C H A N M . P A R K * Y U N G H . LEE*

B E R N A R D B . S C H E P S ~ University of M a r y l a n d

College P a r k , Md. 20740

Slope Measurement from Contour Maps Approximate slope computation from a digital map is feasible at reasonable computer times.

INTRODUCTION RELATED PRIOR WORK

C ONTOUR MAPS of many kinds are pro- Fischer' has tested a n optical-mechanical duced t o describe spatially distributed method of producing a slope map from a

data . Such maps contain isolines, which are given contour map. H e used a positive and a the loci of points a t which some measured negative transparency of the contour sheet quant i ty takes on a given discrete set of separated by a diffusing transparency (semi- values. For example, in a topographic con- mat te translucent acetate). An eccentric turn- tour map, the isolines are lines of equal ter- table was constructed and used t o nutate the rain elevation; in a weather map, they may positive transparency relative t o the nega- be lines of equal pressure (isobars), temper- tive a t a given nutation radius.$ A controlled,

ABSTRACT: T h i s paper describes two methods of estimating slope gradients f rom a digitized contour m a p . The first methpd uses amount of contour l ine per uni t area as a slope measure, whereas the* ~ c o n d computes slope b y measuring dis- tances to nearest contour lines. Both"methods have been implemented in P A X I1 o n the Univac 1108 computer.

a ture (isotherms), rainfall (isohyets), etc.; diffuse light source illuminated the trans- and in other examples they may represent parencies from below, and a camera, mounted any of a wide variety of d a t a (stress, magnetic above, recorded the results in a time exposure variation, radiation, etc.). taken over a period of one or more complete

Users of contour maps often desire to nutations. I t is evident t h a t in this arrange- know the rate of change (i.e., first derivative, ment, more light will pass through a region or slope) of the data . Calculation of slope having many contour lines per unit area than along a particular path is a straightforward through a region having few or none. Because matter. I t is less trivial, however, to deter- slope is also high where there a re many con- mine the slope gradient (i.e., the maximum tours per unit area, the photographic re- slope in a n y direction) a t a point, let alone cording can thus be regarded a s a slope map. a t all points of a region. This paper describes I t is a fairly straightforward matter t o cali- computer programs which measure, "in brate this system so as t o be able t o convert parallel," approximations t o t h e slope gra- a n y density on the photograph into a n dient a t all points of a digitized map, so as t o equivalent number of lines per uni t area yield a slope m a p of the given region. An example of the results obtained by this

method is shown in Figure 1.

* Computer Science Center, University of Mary- $ Fischer suggested that the nutation radius land. should be CIS, where C is the contour interval

i. Army Engineer Topographic Laboratories, Ft. divided by the scale of the map, and S is the small- Belvoir, Virginia. est slope to be detected.

FIG. 1. Fischer's nutation method. Input contour[map (top) and Output slope map (bottom).

Monmonier, Pfaltz and Rosenfeld2 re- ported a computer program called SAMP which estimated surface area from a contour map. T o arrive a t area, the program first computed slope a t each point by determining distances to the nearest contours in two orthogonal directions. Different procedures were used depending on whether or not the given point was itself on a contour line, and on whether or not another contour line existed in one or both directions within a specified distance (after which the terrain

was regarded as flat). The slope computation also depended on whether the contours reached were the same or different. The methods reported in this paper are simpler in tha t they consider only distances between contours (whether the same or different), and do not give special treatment to points on contours. On the other hand, the present methods provide map output, which SAMP did not. Figure 2 shows a digital contour map which was used as input to both the SAMP program and the programs reported

SLOPE MEASUREMENT FROM CONTOUR MAPS 279

. c 1 ?

7 1 72 1 2 1 2 I 1 I

hereinafter: one picture element on this map h : >2

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>2 I 1 2?7 1 I 1 I 2 1 2

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2 1

2 I >22 7 1 2 1 THE DIGITAL NUTATION METHOD

I 1 227 1 2 1

2 1 22 1 2 1

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2,2 / I 1 L A simple digital version of Fischer's opti-

2 1111111111 2 I 7 1 2 1 cal-mechanical method is a s follows: T h e 1

2 1 2 1 I

1 2 1 2 1 2 1 positive and negative transparencies a re rep- 4 2 I 2 1

I 22 I I 227~?02,7?2 I 2 1 2 1 resented by the digital contour map and i ts 2 I 2 1

I 2 1 1 2 1 2 1 I 2 1 , 1 1 2 t 2 ~ complement. One of these maps is nutated I 7 1 2 1 1 2 1 2 1 2

I I I I 1 I I 1 I 1 I , 7 1 2 ; i 3, :;; 22 : 2 : :: about t h e other; for each relative shift, they I 2 1 . 11 : ; : : : are ANDed, and the results a re summed over

I 2 1 . 1 2 1 , : ; : : : : a complete nutation. Figure 3 shows the re- I , 1 2 2 1 2 I ? 2 1 7 1 2 2 1 sulting sums for a nutation amplitude of 10. 1 1 2 2 1 2 1 2 2

9 2 I 2 - 1 2 I 2 2 1 AS can be seen, the results a re too discrete, I 2 I 2 r 2 1 2 1 2 2 1 F I ? ,

1 2 1 : : ; ; : : perhaps because the nutation amplitude is not 1 7 l * ? I : : : : : great enough. Furthermore, there is a ten-

??2 I r 1

r 122 1 ,' : : : : : dency toward discontinuity both along and

1 - 1 I 2 1 2 2 1 3 2 1 I 1111 2 1 2 across contours. This may be due t o the

77 I 2 ? 2 1 2

2 1 2 2 1 .

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1 2 1 2 22 : : : : the optical process was not simulated in the 1111 ?

1111 2 2 ~ 7 2,2r2122 : $ : 11 digital method. 2 2

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1 2 2 1

22 1 2 2 1 I t was decided t h a t a closer equivalent t o

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! l 1 1 iP 7 L 1 ~ I l , l : : : smoothing or blurring the results of t h e nuta-

7 1 2 L

, 2 - I. ?. : : : tion, i.e., averaging over a fixed neighborhood a t each point. The results of such a smooth-

FIG. 2. Digital contour map. ing, by averaging over a circular neighbor-

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FIG. 3. Digital nutation method (no averaging; FIG. 4a. Digital nutation and averaging nutation radius 10) (blur radius 4), printout,

FIG. 4b. Picture output for Figure 4a.

hood of a radius of four a t each point, are shown in Figure 4. However, quantitative evaluation for eight test points (Table 1) shows deviations of several percent of slope, which would be too rough a n approximation for most purposes. Further s tudy is needed to determine the optimum nutation radius and blur radius for this method.

Qualitatively, a simple smoothing of the original contour map should also yield high values in areas of high slope. T h e iesults of such a smoothing, using circular neighbor- hoods of radius 10, are shown in Figure 5 .

T h e second digital method tested uses straightforward measurement of distance be- tween contours on the digitized map. As

Result of Result of ' lope nutation digital slope cOmp'Lted as

Point gradient SWa'e root

(X3) of s u m

method of squares

* This discrepancy is probably due to interaction between the nutation and the small closed contour

FIG. 5a. Averaging of original contour map (radius lo), printout.

the map scale and contour interval are known, slope can be easily computed from such a distance measurement. This was done in both the x and y directions (Figure 6a-d).*

* It would have been desirable, either as a check or to provide a closer approximation, also to com- pute slopes in the 45' directions. However, as the contour lines on the digitized map are thin, searches along 45' lines would often cross them without detecting them.

just surrounding Point C. FIG. 5b. Picture output for Figure 5a.

SLOPE MEASUREMENT FROM CONTOUR MAPS 2 8 1

FIG. 6a. Digital slope computation. Distance be- FIG. 6b. X-component of slope for Figure 6a. tween co~ltour lines in X-direction

FIG. 6c. Similar to Figure6a but in the Y-direction. FIG. 6d. Similar to Figure 6b but in the Y-direction.

282 PHOTOGRAMMETRIC ENGINEERING. 1971

The slope gradient can then be computed as the square root of the sum of the squares of these x and y slope components. T o simplify the computation, an approximation to the square root was used, namely u+v/2, where u is the larger and v the smaller of the two components. (Note tha t for v = O we have

whereas for v = u we have

and

so tha t this approximation is reasonably good a t both ends of the range O<v_<u.) The result is shown in Figure 6e. A quantitative evaluation of the results for the eight test points is given in Table 1.

A slope-class map was constructed by thresholding these results, using the slope intervals 0-2%, 2-5y0, 5-lo%, 10-2095, and >20y0 (Figure 7a). The map has a some- what blocky, grainy appearance. One cause of this is the fact tha t the estimated slope usually has discontinuities a t contour lines. T o combat this, the slope map was smoothed by assigning to each point on a contour line the average of the slopes a t the four neigh- boring points, and further by averaging over neighborhoods. The results are shown in Figure 7b. Analysis of the printout shows tha t scattered errors still survive, but if de- sired, many of these could be eliminated using standard noise-cleaning techniques.

The errors for the gradient method are of about the same magnitude as in the nutation method, but are differently distributed. An interesting difference between the two methods is that near the edges of the map, the nutation method underestimates the slope, because the map is blank beyond its edge, whereas the gradient method overesti- mates the slope, because i t treats the edge of the map as a contour. I n general, the gradient nlkthod is more flexible,because i t can pro- vide a variety of intermediate products, stich as slopes in specific directions.

The results of these studies indicate that approximate slope computation i n parallel from a digital contour map is feasible. The computation times required were not exces- sive (of the order of one minute), and on a parallel computer such as ILLIAC 111 i t should

FIG. 6e. Approximation to the gradient for Figure 6a.

be possible to do the computations for an entire map in times of tha t order.

An important limitation on such ap- proaches t o slope computation is imposed by the fact tha t the contour lines must be one picture element wide. (In the optical analog application too, the contour lines on the transparencies must have finite width.) Thus the accuracy of slope estimates can never be greater than 1/D, where D is the average distance between contours. This limitation is especially serious for steep ter- rain; thus the input map used in the tests represents a worst case. In spite of this, the results are believed to be encouraging, and the methods warrant serious consideration for use in practical applications.

ACKNOWLEDGMENTS

The work reported was initiated by Mr. Scheps in connection with geographic data processing studies under a Secretary of the Army Research and Study Fellowship. Mr. Lee implemented the first method. The guid- ance of Dr. Azriel Rosenfeld of the Uni- versity of Maryland, and the support of the National Aeronautics and Space Administra- tion and the Atomic Energy Commission, are gratefully acknowledged.

SLOPE MEASUREMENT FROM CONTOUR MAPS 283

FIG. 7. Digital slope class map. (a) Unsmoothed (top), and (b) Smoothed (bottom).

REFERENCES 2. M. S. Monmonier, J. L. Pfaltz and A. Rosen- 1. W. A.Fischer, Personal communication to B. B. feld, Surface area from contour maps, Photo-

Scheps, 29 October 1963. gram. Eng. 32, May 1966,476-482.

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