Abstract
Shannon entropy is the most popular method for quantifying information in a system. However, Shannon entropy is considered incapable of quantifying spatial data, such as raster data, hence it has not been applied to such datasets. Recently, a method for calculating the Boltzmann entropy of numerical raster data was proposed, but it is not efficient as it involves a series of numerical processes. We aimed to improve the computational efficiency of this method by borrowing the idea of head and tail breaks. This paper relaxed the condition of head and tail breaks and classified data with a heavy-tailed distribution. The average of the data values in a given class was regarded as its representative value, and this was substituted into a linear function to obtain the full expression of the relationship between classification level and Boltzmann entropy. The function was used to estimate the absolute Boltzmann entropy of the data. Our experimental results show that the proposed method is both practical and efficient; computation time was reduced to about 1% of the original method when dealing with eight 600 × 600 pixel digital elevation models.
Highlights
Raster data representation offers a number of important advantages over other options, the technology predominates in satellite imagery, digital elevation modeling, landscape gradient mapping, and other applications
Information held in raster form has been used to evaluate the performance of image fusion [2,3,4], and it is regarded as an essential reference for band selection in hyperspectral imaging [5,6]
Information production may be quantified through Shannon entropy [7], which is commonly used in many domains, such as computer graphics [3] and landscape ecology [8,9]
Summary
Raster data representation offers a number of important advantages over other options, the technology predominates in satellite imagery, digital elevation modeling, landscape gradient mapping, and other applications. By assuming that the shape of the curve, and its function, can be estimated from several points, we were able to predict the values of the other points and use these to estimate the absolute Boltzmann entropy of the numerical raster data. The points that we used to estimate the shape of the curve were close to the y-axis This was the result of a fundamental limitation of the iterative process, which always starts with the original gradient and proceeds in sequence. It was reasonable to estimate the shape of the curve from these critical points This solution was developed to avoid having to compute the relative Boltzmann entropies at all levels, which should improve the computational efficiency of the entire process. The approach of head/tail breaking has been used for estimating the relative Boltzmann entropies of each class for a given numerical raster dataset. Inclathssecsaanmbee wcoamy,ptuhteda,cicnutrhaetoerrye.lHatoivweeBveorl,tzifmwaenn entrwopeireestofctohme (pNut−e t1h)ethacclcausrsaatne dretlhaetiv(Ne B−o2lt)ztmh calnanssecnatnrobpeycofmepvuerteydc,liansst,htehoerny.wHeowwoeuvledr,nifowt heawveere to cosomlvpeudteththe einaecfcfiucrieantecyreplarotibvlemBo. ltzmann entropy of every class, we would not have solved the inefficienIncystepardo,bwleemu. sed the accurate relative Boltzmann entropies of several classes to estimate the reIlnasttiveaedB,owltzemuasnendenthtreopaciecsuorfaotethreerlcaltaisvseesB. oTlhtizsmisavnanlidenbtercoapuiseesthoefsseerveelartaivl eclBaoslstezsmtaonnesetnimtroapteietshe relatciavnebBeodltezsmcraibnendeanptrporpoixeismoafteoltyhtehrrcoluagssheas.liTnheiasrifsuvnacltiidonb.eTchaiusspeatpheersechroeslaettihvee lBinoeltazrmfuanncntieonntryopies canabe xd.eRscerciablledthaatptphreofxirismt acltaeslsyththrorouugghhthae liNnear1funccltaiossna. llTbheilsonpgatpoetrhcehtoaisle, wthheilelitnheealrasftucnlacstison btyhe=elolabininacengtc×leoousanxrttrha.ogftesReuthtleniroecnecatetlhahlialoeteriatnvhhfdeu(ae.inta.BWecdto.ht,.eileottWzhsnfimueer(baiiss.nstneutd.ncib,tleustaephttsnieeetstundrtitodnhetpdehrdonyeeutpthaogevcefnhactadruhtciehraecnaebuttlre(NevaNrtaieesr−lira[1aNet1bli)alv−ettehiicv1csBle]lao,asBalNsstons)z.ldmatWzl1tlamhenb,eaeniannldnoenvednnepegtrtnershontteeorpdloydytephensyoepotfloetvvtanfhaeidtrledh,iea(ewntbNhthleveN−ilapiers1ait)arth1thabhemelcelaelaaictsscesctlsrtuachsirlαensaatstoes relaatthinvederdeBmeorlaitviznemidnagthncenlaeesxnspetrsreo, swpsyihoinochfotfbhetehloe[nNfgu−endc1tt]iotohntch.leTasthasei)nl.s,.Wwee icnovmeprsuetleydstohleveredlatthiveepBaorlatmzmeatenrnαenatnrdopdieesriovfed
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