SAMPLING and quantization are fundamental processes

Transcription

1 Optmal Hgh-Resoluton Adaptve Samplng of Determnstc Sgnals Yehuda Dar and Alfred M. Brucksten arxv:6.85v [cs.it] Jun 7 Abstract In ths work we study the topc of hgh-resoluton adaptve samplng of a gven determnstc sgnal and establsh a connecton wth classc approaches to hgh-rate quantzaton. Specfcally, we formulate solutons for the task of optmal hghresoluton samplng, counterparts of well-known results for hghrate quantzaton. Our results reveal that the optmal hghresoluton samplng structure s determned by the densty of the sgnal-gradent energy, just as the probablty-densty-functon defnes the optmal hgh-rate quantzaton form. Ths paper has three man contrbutons: the frst s establshng a fundamental paradgm brdgng the topcs of samplng and quantzaton. The second s a theoretcal analyss of samplng, for arbtrary sgnal-dmenson, relevant to the emergng feld of hgh-resoluton sgnal processng. The thrd s a new approach to nonunform samplng of one-dmensonal sgnals that s expermentally shown to outperform an optmzed tree-structured samplng technque. Index Terms Hgh-resoluton samplng, adaptve samplng, hgh-rate quantzaton. I. INTRODUCTION SAMPLING and quantzaton are fundamental processes n sgnal dgtzaton and codng technques. Each of them s a feld of research rch n theoretcal and practcal studes. Quantzaton addresses the problem of dscretzng a range of values by a mappng functon, usually based on decomposton of the range nto a fnte set of non-ntersectng regons. Samplng a gven sgnal, defned over a contnuous and bounded doman, s the task of dscretzng the sgnal representaton for example, representng a fnte-length onedmensonal sgnal as a vector). In ths paper we consder nonunform samplng that reles on segmentaton of the sgnal doman nto non-overlappng regons, each represented by a sngle scalar coeffcent we do not consder here generalzed samplng that uses projectons of the sgnal onto a dscrete set of orthonormal functons). The sgnals consdered n ths work are determnstc n the sense that ther contnuous forms are accessble to the sampler for ts operaton. Our man settngs address a gven one-dmensonal sgnal defned over the nterval [, ), and ts representaton usng a pecewse-constant approxmaton that reles on a hgh-resoluton nonunform segmentaton see Fg. ). The dscretzatons n quantzaton and samplng were frst mplemented n ther smplest forms relyng on unform dvsons of the respectve domans. Then, the quantzer desgns progressed to utlze nonunform structures, explotng nput-data statstcs to mprove rate-dstorton performance. In hs fundamental work, Bennett [] suggested to mplement The authors are wth the Department of Computer Scence, Technon, Israel. E-mal addresses: {ydar, freddy}@cs.technon.ac.l. Fg.. The general procedure for nonunform samplng and reconstructon of a one-dmensonal determnstc sgnal ϕ t) defned for t [, ). nonunform scalar quantzaton based on a compandng model where the nput value goes through a nonlnear mappng compressor), the obtaned value beng unformly quantzed and then mapped back va the nverse of the nonlnearty expander). Moreover, under hgh-rate assumptons, Bennett derved a formula for approxmatng the quantzer dstorton based on the source probablty-densty-functon and the dervatve of the nonlnear compressor functon. Ths mportant formula s often referred to as Bennett s ntegral. Bennett s work was followed by a long lne of theoretc and algorthmc studes of the nonunform quantzaton problem. A promnent branch of research addressed the scenaro of nonunform quantzaton at hgh-rates for example, see [], []), where the quantzer has a large number of representatonvalues to be wsely located at the quantzer-desgn stage. The popularty of hgh-rate studes s due not only to ther relevance n addressng hgh-qualty codng applcatons, but also to the possblty to gan useful theoretcal perspectves. Specfcally, reasonable assumptons made for hgh-rate quantzaton often led to convenent closed-form mathematcal solutons that, n turn, provded deep nsghts nto rate-dstorton trade-offs. Samplng has been prevalently studed for the purpose of acqurng sgnals based on coarse characterzatons such as ther bandwdth. The classcal unform samplng theorems see the detaled revew n [4]) were also extended to the nonunform settngs where the varable samplng-rate s adapted to, e.g., the sgnal s local-bandwdth estmate [5] [8]. In [6] [8], a nonunform samplng grd was desgned by mappng the sgnal s tme axs to expand portons correspondng to hgh local-bandwdth at the expense of segments contanng content of lower local-bandwdth. After applyng ths transformaton, sgnal acquston could subsequently be carred out usng unform samplng. Whle these works refer to acqurng sgnals of unknown content, we consder here samplng of fully-accessble determnstc sgnals for representaton-orented tasks such as compresson as wll be descrbed below). Despte the above dscussed essental dfference n the samplng and reconstructon procedures, some results presented n ths paper conceptually resemble samplng methods for acquston of unknown sgnals usng coordnate

2 transformaton. The results presented here for samplng gven determnstc sgnals may also be nterpreted as complements of the adaptve samplng paradgms prevously proposed for sgnal acquston. In our determnstc settngs, the analyss and samplng-rate desgn drectly rely on propertes obtaned from the contnuous sgnal nstead of coarse, local averagedbased, spectral characterzatons. Practcal sgnal compresson see, e.g., [9], []) usually consders a dgtal hgh-resoluton nput functon that s gong through a nonunform adaptve parttonng of the doman followed by a quantzaton procedure that suts the partton sze and shape. Analytcally, the dgtal nput can be regarded as a sgnal defned over a contnuous doman that needs to be nonunformly sampled and quantzed. In practcal codng procedures, the nonunform doman segmentaton often reles on structured parttons of the doman n order to reduce the bt-cost and computatonal complexty examples for the common use of tree-structured parttonng, e.g., defned based on bnary or quad trees, are avalable n [], []). Ths applcaton s one of the man motvatons to the study of adaptve samplng of determnstc sgnals. Nevertheless, we consder here the nonunform samplng problem n ts most general form n order to provde nsghts to the basc samplng problem, that may be useful to problems beyond compresson. For example, adaptve samplng s often used n computer graphcs n the task of Halftonng e.g., [], []) and n renderng an mage from a D/D-graphcal model, where the nonunform samplng pattern s descrbed by pont dstrbutons e.g., [4] [8]). It should be noted that the sgnal processng and computer graphcs contexts of the samplng problem are qute dfferent. Interestngly, t was suggested n [6] to practcally employ the Lloyd algorthm [9], orgnally ntended for quantzer desgn, for mprovng the samplng pont dstrbuton. Recent studes [7], [8] extended the pont-dstrbuton computaton to rely on kernel functons, so that the resultng task resembles a generalzed sgnal-samplng procedure. Whle the conceptual relaton between vector-quantzaton and data-representaton has been understood and used n graphcs applcatons, we argue that a clear mathematcal analyss that demonstrates the connecton between quantzaton and hgh-resoluton sgnalsamplng has not been provded yet. Some recent papers [] [] explore the samplng task from a stochastc, nformaton theoretc rate-dstorton tradeoff perspectve, consderng lossy compresson of the sampled values. Ther focus s on unform [], [] and nonunform [] samplng where the samplng desgn s based on spectral characterstcs of statonary sgnals. Another nterestng drecton of nonunform samplng was explored n [], [4] where the samplng ntervals are recursvely determned based on prevously recorded data. Ths process does not requre the codng of the samplng ntervals. The samplng desgn n [], [4] s based on ether the local propertes of the stochastc process whose realzatons are sampled, or the local behavor of determnstc sgnals exhbted n ther Taylor expanson. In ths paper we theoretcally explore the task of hghresoluton adaptve samplng of a gven determnstc sgnal. The samplng analyss provded emerges from deas smlar to the ones that were appled to the study of hgh-rate quantzaton [5], thereby lnkng samplng and quantzaton n a new and enlghtenng way. We analytcally formulate the optmal hgh-resoluton samplng of one-dmensonal sgnals, based on the Mean-Squared-Error MSE) crteron, showng that the optmal parttonng s determned by the cube-root of the sgnal-dervatve energy. Ths result corresponds to the work by Panter and Dte [], where the optmal onedmensonal quantzer s desgned based on the cube-root of the probablty-densty-functon. We also connect ths result to the fundamental analyss gven by Bennett [] for hgh-rate nonunform quantzaton based on compandng. We contnue our analyss by addressng the problem of samplng K- dmensonal sgnals. We show that the optmal samplng-pont K K+ densty s determned by the densty of the -power of the sgnal-gradent energy, a generalzaton of the one-dmensonal result. We obtan ths result based on assumptons that parallel a famous conjecture gven by Gersho [] n hs analyss of hgh-rate quantzaton. Gersho s conjecture states that, for asymptotcally hgh rate, the optmal K-dmensonal quantzer s formed by regons that are approxmately congruent and scaled versons of a K- dmensonal convex polytope that optmally tessellates the K- dmensonal space where the polytope optmalty s n the sense of mnmum normalzed moment of nerta []). The latter holds for a gven K only f the optmal tessellaton of the K-dmensonal space s a lattce, and therefore constructed based on a sngle optmal polytope. Ths assumpton sgnfcantly smplfes the explct calculaton of the quantzer s dstorton. Ths conjecture draws ts credblty from two promnent sources. Frst, t s known that the best tessellaton s a lattce for K = based on equal-szed ntervals) and for K = based on the hexagon shape [6]). Whle not proven yet for K =, t s also beleved that the optmal threedmensonal tessellaton s the body-centered cubc lattce [7], [8]. Second, Gersho s dstorton formula conforms to the structure of the expresson rgorously obtaned by Zador [9]. Moreover, Gersho s conjecture determnes the value of the multplcatve-constant left unspecfed) n Zador s formula. Hence, the possble naccuracy n Gersho s conjecture wll affect only the multplcatve constant, and the devaton s assumed to be moderate [5]. Due to ths, the conjecture, stll unproved for K, s wdely consdered as a valuable tool for analyss of hgh-rate quantzaton see the thorough dscusson n [5]). The analyss provded n ths paper to hgh-resoluton multdmensonal samplng s based on two man assumptons. Frst, the sgnal s assumed to be approxmately lnear wthn each of the samplng regons. Second, samplng regons are assumed to be approxmately congruent and scaled forms of the optmal K-dmensonal tessellatng convex polytope just as n Gersho s conjecture for hgh-rate quantzaton. These hgh-resoluton assumptons yeld our man result that the optmal samplng-pont densty s determned by the densty of K K+ the -power of the sgnal-gradent energy. We emphasze the mportance of the sgnal s local-lnearty assumpton as a prerequste stage that mathematcally connects the sgnalsamplng problem to the conjecture on the hgh-resoluton cell

3 arrangement. Ths paper s organzed as follows. In secton II we mathematcally analyze the optmal samplng of one-dmensonal sgnals. In secton III we provde an expermental evaluaton of the proposed samplng method for one-dmensonal sgnals. In secton IV we generalze our study by theoretcally addressng the optmal samplng of multdmensonal sgnals. Secton V concludes ths paper. II. ANALYSIS FOR ONE-DIMENSIONAL SIGNALS A. Optmal Hgh-Resoluton Samplng Let us consder a one-dmensonal sgnal ϕ t) defned as a dfferentable functon ϕ : [, ) [ϕ L, ϕ H ], ) defned for t n the nterval [, ) and havng values from a bounded range [ϕ L, ϕ H ]. Ths sgnal s sampled based on ts partton to N N non-overlappng varable-length segments, where the th subnterval [, a ) s assocated wth the sample ϕ for =,..., N. We assume a segmentaton structure satsfyng a =, a N =, and < a for =,..., N. The samplng procedure s coupled wth a reconstructon that provdes the contnuous-tme pecewseconstant sgnal ˆϕ t) = ϕ for t [, a ). ) The samplng s optmzed to mnmze the mean-squared-error MSE), expressed as ) MSE {a } N, {ϕ } N = N ϕ t) ϕ ) dt exhbtng the roles of the sgnal parttonng and sample values. Note that n ), as also n ths entre secton, averagng over the unt-nterval length s mplct. Optmzng the samplng coeffcents, {ϕ } N, gven a parttonng {a } N = s a convex problem that can be analytcally solved to show that the optmal th sample s the sgnal average over the correspondng subnterval, namely, ϕ opt ) = ϕ t) dt 4) where a s the length of the th subnterval. We contnue the analyss by assumng hgh-rate samplng, meanng that N s large enough to result n small samplngntervals that, however, may stll have dfferent lengths. Furthermore, the samplng ntervals are assumed to be suffcently small such that, wthn each of them, the sgnal s well approxmated va a local lnear form an argument that s analyzed next. Accordngly, for t [, a ) =,..., N), we consder the frst-order Taylor approxmaton of the sgnal about the center of the th samplng nterval, t + a ), ϕt) = ϕ t ) + ϕ t ) t t ) + o t t ) 5) where the remander term o t t ) corresponds to our hghresoluton assumpton n descrbng the approxmaton error for t t. Usng the lnear approxmaton, and by 4), the optmal sample n the th subnterval s gven by ϕ opt = ϕ t ) + o ) 6) due to the fact that the average of a lnear functon over an nterval s ts value at the nterval s center. The term o ) descrbes the error n the sample value due to the lnear approxmaton. The sze of the error term as s provded n Appendx A and reles on assumng that the second dervatve of the sgnal exsts and bounded over the samplng nterval. Then, the MSE of the th subnterval s expressed as MSE, a ) = ϕ t) ϕ opt) dt 7) = ϕ t )) + o ) 8) revealng the effect of sgnal-dervatve energy on the samplng MSE. The dervaton of 8) s detaled n Appendx A. Returnng to the total samplng MSE, correspondng to optmal coeffcents, and relyng on ts relaton to the subntervals MSE yelds recall the mplct normalzaton to unt-nterval length) ) N MSE {a } N = MSE, a ) = N ϕ t )) + o ) max where max max{,..., N } s the largest subnterval. Accordngly, the naccuraces n evaluatng the MSE usng the sgnal lnearty assumpton are of sze o max). Let us connect our dscusson to the classcal approach of studyng hgh-rate quantzaton based on the reproductonvalue densty functon see examples n [], [9], [5], []). Followng our scenaro of hgh-resoluton samplng we assume that the samplng-pont layout can be descrbed va a samplngpont densty functon, λ t), such that a small nterval of length around t approxmately contans λ t) samplng ponts. Moreover, the samplng-pont densty s related to the samplng ntervals va 9) λ t) N, for t [, a ). ) As we consder arbtrarly large values of N, the densty λ t) s assumed to be a smooth functon. Then, pluggng the relaton / N λ t )) nto the samplng-error expresson 9), n addton to approxmatng the sum as an ntegral and omttng the explct naccuracy term, yelds MSE λ) N ϕ t)) λ dt. ) t)

4 4 Here the samplng structure and the resultng MSE are determned by the samplng-pont densty λt). The last MSE expresson can be nterpreted as the samplng equvalent of Bennett s ntegral for nonunform quantzaton []. Commonly, the expresson form n ) s mnmzed va Hölder s nequalty see examples n [], [5], []). For our problem, see detals n Appendx B, we have the optmal samplng-pont densty gven by λ opt t) = and the optmal samplng MSE s MSE λ opt) N ϕ t)), ) ϕ z)) dz ϕ t)) dt. ) The error expresson n ) s a product composed of two parts: samplng error for a smple lnear sgnal wth a unt slope), and a term expressng the nonlnearty of the gven sgnal based on ts dervatve energy. Moreover, Hölder s nequalty also shows that ) s the global mnmum. The MSE expresson n ) s the samplng counterpart of the famous Panter-Dte formula for hgh-rate quantzaton MSE []. Evdently, whle the quantzaton dervatons were determned by the probablty-densty-functon of the source, the samplng analyss provded here depends on the sgnal-dervatve energy. Usng the samplng-pont densty, λt), we can mplement our nonunform samplng va the compandng desgn. Compandng [] s a wdely-known technque for mplementng nonunform quantzaton based on a unform quantzer. Ths s acheved by applyng a nonlnear compressor functon on the nput value, then applyng unform quantzaton and mappng the result back va an expander functon the nverse of the compressor). Snce n samplng we address the problem of dscretzng the sgnal doman, the correspondng compressor and expander functons operate on the sgnal doman.e., as a nonlnear scalng of the tme axs). The optmal compressor functon s defned based on the optmal densty ) as t t ϕ z)) dz u t) = λ opt z)dz =. 4) ϕ z)) dz The correspondng expander, v τ), s the nverse functon of the compressor, hence, t can be defned va the relaton vτ) λ opt z)dz = τ. 5) The last equaton has a unque soluton for strctly monotonc sgnals where the sgnal-dervatve energy s postve over the entre doman, leadng to a strctly-monotonc ncreasng compressor functon 4) that s nvertble and, thus, defnes the expander functon. Two suggestons for the treatment of non-monotonc sgnals wll be descrbed next. ) Usng a Strctly-Postve Extenson of the Sgnal s Dervatve-Energy: Consder the expander functon n 5) to construct the boundares of the nonunform segmentaton of [, ) va a opt = v N ), =,..., N, 6).e., evaluatng the nverse of the compressor functon at N equally-spaced ponts. However, for non-monotonc sgnals that also may have constant-valued segments), t s lkely to need expander values at ponts where the compressor functon s not nvertble an ssue that can be solved as follows. Snce the problem occurs where the sgnal dervatve s zero, we defne the followng extenson of the dervatve energy: { gε ϕ t)) for ϕ t)) > ε t) = 7) ε otherwse where ε > s an arbtrarly small constant. The correspondng extenson of the optmal samplng-pont densty ) s λ opt g ε t) = ε t) 8) g ε z)dz Accordngly, the densty λ opt ε t) enables treatment of nonmonotonc sgnals, whle closely approxmatng the densty λ opt t) n ). Replacng λ opt t) wth λ opt ε t) n 4)-5) provdes a practcally useful compressor-expander par, n the sense that the compressor functon s nvertble everywhere n the doman, assurng the computatons n 6). ) Sequental Soluton va Integraton Thresholds: Eq. 5) defnes the segment level a opt =,..., N) as can be evaluated gven the former part- mplyng that a opt tonng level a opt a opt a opt va a opt λ opt z)dz = N 9) ϕ t)) dt = N ϕ z)) dz ).e., we just need to contnuously ntegrate ϕ t)) startng at t = a opt untl we reach the threshold level defned by the rght sde of the last equaton. Then, the tme of achevng the threshold level defnes a opt. Snce a parttonng level s placed at the frst) tme the threshold level s obtaned, ntervals of zero sgnal dervatve-energy do not cause ambguty n segmentaton defntons. Importantly, one should note that the optmal) threshold, defned accordng to Eq. ) as T opt = N ϕ z)) dz, ) requres knowng the sgnal dervatve over the entre [, ) nterval. Note however that the segmentaton levels are here well-defned for any gven nonzero sgnal.

5 5 B. The Samplng-Quantzaton Dualty at Hgh Resoluton The above developments for optmal hgh-resoluton samplng show a clear correspondence to well-known results from hgh-rate quantzaton studes. The connecton between samplng and quantzaton, n the hgh-resoluton settngs, emerges from local lnear approxmaton of the sgnal wthn each of the samplng ntervals, mplyng a constant sgnaldervatve wthn each nterval. Ths mrrors the assumpton of locally constant probablty densty n the hgh-rate quantzaton problem. In turn, our constructon yelds an optmal samplng procedure determned by the sgnal s dervatveenergy densty. Specfcally, the MSE forms n ) and ) and the correspondng compandng approach are as n the classcal hgh-rate quantzaton results, however, here they correspond to the sgnal s dervatve-energy densty nstead of the probablty densty functon of a source to be quantzed. In the sequel, let us consder the scalar quantzer desgn for a random varable X, defned by the probablty densty functon p X x) that may be postve only for x [x L, x H ). Then, recall Bennett s ntegral [] for the MSE of hgh-rate quantzaton usng N reproducton values: MSE Q λ Q ; p X ) N x H x L p X x) λ dx ) Q x) where, as nterpreted n [9], λ Q x) s the reproducton-pont densty that defnes the quantzer structure to optmze, and p X x) s the gven nput probablty densty functon. In ths subsecton we dstngush between optmzatons orgnatng n samplng and quantzaton procedures by the followng notatons: the quantzer desgn procedure addresses Bennett s ntegral ) to mnmze the quantzaton dstorton, MSE Q, as a functon of the reproducton-pont densty λ Q. Smlarly, the samplng procedure consders the samplng counterpart of Bennett s ntegral presented n ), that usng ths subsecton s notatons s formulated as MSE S λ S ; ϕ) N ϕ t)) λ dt, ) S t) where MSE S, the dstorton of samplng the sgnal ϕt), s to be mnmzed by optmzng the samplng-pont densty λ S. ) Optmal Samplng va Optmal Quantzer Desgn: For a gven dfferentable sgnal, ϕ t) defned for t [, ), we can defne a probablty densty functon, p ϕ x), defned for x [, ) va p ϕ x) = ϕ x)), 4) ϕ z)) dz.e., the probablty densty s the sgnal s local squareddervatve normalzed by the total dervatve energy. Evdently, p ϕ x) s non-negatve valued and ntegrates to one, hence, t s a vald probablty densty functon. Bennett s ntegral for hgh-rate quantzaton of a source dstrbuted accordng to p ϕ s obtaned by pluggng the defnton of p ϕ x) from 4) nto ), resultng n MSE Q λ Q ; p ϕ ) N E ϕ ϕ x)) λ dx. 5) Q x) where λ Q ) s the reproducton-pont densty defnng the hgh-rate quantzer structure, and the total dervatve-energy of the sgnal s denoted here as E ϕ ϕ z)) dz. 6) Then, the hgh-resoluton samplng MSE, MSE S, formulated n ) s related to the hgh-rate quantzaton MSE n 5) va MSE Q λ Q ; p ϕ ) MSE S λ Q ; ϕ), 7) E ϕ.e., the MSE of quantzng a source defned by p ϕ usng a quantzer structured accordng to the parttonng nduced from λ Q s equvalent, up to a normalzaton n the total dervatve-energy of the sgnal, to the MSE of samplng the sgnal ϕ t) accordng to the segmentaton defned by λ Q. The MSE relaton 7) shows that the optmal λ Q for the above quantzaton problem, s also optmal for samplng of the sgnal ϕ t). Indeed, Bennett s ntegral for quantzaton of p ϕ x) s mnmzed for λ Q formulated exactly as the optmal samplng-pont densty n ). ) Optmal Quantzer Desgn va Optmal Samplng: Gven a random varable X correspondng to the probablty densty functon p X x) defned for x [x L, x H ), we can construct a sgnal ϕ X t) defned for t [, ) va ϕ X t) = x H x L x L +t x H x L ) x L px x)dx. 8) We take the postve square-root of p X x), hence, the sgnal ϕ X t) s monotoncally non-decreasng. The dervatve of ϕ X t) wth respect to t) s ϕ X t) = p X xl + t x H x L ) ). 9) Then, accordng to ), the samplng MSE of ϕ X t) for a samplng-pont densty λ S s that here equals to MSE S λ S ; ϕ X ) N MSE S λ S ; ϕ X ) N ϕ X t)) λ S t) dt ) p X xl + t x H x L ) ) λ S t) dt. ) By changng the ntegraton varable to x = x L +t x H x L ) we get MSE S λ S ; ϕ X ) N x H x L ) x H x L p X x) λ S x x L x H x L ) dx. )

6 6 By referrng to ), the last form shows the followng relaton between the samplng MSE and the quantzaton MSE MSE S λ S ; ϕ X ) x H x L ) MSE Q λ Q ; p X ) ) where, for x [x L, x H ), ) x xl λ Q x) = λ S. 4) x H x L The last result means that for a random varable X, wth a gven probablty densty functon p X x), one can desgn the optmal hgh-rate quantzer by consderng the hgh-resoluton samplng problem of the sgnal ϕ X t) defned n 8). Equatons )-4) show that the MSEs of the two procedures are equal up to a normalzaton by the wdth of the valuerange of X, and a lnear transformaton of the coordnates of the parttonng structure see Eq. 4)). Hence, mplementng the optmal samplng-pont densty ) for the sgnal ϕ X t), together wth the approprate lnear-transformaton of the coordnates, provdes the structure that mnmzes the hghrate quantzaton MSE for p X x). C. Numercal Demonstratons We now turn to study our theoretcal results by applyng them for analytc sgnals. ) Exponental Sgnals: We consder an exponental sgnal of the form ϕt) = e αt, t [, ) 5) where α > s a real-valued parameter determnng the growng rate see examples n Fg. a). The sgnal-dervatve energy s expressed as ϕ t)) = α e αt, whch s postve valued for t [, ) Fg. b). Therefore, the optmal samplng-pont densty for ths monotonc sgnal s expressed as e αt λ opt t) = α e α 6) Pluggng 6) nto 5) defnes the optmal expander, v τ), va that yelds e αvτ) = τ 7) e α v τ) = ) ) α log e α τ +, 8) and the correspondng optmal nonunform parttonng s determned va 6) as a opt = ) ) α log e α N +, =,..., N.9) The MSE correspondng to the optmal nonunform samplng s calculated usng ) and expressed as {a opt} N ) 9 MSE = αn e α ). 4) We evaluate the gan of our approach wth respect to the unform samplng. for =,..., N). The MSE of unform hghresoluton samplng s calculated by settng a unform = N n the MSE expresson n 9), yeldng { } ) N MSE a unform = α e α 4N ). 4) The MSE of the two samplng methods are compared n Fg. c and Fg. d for varous exponental sgnals va the α parameter) and samplng resolutons the parameter N), respectvely. Note that the MSE values are normalzed by the sgnal energy, here expressed as ϕ t) = α t= Fgure presents an example for the compressor functon and the correspondng nonunform samplng of an exponental sgnal. ) Cosne Sgnal: Let us demonstrate our approach for a non-monotonc sgnal of the form: e α ). ϕt) = cos παt), t [, ) 4) where α > s an nteger determnng the number of perods contaned n the [, ) nterval see example for α = 5 n Fg. 4a). The dervatve energy of the cosne sgnal n 4) s ϕ t)) = 4π α sn παt), t [, ). 4) As demonstrated n Fg. 4b, the sgnal-dervatve energy s zero only at the ponts t = j α for j =,,..., α. The compressor functon does not lend tself here to a smple analytc form, nevertheless, t can be constructed numercally va ts defnton as a cumulatve-densty-functon see Eq. 4)) provdng the compressor curve n Fg. 4c. The α ponts of zero sgnal-dervatve energy are not a real obstacle n our numercal constructon and, anyway, ther correspondng values can be replaced by an arbtrarly small ε value as suggested above. The expander functon s numercally formed as the nverse of the compressor curve. The resultng nonunform samplng structure Fg. 4d) shows ts adaptaton to the local sgnal dervatve and to the perodc nature of the sgnal. ) Chrp Sgnal: The cosne sgnal n 4) can be extended to the followng chrp sgnal wth a lnearly ncreasng frequency: ϕt) = cos πt + αt)), t [, ). 44) Here the α > parameter determnes the lnear growth-rate of the frequency Fg. 5a exemplfes ths for α = 5). The sgnal-dervatve energy of the chrp 44) s ϕ t)) = 4π + αt) sn πt + αt)), t [, ). 45) The nonunform samplng of the chrp s demonstrated n Fg. 5d. Comparson between the nonunform samplng of the cosne sgnal Fg. 4) and the chrp sgnal Fg. 5) reveals the nfluence of the varyng frequency emboded n the chrp. In the numercal demonstratons provded here we set the value of ε to the smallest postve floatng-pont value avalable n Matlab va the command eps, whch returns the value 5.

7 7 a) b) c) d) Fg.. Demonstraton for an exponental sgnal ϕt) = expαt) for α >. a) The exponental sgnal for several α values. b) The sgnal-dervatve energy for several α values. In c)-d), theoretcal reconstructon-mse obtaned va nonunform and unform samplng procedures are compared. c) evaluated for varous α values and N = 5. d) evaluated for a range of samplng resolutons N) and α =. 46) a) Compressor Curve and the correspondng sgnal-dervatve energy s 5 for t [, α) [ α, ) ϕ t)) = for t [α, α) [ α, α) for t [α, α). 47) Fgure 6 demonstrates the samplng procedure for a pecewselnear sgnal constructed for α = 8. Fg. 6d shows that usng the extended samplng-pont densty functon from 8), where small dervatve-energy values are clpped to ε, provdes a good soluton to the constant-valued regon of ths sgnal. III. EXPERIMENTAL RESULTS: SAMPLING OF ONE-DIMENSIONAL SIGNALS In ths secton we present expermental evaluatons of the procedure proposed n secton II for nonunform samplng of one-dmensonal sgnals. b) Nonunform Samplng Fg.. Optmal nonunform samplng N = 5) of an exponental sgnal ϕt) = expt). a) the mappng between unform to nonunform samplngspacng. b) shows the orgnal sgnal magenta), the reconstructed sgnal from nonunform samplng blue), and the parttonng to samplng ntervals red). 4) Pecewse-Lnear Sgnal: Now we turn to evaluate our samplng method on a pecewse-lnear sgnal contanng a sgnfcant porton of a constant value. Ths wll show the sutablty of the proposed method also to sgnals wth a zero dervatve-energy over a large contnuous sub-nterval. We defne here a specfc sgnal form as 5t for t [, α) t α) + 5α for t [α, α) ϕt) = 6α for t [α, α) t α)) + 6α for t [ α, α) 5 t α)) + 5α for t [ α, ) A. The Man Competng Method: Tree-Structured Nonunform Samplng Let us consder samplng based on a nonunform parttonng that s represented usng a bnary tree. The approach examned here s nspred by the general framework gven n [] for optmzng tree-structures for varous tasks, and s also nfluenced by the dscrete Lagrangan optmzaton approach [] and ts applcaton n codng [], [4]. The suggested approach reles on an ntal tree, whch s a full d-depth bnary-tree, representng a unform segmentaton of the nterval [, ) nto d samplng ntervals of d length see example n Fg. 7a). The segmentaton of the nterval [, ) s descrbed by the leaves of the bnary tree: the nterval locaton and length are defned by the leaf poston n the tree, specfcally, the tree-level where the leaf resdes n determnes the nterval length. The examned nonunform segmentatons are nduced by all the trees obtaned by repeatedly prunng neghborng-leaves havng the same parent node examples are gven n Fgures 7b-7c). The ntal d-depth full-tree together wth all ts pruned subtrees form the set of relevant trees, denoted here as T d. The leaves of a tree T T d form a set denoted as LT ), where the number of leaves s referred to as LT ).

8 8 a) b) c) d) Fg. 4. Demonstraton for a cosne sgnal ϕt) = cosπt). a) The sgnal. b) The sgnal-dervatve energy. c) The optmal compressor curve. d) Optmal nonunform samplng usng N = samples the parttonng of the sgnal doman s n red). a) b) c) d) Fg. 5. Demonstraton for a chrp sgnal ϕt) = cos πt + 5t)). a) The sgnal. b) The sgnal-dervatve energy. c) The optmal compressor curve. d) Optmal nonunform samplng usng N = samples the parttonng of the sgnal doman s n red). a) b) c) d) Fg. 6. Demonstraton for a pecewse-lnear sgnal as n Eq. 46) wth α =. a) The sgnal. b) The sgnal-dervatve energy. c) The optmal compressor 8 curve. d) Optmal nonunform samplng va the extended samplng-pont densty usng N = samples the parttonng of the sgnal doman s n red). Accordngly, the tree T represents a possbly) nonunform parttonng of the [, ) nterval nto LT ) samplng ntervals. A leaf l LT ) resdes n the hl) level of the [ tree and corresponds to the nterval a left l), arght l) ) of length l) = hl). Followng the analyss n secton II, the optmal sample correspondng to the leaf l LT ) s expressed va 4) as ϕ opt l) = l) a rght l) a left l) ϕ t) dt. 48) Consequently, as the tree leaves correspond to a segmentaton of the [, ) nterval, the samplng MSE nduced by the tree T T d s calculated based on the leaves, LT ), va MSE T ) = l LT ) a rght l) a left l) ϕ t) ϕ opt l) ) dt. 49) For a sgnal ϕt) and a budget of N samples, one can formulate the optmzaton of a tree-structured nonunform samplng as mnmze T T d MSE T ) subject to LT ) = N, 5).e., the optmzaton searches for the tree wth N leaves that provdes mnmal samplng MSE. The unconstraned Lagrangan form of 5) s mn {MSE T ) + µ LT ) }, 5) T T d where µ s a Lagrange multpler that reflects the constrant LT ) = N. However, t should be noted that due to the dscrete nature of the problem such µ does not necessarly exst for any N value for detals see, e.g., [], []). The

9 9 problem 5) can also be wrtten as a rght l) ) dt mn ϕ t) ϕ opt T T l) + µ LT ). 5) d l LT ) a left l) Note that, due the non-ntersectng samplng ntervals, the contrbuton of a leaf, l LT ), to the Lagrangan cost s C l) = a rght l) a left l) ϕ t) ϕ opt l) ) dt + µ, 5) evaluated for the correspondng samplng nterval. The dscrete optmzaton problem 5) of fndng the optmal tree for a gven sgnal and a Lagrange multpler µ s addressed by the followng procedure. Start from the full d- depth tree and determne the correspondng samplng ntervals and ther optmal samples, squared errors, and contrbutons to the Lagrangan cost 5). Go through the tree levels from bottom and up, n each tree level fnd the pars of neghborng leaves havng the same parent node and evaluate the prunng condton: f C left chld) + C rght chld) > C parent) 54) s true, then prune the two leaves mplyng that two samplng ntervals are merged to form a sngle nterval of double length thus, the total samples n the parttonng s reduced by one). If the condton 54) s false, then the two leaves and the correspondng samplng ntervals) are kept. Ths procedure s contnued untl reachng a level where no prunng s done, or when gettng to the tree root. B. Evaluaton and Comparson The proposed samplng method s compared to two other samplng approaches. The frst s the trval, however commonly-used, unform samplng, where the sgnal doman s parttoned nto equal-sze samplng-ntervals. Specfcally, for a budget of N samples the sgnal doman [, ) s segmented accordng to a = N for =,,..., N. The samples are determned as the averages of the correspondng samplng ntervals. The second competng method s a nonunform samplng based on a bnary-tree structure that s adapted to the sgnal va a Lagrange optmzaton, as presented n the prevous subsecton. We examned samplng of several sgnals defned analytcally. A grd of µ values was set for the tree-structured Lagrange optmzaton, ths determned the number of samples to consder. Frst, the samplng of the cosne sgnal ϕ t) = 55 cos πt) was examned, and showed that our method consstently outperforms the unform and the treestructured technques for varous amounts of samples Fg. 8a). These observatons were further establshed by examnng Note that the proposed method and the unform samplng do not rely on µ and operate drectly based on a gven number of samples, however, we defned the examned sample budgets based on the µ grd of the treestructured samplng n order to mantan an accurate comparson between all the samplng methods. b) a) Fg. 7. Segmentatons of the [, ) nterval produced by bnary trees. The leaves, whch determne the parttonng, are colored n green. a) A full bnary tree of depth and the correspondng unform segmentaton nto 8 sub-ntervals. b) A tree obtaned by a sngle prunng of the full tree, thus the parttonng ncludes 7 sub-ntervals. c) A tree obtaned by several prunngs resultng n 4 leaves/segments. samplng of the chrp sgnal, ϕ t) = 55 cos πt + 5t)), and the pecewse-lnear sgnal from Eq. 46) that was scaled by a factor of 55 Fgures 8b-8c). IV. ANALYSIS FOR MULTIDIMENSIONAL SIGNALS In ths secton we dscuss the theoretc analyss of optmal hgh-resoluton samplng by studyng the problem for samplng multdmensonal sgnals. The analyss for the onedmensonal case s relatvely smple snce the partton s characterzed by samplng-nterval lengths see secton II). However, when consderng the multdmensonal problem the requred analyss becomes more ntrcate as the samplng regons are, n general, of arbtrary shape and sze. Accordngly, our analyss n ths secton s conceptually and mathematcally smlar to the study of multdmensonal hghrate quantzaton provded by Gersho [], whch generalzed the one-dmensonal theory of Bennett [] and the dstorton formula of Panter and Dte []. Thus, we provde a theoretc framework for optmal multdmensonal sgnal samplng at hgh resoluton. The gven dfferentable sgnal, ϕ x), s defned over a K-dmensonal unt cube, C K [, ] K, and s scalar realvalued from a bounded range,.e., ϕ x) [ϕ L, ϕ H ] for any x C K. The sgnal goes through a samplng procedure n order to provde a dscrete representaton usng N N scalar valued) samples, {ϕ } N, correspondng to a parttonng of C K to N dstnct multdmensonal regons, {A } N, such that N A = C K. Agan, we consder a reconstructon procedure provdng the contnuous-doman pecewse-constant sgnal ˆϕ x) = ϕ for x A, 55) c)

10 a) Cosne Sgnal b) Chrp Sgnal c) Pecewse-Lnear Sgnal Fg. 8. Performance comparson of the proposed samplng method, the unform samplng approach, and optmzed tree-structured samplng. The curves present the samplng MSE obtaned for varous sample budgets. and optmzaton n the sense of mnmzng the overall MSE, here formulated as note the mplct normalzaton n the untcube volume) ) MSE {A } N, {ϕ } N = N ϕ x) ϕ ) dx. A 56) Moreover, as before, the optmal samplng coeffcents gven the parttonng {A } N can be analytcally determned, showng that ϕ opt = ϕ x) dx, =,..., N 57) V A ) A where V A ) s the volume of the regon A. We consder the case of optmal hgh-resoluton samplng.e., N s very large), and assume that the optmal samplng regons are all small enough and approprately shaped such that we can further presume that the sgnal ϕ x) s well approxmated wthn each regon by a lnear form that s locally determned. The last assumpton emerges as a generalzaton of the assumptons for the one-dmensonal case that were presented and analyzed n detal n Secton II-A. Specfcally, here ϕx) β T x + γ for x A, 58) where β s a K-dmensonal column vector and γ s a scalar value. Moreover, the above lnear form can be determned n the th regon by the frst-order Taylor approxmaton of the sgnal around the regon center, x V A ) xdx, namely, A ϕx) ϕ x ) + ϕ x) x x ) 59) where ϕ x) s the sgnal gradent, evaluated at the regon center, here havng the form of a K-length row vector consstng of the K partal dervatves,.e., [ ϕ x) x=x ϕ x) =, x ϕ x) x x=x,..., ϕ x) ] x=x, x K 6) where ϕx) x j x=x s the partal dervatve of the sgnal n the j th standard drecton j =,..., K) measured at the regon central pont x. Accordngly, the lnear-form parameters of the th regon are set to β T = ϕ x) 6) γ = ϕ x ) ϕ x)x. 6) The local lnear approxmaton wthn each regon 58) yelds an optmal samplng coeffcent that s approxmately the sgnal value at the regon center,.e., ϕ opt V A ) A β T x + γ ) dx = β T x + γ ϕ x ). 6) Then, the samplng MSE of the th regon s MSE A ) = ϕ x) ϕ opt) dx 64) V A ) A [ β T V A ) x + γ β T )] dx x + γ A = V A ) β x x dx. Inspred by the analyss gven by Gersho [] to hgh-rate quantzaton, we turn to nterpret the last error expresson usng the normalzed moment of nerta of the regon A around ts center x, defned as x x dx A MA ), 65) K V A ) + K where ths quantty s nvarant to proportonal scalng of the regon. Now the regon MSE becomes A MSE A ) K β MA )V A ) K, 66) and the total MSE s expressed as ) N MSE {A } N = V A ) MSE A ) K N β MA )V A ) + K. 67) We now assume that n the hgh-resoluton scenaro there s a samplng-pont densty functon, λ x), such that for any

11 small volume A that contans x, the fracton of samplng ponts contaned n t s approxmately λ x) V A). Furthermore, the densty functon satsfes the approxmate relaton λ x) N V A ), for x A. 68) Hence, the above assumpton mples that adjacent samplng regons have smlar densty values. Na and Neuhoff [] ntroduced n the context of vector quantzaton) the mportant noton of the nertal profle, denoted here as m x). The functon m x) s assumed to be smooth and to approxmate the normalzed moment of nerta of the cells around ther mass centers) n the neghborhood of x. The smoothness of the nertal profle s based on the assumpton that neghborng regons have smlar values of normalzed moment of nerta n a hgh-resoluton segmentaton. The defntons of the samplng-pont densty functon n 68) and the nertal profle let us to express the MSE n 67) as ) MSE {A } N K N K N β m x ) V A λ x ) ). K 69) Furthermore, due to the hgh-resoluton assumpton we approxmate the prevous sum by the followng ntegral, MSE λ, m) K β m x) x) dx. N K λ x) K where x C K 7) β x) ϕx). 7) The error expresson n 7) can be nterpreted as Bennett s ntegral for multdmensonal samplng. Ths formula shows that the samplng MSE for a gven sgnal, whch s represented by ts gradent-energy densty, s determned by the samplng pont densty and the nertal profle of the samplng structure. We now argue that optmal hgh-resoluton samplng of a multdmensonal lnear sgnal wth a unform gradent-energy densty evaluated as everywhere n C K s obtaned by parttonng the sgnal doman, C K, based on a tessellaton generated by a sngle optmal n the sense of mnmum normalzed moment of nerta) convex polytope, A K, such that all the regons n the segmentaton are congruent to t. Accordngly, ths optmal dvson results n a constant normalzed moment of nerta to all of the samplng regons, namely, MA ) = MA K) for =,..., N, 7) or n nertal profle terms: m x) = MA K ),.e., a constant functon. We further assume that optmal hgh-resoluton samplng of a nonlnear sgnal s obtaned by regons that are Based on the hgh-resoluton assumpton, we neglect cells that are ntersected by the boundary of the sgnal doman, C K, and thus may not be congruent to the optmal tessellatng polytope. approxmately congruent and scaled versons of the optmal polytope used for samplng a lnear sgnal. Then, Eq. 7) s satsfed also n our case of samplng a nonlnear sgnal, snce scalng does not change the normalzed moment of nerta. The latter hypothess mrrors the conjecture made by Gersho n [] for hgh-rate quantzaton. Consequently, the MSE expresson 7) s reformed to MSE λ) K MA K ) N K x C K β x) dx. λ x) K 7) We optmze the samplng procedure by characterzng the best samplng-pont densty, λ opt x), that mnmzes the MSE as expressed n 7). Smlar to the optmzaton n the onedmensonal case see Appendx B), we rely on Hölder s nequalty that provdes us a lower bound to the MSE n 7) n the form of MSE λ) 74) K MA K ) β x) ) + K K N K x C K K+ dx Here, followng the applcaton of Hölder s nequalty, the MSE lower bound s attaned when λx) s proportonal to β x) mplyng that the optmal samplng pont densty s λx)) K λ opt x) = β x) ) K K+ z C K β z)) K K+ dz. 75) Note that we used the fact that ntegratng a densty functon over the entre doman should be. The optmal samplng-pont densty demonstrates that n regons where the dervatve energy s hgher, the samplng should be denser by reducng the volumes of the relevant samplng regons. Moreover, returnng to the dscrete formulaton for hgh-resoluton samplng MSE n 67) and by utlzng 68) together wth 75) and 7) shows that n the optmal soluton all the samplng regons contrbute the same amount of MSE. In addton, the results n ths secton are generalzaton of those obtaned n the analyss of one-dmensonal sgnals n the prevous secton, ths can be observed by settng K = and MA ) =, whch s the normalzed moment of nerta for one-dmensonal ntervals or any other K-dmensonal cube) around ther center. The hgh-resoluton analyss provdes a theoretc evaluaton of a sampler based on ts samplng-pont densty functon. As we descrbed above, the suggested framework lets to determne the optmal samplng procedure n terms of the best samplng-pont densty functon. In the one-dmensonal case the samplng-pont densty can be drectly translated to a practcal samplng procedure va the compandng model see sectons II and III-B). However, n the multdmensonal case, n general, there are no drect ways to mplement a sampler based on a gven samplng-pont densty functon. Ths concluson s based on the followng results from the

12 quantzaton feld [], [5], [6]. In [5], [6] t was shown that optmal compandng requres a compressor functon that s a conformal mappng. Ths result mples that a drect mplementaton of multdmensonal sampler based on a gven pont densty s lmted to a mnor class of sgnals wth a sutable gradent-energy densty thus, n general, optmal multdmensonal compandng s mpractcal. Ths result was followed by a treatment of multdmensonal compandng for the vector quantzaton problem n lmted settngs that consder suboptmal solutons and/or partcular source dstrbutons [7] [4]. Consequently, as n the hgh-rate quantzaton lterature, the analyss provded here for the multdmensonal case s a theoretc framework for studyng samplng of multdmensonal sgnals. Specfcally, t descrbes the optmal sampler and allows to assess ts theoretc performance. Ths together wth Bennett s ntegral for multdmensonal samplng Eq. 7)) can be used to evaluate the performance of practcal suboptmal samplng procedures smlar to the analyss of practcal vector quantzers n []). V. CONCLUSION We analyze the topc of nonunform samplng of determnstc sgnals as a mrror-mage of nonunform quantzaton. Wth the advent of new technologes, adaptve samplng becomes a vable alternatve to be consdered n data compresson applcatons. In all the above developments, the crucal localdensty controllng parameter turns out to be the local energy of the sgnal gradent. A new adaptve samplng method for one-dmensonal sgnals s proposed and expermentally establshed as a leadng nonunform-samplng approach. APPENDIX A ANALYSIS OF INACCURACIES DUE TO THE SIGNAL LINEARITY ASSUMPTION Our man hgh-resoluton assumpton suggests to consder the sgnal va ts local lnear approxmaton. Let us assume that the second dervatve of the sgnal, ϕ t), s contnuous and bounded,.e., ϕ t) M for some postve constant M. Expressng the remander of the frst-order Taylor approxmaton usng ts ntegral form, lets us to rewrte 5), for t [, a ) recall that t s the nterval center), as ϕt) = ϕ t ) + ϕ t ) t t ) + R t), where the remander for the th samplng nterval) s R t) = t 76) t t z) ϕ z) dz 77) Consderng the remander n ts ntegral form wll be useful for the analyss n ths appendx. The absolute value of the remander s bounded for t [, a ) as follows R t) M t t ) 78) where the last nequalty conforms wth R t) = o t t ) for t t, as n Eq. 5). As explaned n Secton II, the optmal sample value s the sgnal average over the samplng nterval see Eq. 4)). Therefore, averagng the sgnal n ts form from 76) gves ϕ opt = ϕ t ) + R t)dt. 79) Hence, the remander average s the amount of naccuracy n the optmal sample value due to the sgnal lnearty assumpton. We analyze ths quantty by boundng t a R t)dt R t) dt 8) M t t ) dt = M 4 where we used the remander bound from 78). Usng the bound n 8) we can state that the naccuracy n the sample value s R t)dt = o ) 8) for, as was presented n 6). Now we proceed to analyzng the samplng MSE n the th nterval. The basc expresson gven n 7) s equvalent to MSE = ϕ t)dt ϕ opt) 8) Then, usng the expressons from 76) and 79) the nterval MSE becomes MSE = ϕ t )) + D 8) where D = R t)dt 84) +ϕ t ) a R t)dt t t ) R t)dt evaluates the devaton from the MSE obtaned for the lnearapproxmaton sgnal. Let us bound the MSE devaton term, D, as follows: D 85) R a t)dt + ϕ t ) t t ) R t)dt.

13 Then, usng the bound 78) we get M R t)dt t t ) 4 dt = 4 M 4, 86) and also t t ) R t)dt t t R t) dt 87) M t t dt = M Pluggng 86) and 87) nto 85), together wth assumng that ϕ t ) < M, yelds D M 4 + M M 88) mplyng that the MSE devaton follows D = o ) as. 89) APPENDIX B SAMPLING-POINT DENSITY OPTIMIZATION VIA HÖLDER S INEQUALITY FOR THE ONE-DIMENSIONAL CASE We am at mnmzng the samplng MSE gven n ) as MSE λ) N Usng Hölder s nequalty we can wrte ) ϕ t)) dt λ t) Smplfyng the left sde of 9) gves ϕ t)) λ t) dt ϕ t)) λ dt. 9) t) ) λ t) dt ϕ t)) dt. λ t) dt ϕ t)) dt. 9) 9) Then, snce λ t) s a densty functon ts ntegraton over the entre doman equals to, reducng 9) nto ϕ t)) λ dt ϕ t) t)) dt. 9) Here, Hölder s nequalty s attaned wth equalty when λ t) s proportonal to ϕ t)) λ t), therefore, the optmal samplng-pont densty s λ opt t) = ϕ t)). 94) ϕ z)) dz Settng the lower bound from 9), whch s acheved by λ opt t), n the MSE expresson from 9) gves the followng optmal samplng MSE MSE λ opt) N ACKNOWLEDGMENT ϕ t)) dt. 95) The authors thank the revewers for very pertnent and useful comments. REFERENCES [] W. R. Bennett, Spectra of quantzed sgnals, Bell System Techncal Journal, vol. 7, no., pp , 948. [] P. Panter and W. Dte, Quantzaton dstorton n pulse-count modulaton wth nonunform spacng of levels, Proceedngs of the IRE, vol. 9, no., pp , 95. [] A. Gersho, Asymptotcally optmal block quantzaton, IEEE Transactons on nformaton theory, vol. 5, no. 4, pp. 7 8, 979. [4] A. J. Jerr, The Shannon samplng theorem Its varous extensons and applcatons: A tutoral revew, Proceedngs of the IEEE, vol. 65, no., pp , 977. [5] K. Horuch, Samplng prncple for contnuous sgnals wth tmevaryng bands, Informaton and Control, vol., no., pp. 5 6, 968. [6] J. Clark, M. Palmer, and P. Lawrence, A transformaton method for the reconstructon of functons from nonunformly spaced samples, IEEE Transactons on Acoustcs, Speech, and Sgnal Processng, vol., no. 5, pp. 5 65, 985. [7] N. Brueller, N. Peterfreund, and M. Porat, Non-statonary sgnals: optmal samplng and nstantaneous bandwdth estmaton, n Proceedngs of the IEEE-SP Internatonal Symposum on Tme-Frequency and Tme- Scale Analyss, 998, pp. 5. [8] D. We and A. V. Oppenhem, Samplng based on local bandwdth, n Conference Record of the Forty-Frst Aslomar Conference on Sgnals, Systems and Computers ACSSC), 7, pp. 7. [9] G. J. Sullvan, J. Ohm, W.-J. Han, and T. Wegand, Overvew of the hgh effcency vdeo codng HEVC) standard, IEEE Transactons on crcuts and systems for vdeo technology, vol., no., pp ,. [] R. Shukla, P. L. Dragott, M. N. Do, and M. Vetterl, Rate-dstorton optmzed tree-structured compresson algorthms for pecewse polynomal mages, IEEE Transactons on Image Processng, vol. 4, no., pp. 4 59, 5. [] E. Shusterman and M. Feder, Image compresson va mproved quadtree decomposton algorthms, IEEE Transactons on Image Processng, vol., no., pp. 7 5, 994. [] R. A. Ulchney, Revew of halftonng technques, n Electronc Imagng, 999, pp [] D. L. Lau and R. Ulchney, Blue-nose halftonng for hexagonal grds, IEEE Transactons on Image Processng, vol. 5, no. 5, pp. 7 84, 6. [4] D. P. Mtchell, Generatng antalased mages at low samplng denstes, n ACM SIGGRAPH Computer Graphcs, vol., no. 4, 987, pp [5], Spectrally optmal samplng for dstrbuton ray tracng, n ACM SIGGRAPH Computer Graphcs, vol. 5, no. 4, 99, pp [6] M. McCool and E. Fume, Herarchcal posson dsk samplng dstrbutons, n Proceedngs of the Conference on Graphcs Interface, vol. 9, 99, pp [7] R. Fattal, Blue-nose pont samplng usng kernel densty model, n ACM Transactons on Graphcs TOG), vol., no. 4,, p. 48.