THE MODELLING OF A REFLECTORLESS RANGE FINDER MAXIMUM RANGE WITH PHONG MODEL

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1 THE MODELLNG OF A REFLECTORLESS RANGE FNDER MAXMUM RANGE WTH PHONG MODEL Jacek Rapiski stitute of Geodesy Uiversity of Warmia ad Mazury i Olszty jacek.rapiski@uwm.edu.pl, Kamil Kowalczyk Chair of Lad Surveyig ad Geomatics Uiversity of Warmia ad Mazury i Olszty kamil.kowalczyk@uwm.edu.pl ABSTRACT This paper gives a propositio of a model of the maximum rage of a rage fider. t is based o the laser rage equatio ad a Phog model. t shows how the icidece agle affects the maximum rage whe measurig to the surface that characterizes with diffuse specular refectio. The results shows that oe ca determie the boudary agle for which the optimum performace of maximum rage is maitaied. 1. NTRODUCTON Reflector-less survey become a commo techique i geodetic practice. The mai advatage of this method is the possibility to perform a survey without target markig. This feature ca be used whe measurig over the obstacle like water reservoirs, uaccessible poits o hills or slopes, areas restricted for etrace. The examples of the use of reflector-less surveys ca be foud i [Kowalczyk ad Kuczyska, 009, Kowalczyk, 011, Suchocki ad Wasilewski, 009]. The major disadvatage of this method is its maximum rage. This parameter is strogly depedet o the targets material, lightig, size of the laser footprit. The descriptio of this issues ca be foud i [Sabatii ad Richardso, 010]. this paper we have focused o the ifluece of the icidece agle o the maximum rage of the rage-fider. Accordig to the laser rage equatio [Sabatii ad Richardso, 010] the maximum rage depeds amog others o the power of retured sigal. this paper we try to model the maximum rage with regard to the icidece agle with use of the Phog model derived from computer 3d graphics [Gregory ad Lader, 009].

2 418. SGNAL REFLECTON AND THE MAXMUM RANGE.1 Laser rage equatio The maximum rage depeds o the proportio derived from the laser rage equatio [Sabatii ad Richardso, 010]: P where: P R received sigal power; P T trasmitter power; σ effective target cross sectio (m ); K a aperture illumiatio costat; R system rage to target (m); λ wavelegth (m); D aperture diameter (m) τ atm atmospheric trasmissio factor; τ sys system trasmissio factor. 4 D τ 4 R τ sys P K a σ T = (1) 16 λ atm R The equatio 1 for the totally reflected beam becomes [Kruapech ad Widjaja, 010]: P D τ atmτ sys P 8R ρ T = () R where ρ stads for the target reflectace. Derivig R from the equatio gives: P D τ 8PR τ ρ T atm sys R = (3) the equatio 3 D, τ sys ad the proportio P T /P R are costat for a certai istrumet. For class lasers the proportio P T /P R is about 4_10-4 ad for class laser it is about 3 _ The term τ atm depeds o the atmospheric extictio coefficiet, ad is costat for the atmospheric parameters durig the survey.. Types of reflectio Oe must distiguish three types of reflectio: specular reflectio, diffuse reflectio, retro reflectio. specular reflectio the etire beam is reflected ad reflectio agle equals icidece agle. diffuse reflectio the beam is diffused accordig to the Lambert's cosie law [Smith et al., 1968]. Retro reflectio occurs whe the reflected beam returs the same way that the icidece beam [Friedma ad Miller, 004]. reflector-less surveys the rage-fider's beam usually reflects ad diffuses simultaeously (Figure 1).

3 419 Fig. 1. G l o s s y d i f f u s e r e f l e c t i o. 3. PHONG MODEL The Phog model ca be used to model the glossy - diffuse reflectio. t's orig is i the computer 3D graphics. t is used to model the reflectio of the light from the object. t assumes that the reflected itesity of light ca be described by the equatio 4. = [ cos Θ + cos Φ] i kd s (4) s Where: the itesity of the reflected beam, i the itesity of the icidece beam, k d the amout of beam that is diffused, k s the amout of beam that is reflected, Θ - icidece agle, Φ - the agle betwee icidece agle ad the viewer directio, - parameter describig the lumiace of the material. Parameters k d, k s ad describes the material type, color ad a structure of the surface. This is depicted o figures (a), (b) ad (c). These parameters ca be used to model almost ay kid of surface. For the example parameters k d = 0:4, k s = 0:6, = 1 ad the icidece perpedicular to the surface, the shape of the itesity plot is show o figure 3. The figure 3(a) shows that there is a rage of agle for which the itesity of reflected sigal grow rapidly. To determie the boudary agle the iflectio of this fuctio must be derived. To do so the secod derivative must be calculated. 1 = cos( si( k (5) s Φ Φ = k s( 1) cos( si( k scos( (6)

4 40 (a) The ifluece of the amout of beam that is diffused (b) The ifluece of the amout of beam that is reflected (c) The ifluece of the lumiace parameter Fig.. M a t e r a l t y p e, c o l o r a d a s t r u c t u r e o f t h e s u r f a c e. Fig. 3. t e s i t y p l o t s. The iflectio is i the poit where the secod derivative is equal to 0. Calculatios for the above example: Φ cos( si( 36 cos( Φ ) = 5 1 (7)

5 41 1 = 0 Φ = a ta 0.98[ rad] = 18.64[ grad] (8) 11 Φ Φ The result of equatio 8 meas that for the optimal performace of laser maximum rage, the icidece agle should be smaller the 18:64grad (of course for the certai material used i this calculatio). f we exceed this agle, the maximum rage will be decreased. 4. THE NFLUENCE OF NTENSTY VARATON ON THE MAXMUM RANGE the equatio 3 the reflectace ρ is a fractio of light reflected by the target. This fractio ca also be derived from the Phog model: Θ + Φ (9) i [ ] = k dcos k scos Sice the ifluece of the atmospheric extictio is take ito cosideratio i equatio 3 we ca rewrite the above equatio as: = Θ + Φ (1 k )cos s s k cos (10) i Namig the right side of the equatio 10 as the rage variatio coefficiet: (1 )cos cos = Substitutig equatios 11 ito 3: R Θ + Φ k s k (11) s PT D atmτ = 8PR τ sys (1) Figure 4 depicts the ifluece of the rage variatio coefficiet o maximum rage of the laser. The red lie presets the above model, while gree lie presets stadard, diffuse reflectio.

6 4 5. CONCLUSONS Fig. 4. R a g e v a r i a t i o c o e f f i c i e t. This paper presets the propositio of the modelig method of the maximum rage of a reflector-less rage fiders. t uses laser rage equatio ad a Phog model. t shows that for the certai agles of icidece ad some materials, the rage ca ehace by 37% (whe the icidece agle is 0 o ). This iformatio ca be useful whe measurig to a distat target. For example the desig of cotrol etwork ca be improved if oe kows the maximum rage of a rage-fider. REFERENCES E. Friedma ad J. L. Miller. Photoics rules of thumb: optics, electro-optics, fiber optics, ad lasers. McGraw-Hill Professioal, 004. J. Gregory ad J. Lader. Game Egie Architecture. A K Peters, Ltd., 009. K. Kowalczyk. The assessmet of the ifluece of material type o the accuracy of hadheld laser rage-fider (i polish). Przegląd Geodezyjy, 5, 011. K. Kowalczyk ad J. Kuczyska. Applicatio of mirrorless total statios for precisio architectural objects documetatio. Reports o Geodesy, (87): 177{185, 009. Sahapog Kruapech ad Joewoo Widjaja. Laser rage fider usig Gaussia beam rage equatio. Optics Laser Techology, 4(5):749{ 754, 010. SSN doi: /j.optlastec R. Sabatii ad M. A. Richardso. RTO-AG-300-V6 Airbore Laser Systems Testig ad Aalysis. NATO Research ad Techology Orgaizatio, 010. A. R. Smith, F. Joes, ad R. P. Chasmar. The detectio ad measuremet of ifra-red radiatio. Claredo Press, C. Suchocki ad A. Wasilewski. Moitorig of shore cliff with the applicatio of scaig istrumet. Reports o Geodesy, (87):373{380, 009.