ARD-A166 416 ANALYSIS OF GRADIENT CHANGE THRESHOLDS IN THE DETECTION i/iOF EDGES OF OBJEC. (U) ARMY ARMAMENT RESEARCH ANDDEVELOPMENT CENTER WATERVIIET NY C.. C N SHEN ET AL.

UNLSSIFIED FEB 86 ARCCB-TR-8607 SI-D-E4 8 317 F/G t7/5 U

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AD-A166 416 _ _ _TECMN#CAL REORT ARCCTR-66007

ANALYSIS OF GRADIENT CHANGE THRESHOLDS IN THE

DETECTION OF EDGES OF OBJCTS FROM RANGE DATA

C. N. SHEN

R. L. RACICOT MAQDacTE

FEBRUARY 1986

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT CENTERCLOSE COMBAT ARMAMENTS CENTER

I-sI ,0 e~9FN9T WEAPONS LABORATORY "-i"

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

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ARCCB-TR-86007 /1- 166 __ _ __ ___ __

4. TITLE (end Subtitle) S. TYPE OF REPORT A PERIOD COVERED

ANALYSIS OF GRADIENT CHANGE THRESHOLDS IN THEDETECTION OF EDGES OF OBJECTS FROM RANGE DATA Final

6. PERFORMING ORO. REPORT NUMBER

7. AUTHOR(e) U. CONTRACT OR GRANT NUMBER(*)

C. N. Shen and R. L. Racicot

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERS

US Army Armament Research & Development Center AM04S No. 6111.01.91A0.OllBenet Weapons Laboratory, SMCAR-CCB-TL PRON No. lA52F59WlAlAWatervliet, NY 12189-4050

It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

US Army Armament Research & Development Center February 1986Close Combat Armaments Center IS. NUMBER OF PAGES

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18. SUPPLEMENTARY NOTES

Presented at the Third Army Conference on Applied Mathematics and Computing,

Georgia Institute of Technology, Atlanta, Georgia, 13-16 May 1985.Published in Proceedings of the Conference.

I. KEY WORDS (Continue on reveree side It necessary mid Identify by block umber)

Gradient Change Scanning SchemeDetect ion Laplacian Method

Threshold Second Residual MethodProbabilities

20. ASTRACT fCWIu m rev oh N newoffemand #~dtfir by block tnusiler)

Edges of objects setting on or within a flat terrain can be defined by the

slope or gradient change occurring at the edge. A simple estimate of slopechange is to calculate the Laplacian using incremental range measurements taken

at different elevation angles. The Laplacian is defined as the second

difference of the range data. Since the range data contains noise in the formof measurement error, consideration must be given to the statistical aspects of

(CONT'D ON REVERSE)

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20. ABSTRACT (CONT'D)

detecting edges in order to differentiate between actual edges and noise

effects. In this report, results are presented which describe the effects onminimum detectable slope change of (1) noise levels, (2) probability of miss,

(3) probability of false alarm, (4) spacing of the measurements, and (5)random distribution of edges near range measurement points.

CO

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TABLE OF CONTENTSPage

INTRODUCTION I

THE LAPLACIAN METHOD 1

THE RAPID ESTIMATION SCHEME AND COMPARISON OF ERROR PROBABILITIES 4

MINIMUM SLOPES OF DETECTABLE OBSTACLES 5

FUNCTION K OF ANGULAR GEOMETRY 7

THE SCANNING SCHEME 9

PROBABILITY OF DETECTION FOR RANDOMLY LOCATED EDGES 10

CONCLUSION 12

APPENDIX 20

TABLES

I. FALSE ALARM PROBABILITIES AND MISS PROBABILITIES FOR 3VARIOUS T/iR AND u*//i

II. ERROR PROBABILITIES 4

III. VALUES OF K AS FUNCTIONS OF GAMMA AND BETA + PHI 8

IV. TOTAL PROBABILITY OF DETECTION FOR RANDOMLY LOCATED EDGES 13

LIST OF ILLUSTRATIONS

1. Distributions and Probabilities Comparing the Two Methods. 15

2. Obstacle Geometry. 15

3. Angular Geometry. 16

4. Projection Geometry. 17

5. Scanning Factors. 18

6. Reference Directions of Laser Rays Near an Edge Point. 19 El

By

SAvailability Codes

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INTRODUCTION

A laser range finder is installed on the top of a mast attached to a land

vehicle or a helicopter. This laser range finder measures the distance

between Itself and the ground points on a terrain with obstacles. For

segmentation, the edges of the boulder and crater must be estimated. In

determining the near edges of a boulder or the far edges of a crater, it is

necessary to locate the first differences of the slopes of the terrain. This

is equivalent to finding the second differences of the terrain range points.

The threshold values of the second differences can be in terms of an angle in

a vertical plane. The probabilities of miss and false alarm can be

ascertained depending on the method of estimation and the scanning scheme.

THE LAPLACIAN ETHOD

The Laplacian Method considers cross-sections of terrain and looks for

changes in slope in the azimuth or radial direction. Since we are only

considering one-dimensional problems, we can write for the measurement

equation

Zi = di + vi (I)

The change in slope is estimated by computing the following sufficient

statistic:A

si - zj+ 1 -2 zi + z_ 1 (2)

This is a digital approximation of the second derivative, or second

difference, of the range.

Due to the presence of measurement noise in the calculation of si, it is

impossible to discern with absolute certainty whether a change of slope

I

exists. The Neyman-Pearson criterion provides a decision rule which we my

use to accurately detect edges of obstacles with a known probability of making

an error. To produce the decision rule, first we must compute the variance of

s i as follows. We assume that the noise components vi of the measurements are

independent Gaussian random variables with zero mean and a variance of 02.

Then, since zi+l, zi, and zi_1 are independent, Eqs. (I) and (2) yield

var(si) - var(zi+l-2zi+zi+l)

m var(zi+l) + var(-2zL) + var(zl+l)

- 02 + 402 + 02 - 602 (3)

Because si is a scalar random variable, the Neyman-Pearson criterion

provides a decision rule identical to that derived using hypothesis testing.

The desired decision rule is

no signal if -T - si C TDECISIONi - (4)

presence of signal if si is otherwise

where T is the threshold in the decision process which can be determined from

the equations

PF - 2#[-T/;6-'07 (5)

PH "[(T-u*)/-60ZJ (6)

where. j z

*(z) e-/2a' • -e 2/2 da (7)

It is noted that the magnitude of sL Is not estimated by the Laplacian Method.

The quantity u* is called the "minimum detectable change of slope," since it

is the smallest change of slope that can be detected with a miss probability

of PH or lower. The quantity PF indicates the probability of a false alarm.

2

The miss probability PM is the probability of not detecting a true change of

slope equal to u*.

Typically, the standard deviation V5 is a known system parameter and u*

is chosen so that suitable values of PF and P14 can be obtained. Table I shows

the trade-off of PF vs. PM4 for different values of the ratio u*/0. These

values are computed using Eqs. (5) and (6).

TABLE I. FALSE ALARM PROBABILITIES AND MISS PROBABILITIES

FOR VARIOUS T/FR AND u*/i

1 I I I rPH For PH For I PMFor I P For P 1 ForIIIIII I I I

T/i P7, u*l /R'2 u* //R-3 I u*l/ I u*/'R;5I u* /R"-6 Ir i r

1.000 .3174 .1587 .0227 .0013 .0000 .0000

1.500 .1336 .3085 .0668 .0062 .0002 .0000

I 1.679 .0932 .3741 .0932 .0102 .0005 .0000

2.000 .0456 .5000 .1587 .0227 .0013 .0000

2.146 .0319 .5580 .1966 .0319 .0022 .0001

2.500 .0124 .6915 .3085 .0668 .0062 .0002

3.000 .0026 .8413 .5000 .1587 .0227 .0013

3.500 .0003 .9332 .6915 .3085 .0668 .0062

4.000 .0000 .9772 .8413 .5000 .1587 .0227

3

.1

THE RAPID ESTIMATION SCHEME AND COMPARISON OF ERROR PROBABILITIES

The variance of si for the Laplacian Method is usually too high when a

change of slope occurs during scanning. An adaptive method called Rapid

Estimation Scheme is used instead to keep the variance lower, thus also

reducing the probabilities of false alarm and miss. A discussion of the

Second Residual Method for the Rapid Estimation Scheme is given In the

Appendix.

The different expressions for the error probabilities of the two methods

are given in Eqs. (5) and (6) and reiterated in Table II. We see that the

expressions for both methods are the same except that 7602 appears with the

Laplacian Method, whereas 7'5i+2 appears with the Second Residual Method.

TABLE I. ERROR PROBABILITIES

I 1 1I Method I Probability of False Alarm I Probability of MissI _ _ _ _ _ I _ _ _ _ _ _ _ _ I_ _ _ _ _ _

! I I ___

I Laplacian pL - 2#[-T//6' ) I pL -*[(T u*)/6ZI II I IM II Second Residual I P5 - 2[-T/Si-+2 I PS - #[(T-u*)/,rSi+21 I -- I F -I

We know that the Second Residual variance is less than or equal to the

Laplacian variance:

7 (i+ -C A- (8)

Since #(z) is a monotonically increasing function and has a negative

argument in all the expressions in Table II (we assume T-u* < 0), we can

conclude that

pS (Second Residual) 4 pL (Laplacian) (9)F F

4

PS (Second Residual) 4 pL (Laplacian) (10)H H

Figure 1 illustrates how the smaller Second Residual variance leads to smaller

error probabilities.

With smaller error probabilities, we expect the Second Residual Method

will have a better performance over the Laplacian Method.

MINIMUM SLOPES OF DETECTABLE OBSTACLES

In this section we will determine the threshold in edging of gradient

changes of obstacles. These may be detected using the Second Residual or the

Laplacian Method.

Obstacles may or may not be detectable depending on the change of slopes

of the terrain at its edges. It will be shown here that the minimum

9 detectable slope change depends upon a given group of parameters.

Figure 2 shows a diagram of an obstacle. C and B are consecutive points

where laser beams emanating from laser range finder bounce off the terrain.

The slope of the obstacle is tan 6, since the slope between C and B is zero.

The mast height is b, the height at which the laser range finder is located.

Point D is the estimate of the range based on the data at point B and previous

points. Thus, the range of point C is measured as zi+2 and point D lies at a

range of di+2 , where di+2 - HFi+lX*i+l is the prediction of the range d1+2

from previous data. We see that the distance from C to D is the residue,

ri+2. It is apparent that the expected value of ri+2 takes its minimum value,

u*. From the geometry, the quantity tan e can be computed by finding x/y.

These quantities are found as follows:

5

x - U* sin B (11)

y- Ap - u* Cos B (12)

then

tan 8 - x/y = (u* sin B)/(Ap - u* cos B) (13)

By extending the above idea, we have two slopes, AB and BC, instead of

one in the previous cases as shown in Figure 3. The angle Y is the difference

of the angle 0 for slope BA and the angle + for slope CB. Now the slope of Y

becomes

atan Y - , (y (14)

b

wherea - u* sin (0+4) (15)

and

b - (Ap) cos - (Ap) sin # cot (U) -u* cos (+ )

sin (B+) cos - cos (0+#) sin *( ) -u* cos (B+*)sin(+)

(AP) sinT u* cos (0 *)

sin (0+#)

1 u* 1- (AP) sin B sin (0++) [----- ----- - ----- (16)

sin2 (0+#) (AP) sin B tan (0+-)

Combining the terms, we have

(u*/T) tan2 ((7)1 + tan2 (0+4) - (u*/T) tan (0+f)

where

- (Ap) sin 0 (18)

The quantity T is the projection of the data spacing in the direction of sin

6

..........

If Eq. (17) is solved for u*/T, we have

A tan y [I + tan2 (0-),-u*l't K('y,84,) --------------- (19)tan (0*4) [tan (C*#) + tan i()

Table III gives the values of K as a function of angle of detection y and

the sum of the elevation angle B and the terrain angle *, (0).

The above equation can also be derived from Figure 4 which gives

Au*/T - K(y,O) . [cot (03*) - cot (0+4jy)] (20)

FUNCTION K OF ANGULAR GEOMETRY

If we can keep the ratio u*/T constant in Eq. (19) or (20), then the

value of K(Y,O+) will be constant in Table III. This can be achieved by

letting both u*//R and T/i be constant. First, we will discuss the value of

u*/rR.

Table I shows the values Of PF for given threshold to noise ratio T/lR

which guarantees the probability of detection. It also lists the values of PM

for both T/FR and the signal to noise ratio, u*//I. For example, let us take

T/' 1.679 and u*/CR 3. We have

PF - 0.0932 (21a)

Pm - 0.0932 (21b)

which are reasonable values for our problem.

The second part is to keep the ratio T/i constant. This is related

to the scanning scheme given in the next section. From Eq. (18), we have

T/I = (Ap)(sin )//IR - constant - L (22)

Then the ratios3- ( y, ) (23)

!. t/i L

7

7I

-- - - ---K --- - - - - - - - - - -%D D 4 t % 4 4 i %D% 0 " r40 0 "%D 4 rV - W

M M ... ... In In %0 D D P. 0 w w 0 C47 C 0 0 C C C

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IC) 0 C o o 0 0 o C) 0 C 0 0 0 C 0 C C 0 C C C C 4 P-. C4 P4 P4 C CC

- - - - - - - - - - -0- - - --Iq-4--4----- - - - -~

- ~ ~ ~ ~ ~ ~ ~ 0 01 VC C C C -C C4 C CCCCCCC C- C4 C4 M M

o 00 00 00 00 00 0 4I- o4u- 4

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ffl M0 4% %*% l % 0WrMw. 41-r -% D%H4Mr AWt Tf .MWr M- P 40r %% M0I

Ln0 )% L (% ' O fb - l r4%D0 *W % % %0 - f r - W

en C C4 &M %0 CC P- C W W a% M C C C C 4 C A C C C C C

- -- - - - - - - - - - - - - - - - - - - - - - - - - - -

99 Cd C C C C C * C C; C; C C; 4 C: C: C: C : 1: C : C: C: C: C: C C: C4104~c

-- - - - - - - - - - - - - - - - - - - - - - - - -C ~ ~ & CD -V LM C C " P4 U% C C C C C C C C

P4H DF *"L 4 gnM0L DM0% 4PH nW* 0r 400 -- 4r Y %*0 -"% r nWNL

H4 0 -M0 %0P4"" M-00I n DF.P %0

8 008 8 8 8 4 M:CO

-- - - - - - - - - - - - - - - - - - - - - - - - - -

O (; C C C 8 4 C4 C I. C C .; C I. C 1. C 0C 0 C C C C A C C4 C A 0 A C4

1- - - - - - - - - - - - - --- - -~ - - - - - -,

If L is chosen as 4, then one will look at the points for K - 0.75 in Table

ItI. If L is chosen as 1.6, then one will look at the points for K - 1.875 in

Table Ill. For K - 0.75, the follwing set of angles appears:

804 15@ 200 25' 300

y 3.80 6.70 10.2' 15.8 °

In summary, the above value of Y is guaranteed to be detected for the

conditions

u*/R - 3 , PF - PM 0.0 9 3 2 ,

and the variable 0+# as listed.

THE SCANNING SCHEME

In order for the value of L to be constant in Eq. (22), one will take the

discrete form in Figure 5 as

(AP)1 sin 01 = (AP)2 sin 02 T = I L - constant (23)

If b is the height of the mast, then

b bsin B1 = ..... sin 02 . ... . (24)

Thus, we have in Figure 4

(Ap)h (0p)2 T

- ft - - -- (25)

The above equation indicates that the spacing of the horizontal

projection is proportional to the radial distances from the laser to points on

the horizontal plane. For example, let us take b - 2, then

(AP)l (0p)2 vi" +. --- (26)

4+P1Z A+ p29 2 2

9

or

(Ap)i - (/4+pj')&/2 (26)

For example, given FR - 0.125 and L - 1.6 or rR - 0.05 and L - 4.0, in both

cases we have T - r-L - 0.20. Then, from Eq. (26), we have

p 2 m 5 m 10 a 20 .

AO 0.2828 m 0.5353 m 1.1832 m 2.010 a

In this case we may miss a boulder of 0.2 m at 2 a away or a boulder of 2 m at

20 m away.

PROBABILITY OF DETECTION FOR RANDOMLY LOCATED EDGES

In the previous sections it was assumed that range measurements were

available from the range finder to the exact vertex of an edge. This is

represented by point B in Figures 2 and 3. An edge is defined here by a

discrete angular change Y. If the range measurement falls exactly on an edge,

then only a deterministic value for the residue u* in Eq. (19) is obtained.

This, in turn, corresponds to a deterministic value for the PD, probability of

detection, in which PD - 1-PH with PN being given by Eq. (6).

In practice, however, the edges of objects might be randomly distributed

and the range measurements, in general, might not fall exactly on an edge. A

more realisitc approach, therefore, might be to treat the range measurements

to be randomly distributed near an edge represented, for example, by points

A, B, and C in Figure 6. A random residual u* would result for given edge

angular change y instead of the constant deterministic value of u* assumed

previously.

10

The probability of detection in this case can be calculated using Eq. (6)

as a function of the random variable u*:

T-u*P(Detectionlu*) - 1 - P (Misslui*) - 1 - *[--] (27)

From geometric considerations, it can be shown that u* will range

approximately from u*/2 to u* in Figure 6. It can be further assumed that the

distribution of u* will be uniform over this range of values which corresponds

to a purely random distribution of object edges on a given terrain. Equation

(27) can then be used to determine the total probability of detection:

P(Detection) P(Detlu*)f(u*)du*u*/2

2 u*

" f/ P(Detlu*)du*

p.,*1u* u*/2

2 u* T-U* -- - 'u12 (11-[---])du* (28)

in which f(u*) equals uniform distribution on (u*/2 to u*), and u* is given by

Eq. (19).

Equation (28) can be solved as a function of T, ii, and u* where u* is a

function of y, (B+#), and T as in Eq. (19). As an example, let

T/i - threshold level for detection

M 2.146; gives PF n 0.0319 from Table I

/1 - /62 where a - range data noise level

M 0.1225 for a - 0.05

T - 0.20

11

The resulting P(Detection) is shown in Table IV as a function of the angle

change Y and (0+4). Other similar information can readily be generated for

other parameter values depending on the actual problem to be solved.

The behavior of the minimum detectable angular change y can be readily

determined from the type of information given in Table IV. For example, by

requiring P(Detection) - 0.90 and letting 0# - 15 degrees, results in a

minimum detectable value of y to be about 40 degrees. This compares to a

value of y at 13 degrees for the nonrandom case previously considered where

the range is assumed to be measured directly to the edge vertex.

CONCLUSION

For an assigned probability of false alarm and miss, one can determine

the signal to noise ratio and the threshold to noise ratio. If we use a

special scanning scheme such that the spacing of horizontal projections is

proportional to the radial distances from the laser to a horizontal plane,

then the function K of angular geometry is also constant. The detectable

angle Y with certain probability can be found if K and Of- are known. The

angles 0+# are related to the elevation angle 0 and the terrain slope *.

The following results were discussed in this report:

1. Ability to detect the change of slopes in a terrain for navigation of

vehicles or robotic platforms.

2. Determination of the probability of false alarm and the probability

of detection for various signal to noise ratios and for the case of randomly

distributed measurements.

12

r 1 1 r 1 1

-- - - - -- - - ----- - --

0~~~~ a %00t 'Vr0~ - -o ri V-4 %r4V 'i1.

0 00 000000000000000M

I t C V % Dr t % O W o r . " C m m mU y 0

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0 t% a "*e% 0 0 0 0 0 0 0 0% 0 0 8 8 8D8w o--- 0000000-----00000

0'00 0000 000 0000013

3. Determination of required signal to scanning factor ratios for

various slopes and slope changes. The signals relate to the probability of

detection and the scanning factor influences the size of the obstacles.

4. Computation of the probability for detection of an edge if the

reference directions of the laser rays are uniformly distributed over the

slopes near an edge point.

14

Varianccecvf.71

-T 0 T U*x

Figure 1. Distributions and Probabilities Comparing

the Two Methods.

i-Laser Range-FInder A

Projectionu~o-Ap I-% 7 Proiection=p

D%

Figure 2. Obstacle Geometry

15

I

I A.'I

I >.'II

/ - - - - - - - 'I,'I, II- - I/ 4~ j ~hq,~'\ /

I \/9/ I p

A 1j4'* a9-4

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II -. -/ I

~ 3dI A) go.94/ II 4r~~

/'-N

I I'- -/ - - I - - -

I

/I

I16

V)

Figure 4. Projection Geometry

17

WI 41>J

- - -

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- ~~0, *1-(3II

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p*14

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18

41

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004

IS

190

APPENDIX

THE SECOND RESIDUAL METHOD

The Second Residual Method, similar to the case with the Laplacian

Method, considers cross-sections of terrain and looks for changes in slope.

Here, however, a state estimation and decision process is used to perform the

detection. A discrete second order linear time-varying system model is used

to estimate ranges and gradients (slopes) from current and previous data. If

the difference between a range measurement and range prediction is large

enough, a change in slope is indicated and a special estimation scheme is

employed.

THE SYSTEM MODEL

A stabilized system model for a terrain has been proposed. The state

vector is

xi - [di, gijT (Al)

where di indicates the i-th range and gi the i-th gradient (or slope). A

change of slope is modelled by the presence of an unknown input, ui, which

adds to the gradient component of xi through the input matrix. The system

model is

x+l - Fix i + Bui (A2)

where I - - I I - -Ibi 1-c 10

Fi-i I and B- Io qi I Il

I_ _I I__

20

where bi and qi are the time variant parameters and c is a small non-negative

real number necessary for system stability while scanning inward from

skylines. The measurement equation is

zi+ 1 = Hxt+l + vl+ (A3)

where H - [1,0] and vi+ 1 is zero mean Gaussian noise where

0 for all i JE{vivJ) - (A4)

o2 for all i J

Our problem, then, is to detect any small nonzero inputs ui, since

they represent a change in slope.

As in the Laplacian Method, a sufficient statistic is necessary to detect

the change of slopes. This sufficient statistic is called the residue, ri+2,

as

ri+2 = zi+2 - HFi+lx*i+1 (A5)

where x*i+1 is optimal estimate of xi+1 given zi+l, zi,...,zl. It can be

shown that

E(ri+l} - HFi+iBui

= (1-c)ui ui (A6)

21

A

*1

I

'I

1q~*h.

S

I'

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