edges and in thecomers of the orthophotos shall not exceedthe limit of 0.1 mm.

Differential rectifiers with frontal projection and very small field angles (usuallybelow 1°) are practically insusceptible tosuch influences.

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Year Numbers

February 1975-Vol. XLIMarch 1975-Vol. XLI

No.2No.3

system errors of differential rectifiers with optical …...horst scholer lenoptiklena g.m.b.h....

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