HARMONIC ANALYSIS FOR OPTICALLY MODULATING BODIES USING THE HARMONIC STRUCTURE FUNCTION (HSF) Dr. R. David Dikema Chief Scietist Mr. Scot Seto Chief Egieer Lockheed Marti Hawaii Abstract Lockheed Marti Hawaii presets a ovel sigal processig algorithm for focal plae array processig. We itroduce the Harmoic Structure Fuctio (HSF) ad demostrate its capability i detectig, classifyig ad coutig rotatig bodies i a sigle pixel. The work preseted here is makig a major impact i the Missile Defese Agecy s Project Hercules Forward Based Sesor (FBS) group but the results preseted here are show i a uclassified form. First, the temporal pheomeology of optically modulatig bodies is discussed. Next, the mathematical defiitio of the HSF ad the atural extesio from itegral to discrete form is detailed. Simulatios of rotatig bodies ad modulatig reflectace used for aalysis are the discussed. These simulatios result i the costructio of time series data for N rotatig bodies with fudametal frequecies f i oisy backgrouds. The HSF is the used to aalyze these simulatios. This aalysis yields importat cosideratios for sesor developers ad operators. Fially, the Geeralized Harmoic Structure Fuctio (GHSF) is illustrated. The GHSF is used to aalyze more complicated situatios where objects have multiple rotatioal degrees of freedom.
I. Itroductio This paper presets a geeral method for extractig temporal sigals i multi-frame focal plae data for the purpose of coutig, typig ad discrimiatig optically modulatig bodies. The so-called Harmoic Structure Fuctio (HSF) is itroduced ad its use i focal plae aalysis is sketched. The paper is orgaized as follows. I Sectio II, we discuss the pheomeology of optically modulatig bodies. I Sectio III, we itroduce ad discuss the HSF. Sectio IV exteds HSF. Sectio V cocludes. II. Pheomeology. To uderstad the source of harmoic sigatures from optically modulatig objects, it is useful to cosider a simple model of a slab rotatig aroud two axes (three degrees provides further but straightforward complicatios). I this case, the passive, reflected sigature is proportioal to the ormal area projected to the sesor, ala Fig.. Fig.. Source to Sesor. I a idealized versio of the modulatio, we get somethig like a sigal, S, proportioal to a siusoidal modulatio of the effective area, A eff : r r S Aeffective A ( R) A si( ft ) si( ft) I geeral, however, these modulatios will be o-siusoidal, ad istead periodic fuctios so that we have ( ft) Where Ψ is a periodic fuctio which could look like, for example, a sawtooth or a square wave type fuctio. Let us ow cosider the simplest cases for the sigal S created by these modulatios. This occurs whe f or f is small so that ( f t) ( f t) S A Ψ Ψ ( ) S A C Ψ ft I this case, we, obviously, get a harmoic sigature. From Fourier, whe we expad a periodic series, we get the tower of harmoics of, i this case, f, see Fig...8.6.4..8.6.4. 4 8 6 4 8 3 Fig.. Spectrum for the case where Ψ ( f t) is a sawtooth with frequecy of 4 Hz. Sample Rates are video.
The situatio complicates cosiderably ot from the theoretical cotrol, but from the obtaied sigal whe we have a appreciable f. Here, we get the well-kow frequecy beats. To see this, write the expasio i Fouriers for both periodic sigals: S A A Ψ( ft) Ψ( f t) ( c si( f t) + c si( f t) +...)( d si( f t) + d si( f t) +...) ad the use trigoometry relatios to get: S ( c si( f t) + c si( f t) +...)( d si( f t) + d si( f t) +...) c d si( f t)si( f t) +... c d cos(( f f ) t) + c d cos(( f + f ) t) +... The sigal, S, the becomes a complicated combiatoric coudrum. Moreover, there exists the possibility of aliasig, etc. so that the beats may wrap aroud the spectra further complicatig matters. Furthermore, we may ecouter the situatio where multiple objects are beatig o a sigle pixel: here we get additive sets of complicated beat structures. Fig. 3. below shows a example spectrum for this case. I geeral, such sigatures may provide applicatios, for example, SSA, with a opportuity to perform uique figerpritig...8.6.4...8.6.4. 5 5 5 3 Fig. 3. Complicated Spectra created from two objects both beatig o a sigle pixel. III. The Harmoic Structure Fuctio. For a give fudametal frequecy, f, the harmoically related compoets will have frequecies f for >, where is the harmoic umber ad is the fudametal. A itegratio of a sigal, s ( f ), is performed over frequecy space ad weighted such that oly frequecies harmoically related to a give fudametal frequecy will cotribute to the itegratio. The itegratio is give by H ( f) S( f ) ( ) δ f f df mi mi S( f where mi ad defie the rage of the harmoic compoets to be itegrated over, ad δ ( f ) is the Dirac delta fuctio. Sice the data are i discrete frequecy bis, the itegratio must be performed as a sum over the correspodig bis. If x [i] is the ith bi of the frequecy spectrum ad Δ f is the frequecy resolutio of spectrum, the the sum over the discrete bis is give by ),
H ( f ) S[it( f / Δf )], mi where it() trucates the argumet to the earest iteger value toward zero. The sum, H ( f ), is computed for frequecies i the specified fudametal frequecy rage, fmi f f. Sice each fudametal frequecy bi will be associated up to bis at the th harmoic, a fudametal frequecy icremet of at most Δ f / must be used to esure a fie eough samplig of H ( f ). This esures that all possible harmoic sets correspodig to fudametal frequecy ad harmoic rage fmi f f ad mi are cosidered. I practice, the fudametal frequecy rage is quatized by Δf f [ j j j] fmi + ( j ), where ( f fmi) j it. Δf The, for a give fudametal frequecy idex, j, ad harmoic umber,, the fuctio f j mi H (, j) x it + Δf defies the harmoic structure fuctio (HSF). Harmoic Detectio Example m j matrix, that is the outcome of applyig HSF to a sigle spectral sca of the time- Let A be a frequecy spectrogram show i Figure. Elemets of the matrix are defied by a ij mi( H ( j j, mi + i, j), T ) i m, where elemets of the jth colum form the harmoic set associated with fudametal frequecy f fmi + jδf /, the umber of harmoics beig cosidered is m mi +, ad mi( ) limits the value of a sigle matrix elemet to at most T. A weighted sum of the elemets i each of the colums is give by z ba, where b is a m-legth row vector defied by b [ / m / m L / m]. The resultig power-versus-frequecy vector, z, yields a fudametal spectrum where the large peak correspods to the fudametal frequecy of the harmoic structure. Note that curretly equal weight is give to each harmoic compoet. If a specific harmoic set predomiace is kow a priori, the compoet weightig i the above equatio should be adjusted accordigly. Figure 4 shows two examples of the HSF i actio. Fig. 4a shows the output for the case with three harmoic sigals. The upper left plot shows a sigle sca frequecy spectrum. The upper right plot shows a lofargram (stacked spectra with time o the y axis) of the sigal. The lower plot is the stacked fudametal spectrum obtaied from HSF i this case we see evidece of the three fudametals. Fig. 4b is more complicated: here we have the case similar to Fig. 4a., but the fudametal frequecy is icreasig. Fig.4 explicitly shows the capability of HSF i coutig multiple objects o a sigle pixel. HSF aalysis could potetially be combied with spatial super-resolutio algorithms for object coutig applicatios.
Fig. 4a. ad 4b. Examples of the HSF. IV. The Geeralized Harmoic Structure Fuctio. The HSF preseted above does the job whe we are dealig with the case where we oly have oe appreciable frequecy of rotatio. Whe we have multiple degrees of freedom i play, the HSF eeds to be modified. I this case, we have the Geeralized Harmoic or Beated Structure Fuctio (GHSF). Such a modificatio is straightforward. The idea is to be able to template match (a example template is show i red i Fig. 5a overlaid with a beated spectrum i blue.) received sigatures to geerate a test statistic. I the example spectrum of Fig. 5a, Fig. 5b shows the GHSF image output. The bright spot i Fig. 5b correspods to the case where the spectrum matches the beated spectrum for f ad f. 4 6. f 8.5 4 6 8 3 5 5 5 3 8 4 6 8 4 6 f Fig. 5a ad 5b. Beated Spectra ad the output of the GHSF. V. Coclusio. The sigal processig methods preseted here represet ovel ad optimal ways for exploitig harmoic ad beated sigatures i the presece of oise i a variety of situatios ragig from EO/IR focal plae applicatios to acoustics. The methods preseted have foud success i a variety of DoD applicatios. It is the author s opiio that such methods may have applicatio to the SSA problem. I coclusio, it is iterestig to look at the case where we have a large umber of modulatig bodies o a sigle pixel of a focal plae. I this case, it ca be show that the temporal sigal received is composed of a large umber of beatig objects ad the resultig spectra is a Gaussia shaped distributio cetered at DC. 35 3 5 5 5 3 4 5 6 7 8 Fig 6. Gaussia shaped distributio of multiple beatig objects.