A Novel SDASS Descriptor for Fully Encoding the Information of 3D Local Surface

Transcription

1 1 A Novel SDASS Descrptor for Fully Encodng the Informaton of 3D Local Surface Bao Zhao, Xny Le, Juntong X Abstract Local feature descrpton s a fundamental yet challengng task n 3D computer vson. Ths paper proposes a novel descrptor, named Statstc of Devaton Angles on Subdvded Space (SDASS), for comprehensve encodng geometrcal and spatal nformaton of local surface on Local Reference Axs (LRA). The SDASS descrptor s generated by one geometrcal feature and two spatal features. Consderng that surface normals, whch are usually used for encodng geometrcal nformaton of local surface, are vulnerable to varous nusances, we propose a robust geometrcal attrbute, called Local Prncpal Axs (LPA), to replace the normals for generatng the geometrcal feature of our SDASS descrptor. For accurately encodng spatal nformaton, we use two spatal features for fully encodng the spatal nformaton of a local surface based on LRA. Besdes, an mproved LRA s proposed for ncreasng the robustness of our SDASS to nose and varyng mesh resolutons. The performance of the SDASS descrptor s rgorously tested on several popular datasets. Results show that our descrptor has a hgh descrptveness and strong robustness, and ts performance outperform exstng algorthms by a large margn. Fnally, the proposed descrptor s appled to 3D regstraton. The accurate result further confrms the effectveness of the SDASS method. Index Terms Local feature descrptor, local reference axs, object recognton, 3D regstraton. 1 INTRODUCTION L B. Zhao, X. Le, and J. X are wth the School of Mechancal Engneerng, Shangha Jao Tong Unversty, Shangha , Chna (e-mal: zhaobao1988@sjtu.edu.cn; lexny@sjtu.edu.cn; jtx@sjtu.edu.cn). (Correspondng author: Juntong X) OCAL feature descrptor beng used for encodng the nformaton of local surface has many applcatons n 3D computer vson areas such as 3D regstraton [1-3], 3D object categorzaton and recognton [4-6], 3D model retreval and shape analyss [7, 8], and 3D bometrc [9], to name a few. In the last few years, wth the development of numerous low-cost scanners and hgh-speed computng systems (e.g., Mcrosoft Knect and Intel RealSense), 3D data (e.g., clouds, meshes and depth mages) becomes easly avalable, whch further mproves the sgnfcance of nvestgatng local shape descrptors n 3D computer vson area. A local shape descrptor s usually constructed by transformng the geometrcal and spatal nformaton of a local surface nto a feature vector representaton [10]. It s worth notng that the descrptor presented n ths paper s appled to rgd objects. Therefore, the fundamental attrbute of local shape descrptor should be nvarant to rgd transformaton. Furthermore, a shape descrptor should have a hgh descrptveness and robustness [11]. The descrptveness of a local shape descrptor s an ablty of encodng the predomnant nformaton on the underlyng local surface. In other words, the descrptveness denotes the ablty of dstngushng one local surface from another. The robustness of a local feature descrptor ndcates an ablty of resstng the mpact of varous nusances ncludng nose, varyng mesh resolutons, etc. Besdes, the compactness and effcency are also mportant to a feature descrptor for some applcatons ncludng robots and moble phones [12]. So, desgnng a local feature descrptor wth overall good performance for dealng wth above mentoned nusances s a tremendous challenge. Over the last two decades, a number of local feature descrptors are desgned for mprovng the ablty of copng wth these nusances. Examples nclude Spn Images [13, 14], sgnature of hstograms of orentatons (SHOT) [10, 15], rotatonal projecton statstcs (RoPS) [16]. For more detals, readers can refer to a recent survey [11]. For a local feature descrptor, local frame and feature representaton are two major elements for determnng ts performance. The local frame can be dvded nto two categores. One s defned as local reference frame (LRF), and another s defned as local reference axs (LRA). The LRF s composed of three orthogonal axes, and the LRA only comprses a sngle orentated axs. Therefore, LRF can provde entre local 3D spatal nformaton ncludng radal, azmuth and elevaton drectons, whle LRA only provde the spatal nformaton n radal and elevaton drectons. It s worth notng that some local descrptors (e.g. THRIFT [17], PFH [18]) are not usng local frame for constructng ther features. These descrptors can encode the spatal nformaton only n radal drecton, whch usually present a lmtng descrptveness owng to the lack of adequate spatal nformaton [11]. For the LRF/A-based descrptors, although LRF can provde entre spatal nformaton, the repeatablty of x/y axs are more vulnerable to nusances (e.g. nose, varyng mesh resoluton and symmetrcal surface) than z axs, and more tme s needed for constructng t than LRA [12, 19]. Therefore, a local descrptor constructed on LRA has a hgh potental wth robust to varous nusances [12]. For LRA-based descrptors, the repeatablty of LRA drectly nfluence ther performance. To acheve a hgh repeatablty of LRA, many methods have proposed for constructng LRA or LRF (the z-axs of a LRF can be used as a LRA). These methods nclude the technques proposed by Man et al. [20], Tombar et al. [15], etc. In these methods, LRA or the z-axs of LRF s usually constructed by covarance analyss. Unfortunately, none of these methods have an overall good performance for copng wth varous nusances ncludng nose, mesh boundary, sgn ambguty, etc. In the vew of the feature representaton of a descrptor, the geometrcal and spatal nformaton of a local surface are usually

2 2 encoded for representng a local shape. Some exstng descrptors only encode the geometrcal nformaton of a local surface [17, 21]. In these methods, the devaton angle between normals or between normal and LRA s a popular way of encodng local geometrcal nformaton. However, owng to low repeatablty of normal, geometrcal nformaton of local surface encoded by the devaton angle presents a lower robustness [11]. Therefore, these descrptors of encodng only geometrcal nformaton often present a poor performance for resstng nose, varyng mesh resoluton, etc. In contrast, some descrptors only encode the spatal nformaton of a local surface [13, 16, 22]. However, some of these descrptors are ncomplete to encode the spatal nformaton by transformng 3D to 2D/1D (e.g. RoPS [16], TrSI [22]), and some of these descrptors are redundant to encode the spatal nformaton by repettvely usng the nformaton of x, y and z coordnates of local ponts (e.g. RoPS [22], TOLDI [23]). In addton, some descrptors encode both geometrcal and spatal nformaton of a local surface [3, 15]. Although, encodng geometrcal nformaton together wth spatal nformaton wll sgnfcantly mprove the descrptveness of a feature descrptor [15], as mentoned above, lower robustness of geometrcal feature and the mperfect encodng spatal nformaton also lmt the descrptveness and robustness of these descrptors. In these regards, we propose a novel local feature descrptor named Statstc of Devaton Angles on Subdvded Space (SDASS). Our SDASS s generated on LRA and encode both geometrcal and spatal nformaton of a local surface. Specfcally, we frst propose an mproved LRA whch s developed from the LRF proposed by Yang et.al [23]. Consderng normal beng vulnerable to varous nusances, to mprove the robustness of encodng geometrcal nformaton, we propose a new geometrcal attrbute named local prncpal axs (LPA), whch s verfed of havng a strong robustness to resst varous nusances, to replace normals for constructng the devaton angle between LPA and LRA. For encodng spatal nformaton, our SDASS use two spatal features for fully encodng spatal nformaton of a local surface on LRA. Dfferent from some prevous local descrptors whch need to process ntal pont cloud such as trangulaton [16, 22], our SDASS s drectly performed on the ntal pont clouds. For verfyng the performance of the SDASS, t s appled to three popular datasets ncludng shape retreval, shape regstraton and object recognton scenaros, and compared to several state-of-the-art methods. The expermental results show that the performance of our SDASS exceeds the extng methods by a large margn. In addton, the SDASS s appled to 3D regstraton n the last of ths paper. The accurate outcomes further confrm the effectveness of our method. The major contrbutons of ths paper are summarzed as follows: Frst, we propose a geometrcal attrbute LPA whch has a sgnfcantly hgh repeatablty compared to normals. We use the proposed LPA to replace normals for generatng devaton angles, whch presents a hgh descrptveness and strong robustness for encodng the geometrcal nformaton of local surface. Second, a novel local shape descrptor named SDASS s proposed. The SDASS s generated on LRA, and descrbe a local surface by the combnaton of encodng geometrcal and spatal nformaton. The expermental results show that the performance of SDASS sgnfcantly surpasses the exstng local shape descrptors. Thrd, an mproved LRA s proposed, whch has a superor performance for resstng nose and varyng mesh resoluton compared to the exstng LRF/A. The rest of ths paper s organzed as follows. Secton 2 provdes a bref revew of related work of generatng LRF/A constructon and local shape descrptors. Secton 3 presents a detaled descrpton of the proposed SDASS method. Secton 4 ntroduces the expermental evaluaton of our method and several state-of-the-art methods on three standard datasets. Secton 5 descrbes the smple applcaton of the SDASS descrptor on 3D regstraton. The conclusons and future works are drawn n Secton 6. 2 RELATED WORK Ths secton presents a bref overvew of the exstng methods for local surface feature descrpton. Consderng that the SDASS method belong to the famly of LRA-based methods. The exstng methods for constructng LRA or the z-axs of LRF are frst revewed. Then, the exstng local shape descrptors are dvded nto three categores to be descrbed respectvely. 2.1 The Methods of Constructng LRA or the Z- Axs of LRF Most exstng methods of constructng LRA or the z-axs of LRF are based on covarance [19]. In these methods, the z-axs or LRA s the normalzed egenvector correspondng to the mnmal egenvalue of a covarance matrx. Man et al. [20] use radus neghbors nstead of k nearest neghbors of a key pont to construct the covarance matrx for mprovng the robustness to varyng mesh resoluton. However, the sgn of the LRF s not defned unambguously. Tombar et al. [10] construct a weghted covarance matrx by frst usng a key pont to replace the barycenter of radus neghbors, and then assgnng smaller weghts to more dstant ponts. The sgn of ths LRF s unambguous by nclnng the barycenter of local surface. Ths method has been proven to be qute robust to nose, whle s vulnerable to varyng mesh resoluton [23]. Later, Guo et al. [16] propose a novel technque for constructng LRF by frst applyng two weghtng strateges to each trangle on a local surface, and then usng all weghted trangles to calculate the covarance matrx. The sgn of ths LRF s dsambguated by algnng the drecton to the majorty of the pont scatter. The man advantage of ths method s to have a hgh robustness to varyng mesh resoluton [16], whle the effcency s very low [23]. Recently, Petrell et al. [1] and Yang et al. [23] pck a small subset of the radus neghbors for constructng covarance matrx. The man purpose of usng less ponts for calculatng the z-axs s to mprove ts robustness to occluson, clutter and mesh boundares. In addton, Andrew E. Johnson et al. [13] and Yang et al. [3] drectly use normal as LRA. Snce normal s vulnerable to varous nusances (e.g. nose, varyng mesh resoluton), the LRA present a low robustness. Some methods mentoned above have ncely solved the problem of sgn ambguous, and are robust to some nusances. However, they do not have an overall good performance for copng wth all nusances (e.g. nose, varyng mesh resoluton, etc.), such as the LRF proposed by Guo et al. [16] has a strong robustness to varyng mesh resoluton and nose whle t s vulnerable to mesh boundary, and the LRF proposed by Yang et.al. [23] s robust to mesh boundary, whle t s vulnerable to nose and varyng mesh resolutons.

3 3 2.2 Local Shape Descrptors Local shape descrptors have been wdely proposed n lteratures over the last two decades [11, 24]. Among these descrptors, some only encode geometrcal nformaton, and some only encode spatal nformaton, and others encode the combnaton of geometrcal and spatal nformaton of a local surface. There are some descrptors for only encodng geometrcal nformaton. Flnt et al. [17, 25] proposed a local feature descrptor called THRIFT whch uses the devaton angles between the normal at a key pont and the normals at ts neghbors to construct a 1D hstogram. The THRIFT s a very effcent descrptor, whle s very senstve to nose and varyng mesh resolutons [11]. Rusu et al. [18] proposed a pont feature hstogram (PFH) by usng several features of pont pars n the support regon. The several features are obtaned on a Darboux frame constructed by the normals and pont postons. The PFH s robust to varyng mesh resolutons, whle s vulnerable to nose [11]. Later on, n order to mprove the effcency, they used the smplfed pont feature hstogram (SPFH) of neghbors to construct the fast pont feature hstograms (FPFH) descrptor [21] whch has a smlar performance wth PFH n descrptveness and robustness. Some descrptors only encode spatal nformaton. Johnson and Hebert [13] proposed a local shape descrptor named Spn Image (SI). The Spn Image use the normal at a key pont as the LRA, and then each pont n the support regon s represented by two spatal dstances. Fnally, the Spn Image s generated by accumulatng the number of local ponts nto each bn of the 2D array. The Spn Image completely encodes the spatal nformaton of a local surface on LRA whle s senstve to nose [11] owng to the low repeatablty of ts LRA. Tombar et al. [26] proposed a unque shape context (USC) whch s developed from 3DSC descrptor [27].The USC s generated on a LRF, proposed by Tombar et al. [10], by dvdng the local 3D space nto bns along azmuth, elevaton and radal drectons. The USC completely encode the spatal nformaton on LRF, and present a hgh robustness to nose, clutter and occluson, whle s senstve to varyng mesh resolutons [11]. Guo et al. [16, 28] proposed the rotatonal projecton statstcs (RoPS) descrptor. In RoPS descrptor, a novel LRF s frst constructed, and then the feature representaton s generated by calculatng a set of statstcs of pont densty wth respect to numerous rotatons of local surface around each axs. The RoPS descrptor s proved to have a hgh descrptveness [11], whle s very tme-consumng. Smlar to the vew-based mechansm n RoPS, Guo et al. [22, 29] proposed a Tr-Spn-Image (TrSI). TrSI s also generated on a LRF whch constructed by a smlar technque as n the RoPS. Then, TrSI s generated by ntegratng and compressng three spn mage sgnatures created on each axs of the LRF. The performance of TrSI to resst varous nusances (e.g. nose, varyng mesh resolutons) s slghtly better than that of the RoPS [11]. Lke RoPS, TrSI s also tme-consumng. Recently, Yang et al. [23] proposed a novel trple orthogonal local depth mages (TOLDI) by frst constructng a LRF, and then concatenatng three local depth mages captured from three orthogonal vew planes n the LRF to generate a feature vector. Although TOLDI acheve a good performance for resstng varous nusances, the feature vector of TOLDI present a low compactness [22]. There are also some descrptors for encodng the combnaton of geometrcal and spatal nformaton. Tombar et al. [10, 15] proposed the sgnature of hstograms of orentatons (SHOT) descrptor based on a LRF. The SHOT descrptor frst encodes spatal nformaton on the LRF by dvdng the sphercal neghborhood space nto several bns along the radal, azmuth and elevaton drectons. Then, for each bn, the geometrcal nformaton s encoded by generatng the devaton angles between the normal at the key pont and the normals at the radus neghbors. Despte SHOT havng a hgh descrptveness, t s senstve to varyng mesh resolutons [11, 23]. Recently, Yang et al. [3] proposed a local feature statstcs hstograms (LFSH) by concatenatng three features, ncludng two spatal features (local depth and horzontal projecton dstance) and one geometrcal feature (devaton angle), nto a feature vector. LFSH s very compact and effcent, whle t suffers from lmted descrptveness [3]. In concluson, encodng geometrcal nformaton combned wth spatal nformaton can mprove the descrptveness of a descrptor, whle some exstng descrptors only encodng geometrcal or spatal nformaton, whch have lmted descrptveness. In addton, for encodng geometrcal nformaton, the devaton angles between normals or between normal and LRA s commonly used. However, the normal generated on a very small local regon [30] has a weak robustness to resst varous nusances. For encodng spatal nformaton, some descrptors (e.g. RoPS, TrSI) don t fully encode the spatal nformaton on LRF by transformng 3D to 2D/1D, and some others aren t compact to encode spatal nformaton by repettve usng the nformaton of x, y and z coordnates of local ponts (e.g. TrSI, TOLDI). Therefore, the descrptveness and robustness of exstng descrptors have the potental for further beng mproved. 3 SDASS-BASED LOCAL SHAPE DESCRIPTION Ths secton detals our SDASS technque for local shape descrpton. Specfcally, we frst ntroduce an mproved LRA and robust LPA. Then, we present the SDASS feature representaton by generatng three features (projected radal dstance, heght dstance and devaton angle between LPA and LRA) based on the proposed LRA and LPA. Fnally, the key parameters of SDASS are quanttatvely selected. 3.1 Constructng Local Reference Axs (LRA) and Local Prncpal Axs (LPA) In ths secton, we construct a LRA developed from the z-axs n the LRF proposed by Yang et al. [23]. To mprove the robustness and descrptveness of encodng geometrcal nformaton, we defne a new geometrcal attrbute named local prncpal axs (LPA) to replace normal for calculatng the devaton angles Local Reference Axs In ths paper, we select LRA rather than LRF as a bass for spatal dvson. Although, LRF provdes the entre local 3D spatal nformaton ncludng radal, azmuth and elevaton whle LRA loses one-dmenson spatal nformaton (.e., the azmuth nformaton), as shown n Fg.1. However, the repeatablty and robustness of the x, y axs are sgnfcantly lower than the z axs n a LRF [12, 19], especally n the presence of nose, varyng mesh resoluton, etc. To provde a more accurate spatal dvson n our feature descrptor, we select LRA as a frame to construct our descrptor. Our LRA s developed from the z-axs n the LRF proposed by Yang et.al. [23]. In below, we frst smply repeat the z-axs proposed by Yang et.al. [23], and then propose our mproved

4 4 LRA based on the z-axs. x LRF z y LRA z Fg.1. LRF and LRA. (ρ, θ, φ) denote the radal, elevaton and azmuth drectons, respectvely [12]. Gven a key pont p and a support radus R, local ponts Q = {q 1, q 2,, q n } can be obtaned wthn the dstance of R from the key pont p. The subset of Q, whch s denoted as Q z = {q z 1, q z 2,, q z 3 }, determned by 1/3 support radus neghbors around the key pont s used for calculatng the drecton of the z-axs. Specfcally, based on Q z, a covarance matrx Cov(Q z ) s constructed as: z z T z z 1 1 q q q q Cov( Q z ) (1) z z z z qn q qn q where n s the sze of Q z, and q z s the centrod of Q z. The egenvector v(p) correspondng to the mnmum egenvalue of Cov(Q z ) s computed. Next, all support radus neghbors are used for dsambguatng the sgn of v(p) as follow: n v( p), f v( p) q 0 ( )= 1 p z p v( p), otherwse (2) where between vectors denotes dot-product, and q p represents the vector between q and p. As opposed to use the subset of radus neghbors for calculatng the drecton and usng all radus neghbors for dsambguatng the sgn of the z-axs (also named LRA) proposed by Yang et.al. [23], we use all radus neghbors to determne both the drecton and sgn of our LRA based on Eqs. (1) and (2). In order to verfy the performance of the two strateges (Yang et.al. [23] and our) as well as provde a bass for generatng our LPA n Sect.3.1.2, a test to the two strateges wth respect to varyng support rad s mplemented. For ncreasng readablty, the strategy (Yang et.al. [23]) of usng dfferent rad neghbors to calculate the drecton and the sgn of a LRA s defned as S1, and the strategy (our) of usng the same rad neghbors s defned as S2. In the strategy S1, a support radus for calculatng the drecton of a LRA s fxed to 20 mesh resoluton (mr) as referred to [12, 31], and a support radus for elmnatng sgn ambguty ncreases from 3mr to 20mr wth a step of 1mr. In the strategy S2, the support rad of calculatng the drecton and elmnatng sgn ambguty are smultaneously ncrease from 3mr to 20mr wth a step of 1mr. The datasets used n ths test are detaled n Sect Specfcally, to test the robustness for resstng nose and varyng mesh resolutons, the three scenes n BR (the Gaussan noses wth the devaton of 0.1, 0.3, 0.5 mr correspondngly combnng wth the decmaton rates of 1/2, 1/4, 1/8) are selected. To test the robustness to mesh boundary, clutter and occluson, UWAOR and UWAOR-IR datasets are selected. The evaluaton crteron s ntroduced n Sect The percentage of angle errors below 5 o s counted for evaluatng the repeatablty of LRA. The results of ths test are presented n Fg.2. Several observatons can be summarzed as follows. (a) Tested on BR dataset wth the combnaton of Gaussan nose (b) Tested on UWAOR and UWAOR-IR datasets. and varyng mesh resolutons. Fg. 2. Two strateges (S1 and S2) for generatng LRA wth respect to dfferent support rad. Frst, n the presence of nose and varyng mesh resolutons (as shown n Fg.2 (a)), the repeatablty of the LRAs generated by S1 and S2 s gradually mproved along wth the ncrease of support radus. The hghest repeatablty values of them are the same appeared wth the support radus at 20mr, although the repeatablty of the LRA generated by S1 s hgher than S2 n the process of the support radus ncreased from 3mr to 20mr. So, we can conclude that, for the mpact of nose and varyng mesh resoluton, the support rad for calculatng the drecton and elmnatng the sgn ambguty by smultaneously takng the maxmum value (.e. 20mr) can get a hgher repeatablty. Second, for the mpact of mesh boundary as tested on UWAOR dataset n Fg.2. (b), the repeatablty of the LRA generated by S1 s hgher than S2, and ths dfference gradually dsappear along wth the ncrease of support radus. In addton, the varyng tendency of them s smlar. Ther repeatablty s gradually mproved along wth the ncrease of the support radus before about 6mr, and then drop after 6mr. When elmnate the nfluence of mesh boundary, as tested on UWAOR-IR dataset n Fg.2 (b), the repeatablty of the LRAs constructed by S1 and S2 sn t fall along wth the ncrease of support radus. From ths observaton, we can conclude that the strategy S1 has a superor performance for resstng the mpact of mesh boundary compared to S2, whle, when the mesh boundary s elmnated (as tested on UWAOR-IR), the best performances of S1 and S2 are smlar n the process of support radus from 3mr to 20mr.

5 5 From the above two observatons and correspondng conclusons, we can fnd that our LRA has a strong robustness to nose and varyng mesh resoluton, whle has a weak robustness to mesh boundary compared to the z-axs proposed by Yang et.al. [23]. The reason of mprovng the robustness to nose and varyng mesh resoluton and sacrfcng the robustness to mesh boundary nclude two aspects. The frst s that, for a LRA, t s dffcult to have a comprehensvely good performance for resstng varous nusances (e.g. nose, varyng mesh resoluton and mesh boundary, et. al.). The second s that mesh boundary s easer elmnated than nose and varyng mesh resolutons from a pont cloud [11, 20]. So, the mproved LRA s approprate appled on a pont cloud wth no boundary exstent (e.g. the BR dataset) or boundary dentfed (e.g. the UWAOR-IR dataset). In the case of mesh boundary beng exstent and not dentfed, the z-axs of the LRFs (e.g. the LRF proposed by Yang et. al. [23]) wth robust to mesh boundary can be used as the LRA to generate our descrptor Local Prncpal Axs The devaton angle between normals or between normal and LRA s commonly used for encodng geometrcal nformaton n prevous descrptors (e.g. FPFH [21], SHOT [15]). In ths process, the repeatablty of normals determnes the performance of encodng geometrcal nformaton on a local surface. At present, there are two popular technques of generatng normal. The frst technque uses all trangular patches attached wth a pont to calculate the normal of that pont [11, 20, 32]. The second technque frst generatng a covarance matrx by usng the radus neghbors of a pont, and then calculatng the egenvector correspondng to the mnmum egenvalue as for the normal of that pont [30, 33, 34]. In the followng, for ncreasng readablty, the normal generated by the frst method s called trangular patches based normal (TN) and by the second method s called radus neghbors based normal (RN). However, the dsambguaton of the normals constructed by these two technques rely on a vewpont [13, 34]. If the vewpont s unknown, manual to dsambguate of normals s needed [34], whch s neffcency. In addton, normal s a geometrcal attrbute to represent a very small local regon (e.g. a local surface for generatng TN has a support radus of 1mr averagely, and a local surface for constructng RN usually has a support radus less than 3mr [30, 35]). The repeatablty of the normal generated on a small local regon s senstve to nose, varyng mesh resoluton, etc. (as presented n Sect.3.1.1). For overcomng the above two weaknesses exstng n normal, we here propose a local geometrcal attrbute called Local Prncpal Axs (LPA) to replace normal for constructng devaton angle between LRA and LPA. The method of generatng our LPA s smlar to the technque of generatng the mproved LRA ntroduced n Sect In opposte to use a small regon for calculatng the drecton of normal, the proposed LPA use a larger local regon for determnng ts drecton, whch sgnfcantly ncrease the robustness to varous nusances. Accordng to the tested results presented n Fg.2, consderng the tradeoff among effcency, the robustness to nose and varyng mesh resoluton and the robustness to mesh boundary, the support radus for generatng LPA s selected as 7mr n ths paper. Based on ths support radus, the drecton and sgn of LPA can be determned by usng Eqs. (1) and (2), respectvely, whch are presented n Sect In ths way, dsambguatng the sgn no longer rely on the vew pont for mprovng the effcency and accuracy. 3.2 SDASS Descrptor Once the LRA and LPA are constructed, we are left wth the task of encodng spatal and geometrcal nformaton contaned on a local surface. Gven a pont cloud or surface, the local ponts are determned by a key pont p and a support radus R. We defne the local ponts as Q= {q 1, q 2,, q m}. As shown n Fg. 4 (b), (c), Q s frst transformed to make p concded wth the coordnate orgn, and the LRA of the key pont algned wth z axs to acheve the rotaton nvarance of the local surface. The transformed local ponts are denoted as Q = {q 1, q 2,, q m }. Then, we seek for an approprate manner of encodng spatal and geometrcal nformaton on the local surface. LRA p q h d Fg. 3. Two ways of encodng spatal nformaton on LRA In general, the spatal nformaton of local surface on LRA can be fully encoded by two ways, as shown n Fg.3. One way uses the radal dstance (ρ) and azmuth angle (θ) to encode spatal nformaton, and another way use the heght dstance (h) and projected radal dstance (d) to encode spatal nformaton. Some descrptors use the radal dstance and azmuth angle to encode spatal nformaton (e.g. SHOT [15], USC [26]), whle the majorty of descrptors select to use the heght and projected radal (e.g. Spn mage [13], TrSI [29], LFSH [3]). In ths paper, we also use the projected radal and heght dstance for comprehensve encodng the spatal nformaton on LRA. Besdes, our SDASS descrptor uses a geometrcal feature (the devaton angle between LRA and LPA) to encode the geometrcal nformaton of local surface. The devaton angles between LRA and the normals of neghbors are usually used for encodng geometrcal nformaton on local surface, such as SHOT [15], LFSH [3]. Consderng that normal represents the attrbute of small local regon, whch s senstve to varous nusances, we propose a geometrcal attrbute LPA (see Sect ), whch has very hgh robustness compared to normals as verfed n Sect , to replace the normal for constructng the devaton angles. Next, we present the process of generatng the three local features of the SDASS descrptor. (1) Projected radal dstance nformaton, as shown n Fg.4 (d): For the transformed local ponts Q, the tangent plane at the key pont p s concded wth XY plane. The Q are projected onto the tangent plane for obtanng 2D projected ponts. The dstances between these 2D ponts to the p are computed. Snce the LRA has algned wth z axs, the dstances are smply computed as: l q q m, x 2 2 (1) + (2) 1,2,..., where q (1) and q (2) denote the x, y coordnate of the pont q, respectvely. The range of l x [0, R]. (2) Heght dstance nformaton, as shown n Fg.4 (e): We translate the tangent plane wth a dstance of R along the negatve drecton of LRA. The heght dstances are defned as the dstances between the local ponts Q and the translated tangent plane, whch are calculated as:

6 6 y l R q(3) 1,2,..., m, where q (3) denote the z coordnate of the pont q. The range of l y [0, 2R]. (3) Devaton angles between LRA and LPA, as shown n Fg.4 (f): The devaton angles between the LRA of the key pont p and the LPA of the local ponts Q are used for encodng geometrcal nformaton. Smlar to normals, the LPA of all ponts on the pont cloud s needed to be precomputed. Based on the precomputed LPA of the local ponts, the devaton angles correspondng to ths key pont are calculated as: arccos( LRA, LPA ) 1,2,..., m The range of θ [0, π]. p Z LRA LRA Z p Y Y (a) (b) (c) X Ɵ p X tangent plane tangent plane p R LRA p R x l q q y l (d) (e) (h) n x n y n Ɵ L y 2R 2 x y ( l, l, ) p(0,0,0) I J K R (g) x y q( l, l, ) L x tangent plane LRA p R Ɵ LPA Fg.4. An llustraton of generatng a SDASS feature descrptor for, (a) object, (b) constructng LRA, (c) transformed local ponts, (d) projected radal dstance, (e) heght dstance, (f) devaton angle between LPA and LRA, (g) dstrbuton of feature space, (h) statstcs. q (f) After the three features are generated, a 3D feature space can be created based on the three features, as shown n Fg.4 (g). In the fgure, the axes L x, L y and θ denote the projected radal dstance, heght dstance and devaton angle, respectvely. Assume that the subdvson numbers of the three axes are N x, N y and N θ, respectvely. Then, the number of bns n the 3D space s N x N y N θ and step szes (d x, d y, d θ) of the subdvson along the three axes can be computed as: d R / N ; d 2 R / N ; d 2 / N, x x y y where d x, d y and d θ are the steps along L x, L y and θ axes, respectvely. Accordng to the three step szes, each local pont q can be mapped to a bn n the 3D space. The ndexes (I x, I y, I θ) of the bn along the three axes are calculated as: x y I x l / d x ; I y l / d y ; I / d. After all ponts n Q are mapped to the 3D space, the number of ponts fallng nto each bn s counted to yeld a 1D hstogram wth the dmenson beng N x N y N θ. To acheve robust to varatons of the pont densty, we normalze the whole descrptor to sum up to 1. In addton, consderng that the two spatal features (projected radal dstance and heght dstance) dvde a cylndrcal space wth the radus of R and the heght of 2R for encodng spatal nformaton whch contaned n the sphere wth the radus of R, there may exst some redundant bns n the hstogram. To mprove the compactness of descrptor, these redundant bns are elmnated from the hstogram. In the followng, we analyze the superor performance of the proposed SDASS n theory. Four ponts are summarzed as follows. Frst, n opposte to use the devaton angle of normals n prevous descrptors (.e. LFSH [3], FPFH [21]), the SDASS use the devaton angle between LPA and LRA for encodng geometrcal nformaton of local surface. The LPA has a hgh repeatablty n the presence of varous nusances (e.g. nose, varyng mesh resolutons, etc.) compared to normal (as verfed n Sect ). The repeatablty of LPA/normal determnes the accuracy of calculatng the devaton angle. So, wth the hgh repeatablty of our LPA, the geometrcal nformaton encoded n our SDASS descrptor has a strong robustness and a hgh descrptveness. Second, the SDASS descrptor encode the combnaton of spatal and geometrcal nformaton of a local surface. The spatal nformaton on LRA s completely encoded by two features (projected radal dstance and heght dstance). The geometrcal nformaton s encoded by the devaton angles between LRA and LPA. In the vew of the components n SDASS descrptor, LFSH s the most smlar descrptor to ours. In contrast, the three features used n LFSH are concatenated to consttute the fnal 1D hstogram. Although, t can reduce the dmensons of the descrptor, t s unable to fully encode the spatal nformaton on LRA, and parttoned statstcs of the geometrcal nformaton. In addton, the LFSH drectly use the normal at a key pont as LRA and encodes local geometrcal nformaton by usng the devaton angles between the LRA and the normals of neghbors, whch greatly reduces the robustness of descrptor to varous nusances (e.g. nose, varyng mesh resoluton, etc.) owng to the low repeatablty of normal (as verfed n Sect ). Thrd, the SDASS descrptor s generated on LRA whch provde more accurate spatal nformaton than LRF [12, 19]. We

7 7 propose an mproved LRA for generatng SDASS descrptor. The LRA has a strong robustness to nose and varyng mesh resoluton whle t s senstve to the boundary of ponts cloud (as verfed n Sect ). The reason of generatng ths LRA s of consderng that mesh boundary s easer elmnated than nose and varyng mesh resolutons from a pont cloud [11]. So, we can select to use the mproved LRA n the case of a pont cloud wth no boundary exstent or boundary dentfed. In the case of a pont cloud havng boundary and not dentfed, z-axs of the LRFs (e.g. the LRF proposed by Yang et. al. [22]) wth robust to mesh boundary can be used as the LRA n our descrptor. So, a strong robustness of LRA can guarantee a good performance of the SDASS descrptor. Fourth, the computatonal effcency of our SDASS descrptor s hgh. The computatonal process of the SDASS descrptor manly nclude three steps: transformng local ponts; computng the three features; mappng local ponts from 3D geometrcal space to 3D feature space. Obvously, these steps are all smple arthmetc. The computatonal complexty of each step s O(k), where k denotes the number of the local ponts. So, the computatonal complexty of SDASS descrptor s also O(k). In addton, unlke to some descrptors (e.g. RoPS [16], TrSI [22]) need the mesh nformaton of ponts cloud, the SDASS descrptor s drectly appled on orgnal ponts cloud. 3.3 Selectng SDASS Parameters There are four parameters to generate our SDASS descrptor. They are respectvely the support radus R and the number of subdvsons of the three features N x, N y and N θ. The support radus R s an mportant parameter because large values of R would affect the computatonal effcency and ncrease the descrptor s senstvty to mesh boundary, clutter and occluson, whereas small values of R would make a descrptor less dstnctve [16]. Accordng to the suggeston n [12, 31], we select 20 mesh resoluton (herenafter mr, computed as ether the average length of the edges of the meshes or, should the dataset consst of pont clouds, the average dstance between neghbor ponts [1]. In ths paper, we consstently use the second method to calculate mr) as the value of support radus R. In ths paper, the dmenson of SDASS descrptor s N x N y N θ. Obvously, the bg values of N x, N y, N θ wll affect the computatonal effcency and consume more memory. If the values are too small, the descrptor would lose many detals about local shape. Accordng to our expermentaton and refer to the settng n [3, 15], the three parameters N x, N y, N θ are set to 5, 5, 15 n ths paper, respectvely. 4 EXPERIMENTS In ths secton, our LRA, LPA and SDASS descrptor are tested on three standard datasets for verfyng the robustness to nose, varyng mesh resoluton, mesh boundary, and etc., The three datasets nclude the Bologna retreval (BR) dataset [36, 37], the UWA object recognton (UWAOR) dataset [6], and the UWA 3D modelng (UWA3M) dataset [6, 38]. Our method s also compared wth several state-of-the-art methods (ncludng Spn Image [13], SHOT [15], RoPS [16], TrSI [22], LFSH [3], TOLDI [23]) for contrastve evaluaton. All tested descrptors are mplemented n MATLAB. The MATLAB code of SHOT s 1 The webste of Spn Image s: and the webste of RoPS and provded by author and the MATLAB code of Spn Image, RoPS and TrSI are avalable n webste 1. The MATLAB codes of LFSH s wrtten by us from ther correspondng C++ versons whch s provded by author. The MATLAB code of TOLDI s wrtten by us of referrng to the author s paper [23]. All the experments n ths paper are mplemented on a PC wth an Intel Core CPU and 12GB of RAM. 4.1 Expermental Setup In the followng, the mplementaton detals, ncludng the descrpton of datasets and the adopted evaluaton crtera, of the experments are ntroduced Datasets Lke [23], consderng the BR, UWAOR and UWA3M datasets can cover varous nusances (e.g. nose, varyng mesh resoluton, clutter and occluson, etc.), we also use these three datasets n ths paper for comprehensve evaluatng the performance of our descrptor. Some exemplars of the three datasets are shown n Fg.5. Specfcally, the BR datasets contans nose, and the UWAOR dataset contans occlusons, clutter and mesh boundary, and the UWA3M dataset contans occlusons and mesh boundary. In addton, for comprehensve verfyng the robustness to nose and varyng mesh resolutons, we generate some new scenes n the BR dataset, and for presentng the nfluence of mesh boundary, we generate two new datasets developed from UWAOR and UWA3M datasets, respectvely. The detals are presented as follows. The BR dataset has 6 models and 18 synthetc scenes. The sx models are nose freely coped from the Stanford 3D Scannng Repostory [36], whch are scanned by a Cyberware 3030 MS scanner. The 18 scenes n the BR datasets are created by addng Gaussan noses wth the standard devaton of 0.1, 0.3, and 0.5 mr, respectvely, n each randomly transformed model. To gve a comprehensve comparson for the robustness of descrptors to nose and varyng mesh resoluton, three group scenes are generated from the BR dataset. Each group ncludes 60 scenes. The frst group s generated by addng Gaussan noses wth ncreasng standard devatons from 0.1 to 1.0 mr wth an nterval of 0.1 mr n each randomly transformed BR model, whch nclude the 18 scenes coped from orgnal BR dataset. The second group s created by resamplng each randomly transformed model from 10/10 (10/10 s the orgnal resoluton) to 1/10 of ther orgnal mesh resoluton wth an nterval of 1/10. The thrd group s generated by correspondngly combnng the all levels of Gaussan nose n the frst group wth the all levels varyng mesh resolutons n the second group (e.g. the standard devaton of 0.1 mr combned wth the decmaton rate of 10/10). The scenes n ths dataset wthout any occluson or clutter. The purpose of employng ths dataset s to verfy the robustness to nose and varyng mesh resolutons. The UWAOR dataset contans 5 models and 50 real scenes. The scenes are generated by randomly placng four or fve models together n a scene and scanned from a specfc vewpont usng a Mnolta Vvd 910 scanner. Mesh boundary, clutter and occluson are the major challenges n ths dataset. The UWA3M dataset contans 22, 16, 16, and D vews scanned respectvely from four objects by usng a Mnolta Vvd 910 scanner. TrSI s:

8 8 Due to that a sngle vewpont cannot capture the structure of an entre 3D model, feature descrpton n ths dataset are confronted wth the nusances ncludng mesh boundary and selfoccluson. Snce every two vews of an object n the UWA3M dataset do not always share an overlap. To guarantee modelscene vews of an object havng an overlap, fve pars of vews from each object wth bgger overlap area are selected to test our descrptor (20 vew pars n total). (a) BR dataset (b) UWAOR dataset (c) UWA3M dataset Fg.5. One exemplar model and One correspondng exemplar scene (shown from left to rght) respectvely taken from the BR, UWAOR and UWA3M datasets. (a) A scene of UWAOR- (b) A scene or model of UWA3M-IR IR dataset dataset Fg.6. An llustraton of two scenes for dstngushng boundary and nner regons from UWAOR-IR and UWA3M-IR dataset, respectvely. The boundary ponts are shown n green color. In addton, to present the nfluence of mesh boundary, we generate two new datasets derved from UWAOR and UWA3M, respectvely. For ncreasng readablty, the two new datasets are named UWAOR-IR and UWAOR-IR, respectvely. The new dataset has the same models and scenes wth ther orgnal dataset. The only dfference between them s that the nner and boundary regon of the scenes n UWAOR-IR and the scenes and models n UWA3M-IR need to be dstngushed. The nner regon n a pont cloud s defned as the regon n whch ponts have a dstance larger than the support radus R from the mesh boundary. One exemplar of scene n UWAOR-IR and scene or model UWA3M-IR are shown n Fg.6. The purpose of dstngushng nner and boundary regon s to avod the nfluence of mesh boundary by samplng key ponts only on the nner regon, whch wll be detaled n Sect Evaluaton Crtera We use the precson-recall curve (PRC) to evaluate the performance of feature descrptors, and drectly use angle error to present the repeatablty of the axes n LRF, LRA, LPA and normals (TN, RN). To verfy the repeatablty of proposed LRA and LPA, the angle error between two axes s used as an evaluaton crteron. The angle error of two arbtrary 3D axes v 1 and v 2 can be smply computed as follow. T v1 v2 e arccos( ) v v 1 2 The PRC s a popular method for evaluatng local feature descrptors. The detaled process of generatng PRC can refer to [11, 16, 23]. If the descrptor feature obtans both hgh recall and precson, the PRC would fall n the top left of the plot. In order to compactly and quanttatvely present the performance of feature descrptors, the area under the PRC curve, defned as AUCpr, s also used n ths paper. AUCpr s a smple and aggregated metrc to measure how an algorthm performs over the whole precson-recall space [11]. In the deal case, the AUCpr s equal to 1 snce the recall s always 1 for any precson. In our experments, for the BR and UWAOR datasets, 1000 key ponts are randomly selected from scene, and then ther correspondng key ponts are extracted from the models/model usng the ground truth transformaton whch are gven by the publshers. For the UWA3M dataset, each par of model-scene has an overlappng regon. The overlappng regon s determned based upon the ground truth transformaton whch s obtaned by frst manually algnng the model-scene par and then refnng usng the ICP algorthm [39, 40]. Then, 1000 key ponts are randomly selected from the overlappng regon of the scene, and ther correspondng key ponts are extracted from the model usng the ground truth transformaton. For UWAOR-IR dataset, the process of selectng key ponts s smlar to that of UWAOR except that the 1000 key ponts on scenes are randomly selected from ther nner regon. For UWA3M-IR dataset, the overlappng nner regons on each model-scene par s constructed. The key ponts n UWA3M-IR are selected on the overlappng nner regons by the smlar method of selectng key ponts n UWA3M. After the key ponts are determned, f we only evaluate the repeatablty of LRF/A, LPA and normals, we just need to generate LRF/A, LPA and normals on these key ponts, and use the angle error to evaluate them. If we evaluate the performance of the local feature descrptors, we need to generate the descrptor features on these key ponts. The PRC curve or ts area AUCpr s used for comparng the performance of the descrptors. 4.2 Performance Evaluaton of the Proposed LRA and LPA In ths secton, the repeatablty and effcency of our proposed LRA and LPA are tested. The proposed LRA and LPA are compared wth fve recent technques of constructng LRFs and two popular methods for generatng normal. The fve technques of LRFs nclude Man et al. [20], Tomabar et al. [10], Petrell et al. [1], Guo et al. [16] and Yang et al. [22]. The two methods for constructng normals are respectvely trangular patches based method (TN) [11, 32] and radus neghbors based method (RN) [30, 33], whch have been ntroduced n Sect For convenent expresson, LRF, LRA, LPA and normals are collectvely called local axes n the followng Repeatablty Performance Our proposed LRA and LPA are tested on the three datasets, whch are detaled n Sect , for evaluatng ther repeatablty. The support radus of constructng these local axes are kept the same as 20 mr for a far comparson. Accordng to the mplementaton detals ntroduced n Sect , we can obtan the angle errors of each method tested on the fve expermental datasets (.e. BR, UWAOR, UWA3M, UWAOR-IR and UWA3M- IR). The percentage of the angle errors below 5 o correspondng to each method s counted, as shown n Fg.7. In order to compare the repeatablty of the x and z axs n a LRF, the angle errors of the x and z axs n a LRF are computed respectvely. In the BR dataset, the robustness of our proposed LRA and LPA to nose and varyng mesh resoluton are verfed. For concseness, we select a few of scenes from the scenes of BR datasets for ths experment. The results of the local axes tested on

9 9 the selected scenes of BR dataset are showed n Fg. 7 (a)-(c). Several observatons can be drawn from these fgures. Frst, the z-axs of Guo et al. and our LRA acheve sgnfcantly strong robustness to all levels noses and varyng mesh resolutons. In partcular, the z-axs of Guo et al. acheves the best performance for resstng varyng mesh resolutons, as shown n Fg.7 (b). It may owe to the z-axs proposed by Guo et al. use all the nformaton of local surface rather than only the mesh vertces. Our LRA acheves the best performance n the presence of Gaussan nose and Gaussan nose combned wth varyng mesh resoluton, as shown n Fg.7 (a), (c). Second, for the fve tested LRFs, the performance of the z axes surpasses the correspondng x axes by a large margn. It mples that the repeatablty of a LRA can sgnfcantly outperforms that of a LRF snce the z-axs n a LRF can be as a LRA, whch s the major concern of us to use a LRA rather than a LRF for constructng our descrptor. Thrd, for the results of our LPA and the two normals (.e. TN and RN), the performance of our LPA sgnfcantly outperforms TN and RN n all the cases, and the gap appears to be more obvous along wth the ncrease of the levels of nose and varyng mesh solutons. Therefore, our proposed LPA can be ntegrated n the descrptors (e.g. LFSH, SHOT, etc.), whch nclude the component generated by the normals, for mprovng ther robustness. (a) Gaussan nose (b) Varyng mesh resoluton (c) The combnaton of Gaussan nose and varyng mesh (d) The four datasets resoluton. Fg.7. comparson of the repeatablty to the local axes (ncludng z and x/y axes n the fve LRFs, our LRA, our LPA and two normals). Note that the x and z axs n a LRF are presented n one bn by two colors. The lght color denotes the repeatablty of ts z axs and the correspondng deep color denotes the repeatablty of ts x axs. Note that the legends of all subfgure are the same, and are presented n the left of ths fgure In the UWAOR and UWA3M datasets, some dfferent challenges (e.g. mesh boundary, clutter, occluson) are exstent. Besdes, n order to verfy the nfluence of mesh boundary, the datasets UWAOR-IR and UWA3M-IR are used for testng these local axes. The UWAOR-IR and UWA3M-IR are developed from UWAOR and UWA3M, respectvely, by dentfyng ther mesh boundary regon. The results of testng on the above four datasets are shown n Fg.7 (d). Several observatons can be drawn from ths fgure. Frst, our LPA and the z-axes proposed by Yang et al. [23] and Petrell et al. [1] have a better performance tested on UWAOR and UWA3M datasets, whle they superor performances are not exstent when tested on UWAOR- IR and UWA3M-IR datasets. The thngs n common of the above three axes dstngushed from the other axes s that the drectons of them are calculated by usng a subset of support radus regon rather than whole support radus regon. So, t s obvous that usng a smaller radus to calculate a local axs wll has a strong robustness to mesh boundary. Second, the performance of each method tested on UWAOR-IR and UWA3M-IR correspondngly outperform that tested on UWAOR and UWA3M. It mples that mesh boundary exstng n a pont cloud has a great nfluence to the performance of these local axes. Thrd, for the comparson of LPA, TN and RN, our LPA outperform TN and RN by a large margn on all the four datasets. Besdes, snce TN and RN are generated on a small local regon, TN and RN have a stronger robustness to resst occluson, clutter and mesh boundary than to resst nose and varyng mesh resoluton. Fourth, Our LRA acheves a good performance on UWAOR-IR and UWA3M-IR datasets whle an nferor performance on UWAOR and UWA3M datasets. It shows that our LRA s vulnerable to the mesh boundary. In concluson, our proposed LRA has a strong robustness to nose and varyng mesh resolutons, whle has a weak robustness to mesh boundary. So, our LRA s approprate to be appled on the dataset wth the boundary beng dentfed or no boundary exstent. In the presence of the boundary exstent n a pont cloud, the z axs of some state-of-the-art LRFs (such as Yang et. al [23]) wth robust to mesh boundary can be used as the LRA n our descrptor. Our proposed LPA has a strong robustness to varous nusances (e.g. nose, varyng mesh resoluton, clutter, occluson, et al.), and sgnfcantly outperform the two normals (.e. TN, RN).

10 Tme Effcency In ths secton, the effcency of the methods for generatng these local axes (ncludng the sx LRFs, our LRA, our LPA, TN and RN) s tested. Snce the effcency s manly correlate to the number of ponts n local regon, we only use the BR dataset to test these local axes. In partcular, we frst randomly select 1000 ponts n each model on the BR dataset (6000 ponts n total). For the test of the sx LRFs and our LRA, the total tme costs of each LRF or our LRA generated on these ponts wth respect to dfferent support radus R are counted. Smlar to [23], R also ncreases from 5 mr to 40 mr wth an nterval of 5 mr n ths paper. For the test of TN, RN and our LPA, snce they denote the local geometrcal attrbute of a surface, we count the total tme costs of RN and our LPA on the selected ponts wth a fx support radus, and the total tme cost of PN on the selected ponts based on the trangular patches attached wth these ponts. The fxed support radus of RN and our LPA are set to 3mr and 7mr, respectvely. Fg.8. Tme effcency of the fve LRF and our LRA. The results of the LRFs and our LRA are shown n Fg.8. It can be observed that the computatonal tme of our LRA and the LRFs proposed by Man et al. [20] and Tombar et al. [10] are smlar, and they acheve the best performance n terms of computatonal effcency compared to others. The LRF proposed by Guo et al. [16] s sgnfcantly nferor to the others n terms of computatonal effcency, whch s because that the LRF s generated based on a local surface rather than the vertces of the local surface. The LRFs proposed by Yang et al. [23] and Petrel et al. [1] acheve a medum performance n terms of tme effcency. For the comparson of TN, RN and our LPA, the computatonal tmes of TN, RN and our LPA are s, s and s, respectvely. We can see that the computatonal effcency of TN outperforms RN and LPA, whle the computatonal effcency of TN and our LPA are comparable. It s manly because that RN and our LPA need to calculate a covarance matrx whle TN do not need to. Although our LAP has the worst performance n terms of computatonal effcency, the gaps of the computatonal tme among them are not obvous. Besdes, our LAP s drectly generated on a pont cloud, whle constructng the TN need a pont cloud wth the nformaton of trangular mesh. 4.3 Performance Evaluaton of the SDASS In ths secton, the proposed SDASS descrptor s tested on the fve expermental datasets (.e. BR, UWAOR, UWA3M, UWAOR-IR and UWA3M-IR) usng the PRC curve and AUCpr (ntroduced n Sect ), and compared to sx state-of-the-art descrptors for a thorough evaluaton. The sx descrptors nclude the Spn Image [14], SHOT [10], RoPS [16], TrSI [29], LFSH [3] and TOLDI [23]. Note that Spn Image s the most cted descrptor n the area of local shape descrpton, and LFSH s the most smlar to ours. The SHOT, RoPS and TrSI are verfed wth an advanced performance [11, 23]. The TOLDI s relatvely new proposed descrptor. The parameter settngs of the seven feature descrptors are shown n TABLE 1. All these parameter settngs, unless otherwse specfed, are used for all the experments n ths paper. To verfy the performance of the proposed LPA, we replace the normal used n SI, SHOT and LFSH wth our LPA, and then generate three modfed descrptors: SI combned wth the proposed LPA (SI-LPA), SHOT combned wth the proposed LPA (SHOT-LPA), and LFSH combned wth the proposed LPA (LFSH-LPA). Note that the parameter settngs of SI-LPA, SHOT-LPA and LFSH-LPA are the same wth ther orgnal descrptors. In addton, for satsfyng some tme-crucal applcatons (e.g. robotcs, moble phones, etc.), the comparson of these descrptors n terms of computng effcency s also tested. TABLE 1 Parameter Settngs for Sx Descrptors, where mr Denotes Mesh Resoluton. Support radus (mr) Dmensonalty Length SI SHOT RoPS TrSI LFSH TOLDI SDASS Performance on the BR Dataset The purpose of testng on the BR dataset s manly to verfy the robustness to nose and varyng mesh resolutons. To compactly present the performance, we only use the AUCpr as an evaluaton crteron n ths secton. For each descrptor, we follow the steps ntroduced n Sect to generate AUCpr on the BR dataset. The results of all descrptors are presented n Fg.9. In the vew of robustness to nose (Fg.9 (a)), we gve a number of observatons. Frst, the proposed SDASS descrptor outperforms the others n terms of all levels of nose by a large margn and the advantage of our descrptor s more obvous wth the ncrease of nose levels. Second, the descrptor SHOT-LPA acheves the second best performance for resstng nose, and SI and LFSH have the worst performance for resstng nose. Thrd, as the nose levels ncreased, the performance of SI, LFSH and SHOT deterorated sharply. The descrptors SI-LPA, LFSH- LAP and SHOT-LPA correspondngly outperform ther orgnal descrptors SI, LFSH and SHOT by a large margn. The man reason for ths phenomenon s that the robustness of our LAP to nose sgnfcantly outperforms that of the normals (TN and RN) (as verfed n Sect.4.2.1). In terms of robustness to varyng mesh resolutons, as shown n Fg.9 (b), several observatons can be made from the results. Frst, our proposed SDASS descrptor outperforms all the other descrptors under all levels of mesh decmaton, and the gap s broadened wth the ncrease of mesh decmaton. Second, n the range of decmaton rate from 10/10 to 4/10, the performance of TrSI descrptor s close to our SDASS descrptor. As the levels of decmaton rate surpass 4/10, the performance of TrSI deterorates sharply and the gap between our SDASS and TrSI s more obvous. Smlar to the TrSI, the performance of RoPS and TOLDI also have a sgnfcant drop n the hgh levels of

11 11 decmaton rate. The common trat of these three descrptors s of usng a vew-based mechansm. Thrd, smlar to the performance of resstng nose, the descrptors SI-LPA, LFSH-LAP and SHOT-LPA also outperform ther orgnal descrptors SI, LFSH and SHOT n terms of resstng varyng mesh resoluton. For the robustness to the mpact of combnng nose and varyng mesh resolutons, as shown n Fg.9 (c), our SDASS descrptor also sgnfcantly outperform the other methods n all levels of combned nose and mesh decmaton (except the hghest level). In the hghest level, all descrptors are falng to work, as shown n Fg.9 (c). The overall performance of SHOT-LAP s the second best. The TrSI has a better performance n the low levels of combned nose and mesh decmaton, and the superorty s no longer exstent n the hgh levels. The performances of SHOT-LAP, TOLDI, LFSH-LAP, RoPS, SHOT and TrSI are comparable aganst the performances of SDASS, SI and LFSH descrptors. The performances of SI and LFSH are sgnfcantly nferor to others. Ther fal to work even under the low level of combned nose and mesh decmaton. It s because that the SI and LFSH descrptors use the normal, whch has a weak robustness to nose and varyng mesh resolutons, as ther LRA. (c) The combnaton of Gaussan nose and varyng (a) Gaussan nose. (b) Varyng mesh resolutons. mesh resoluton. Fg.9. AUCpr performance evaluaton of the ten feature descrptors tested on the BR dataset. Note that the legends of the subfgure (a) and (b) are same wth that of the subfgure (c). The x labels n the subfgure (c) present the Guassan noses n the subfgure (a) correspondngly combnd wth the varyng mesh resolutons n the subfgure (b). (a) UWAOR dataset (b) UWAOR-IR dataset (c) UWA3M dataset (d) UWA3M-IR dataset Fg.10. PRC performance evaluaton of the eleven feature descrptors on the four datasets (UWAOR, UWAOR-IR, UWA3M, UWA3M-IR) Performance on UWAOR, UWAOR-IR, UWA3M and UWA3M-IR Datasets In ths secton, all descrptors are tested on UWAOR, UWAOR- IR, UWA3M and UWA3M-IR datasets for verfyng ther robustness to clutter, occluson and mesh boundary. The propose of usng UWAOR-IR and UWA3M-IR datasets s manly to present the nfluence of mesh boundary to the descrptors. The UWAOR-IR and UWA3M-IR datasets are ntroduced n Sect.

12 In addton, n consderaton of the LRF proposed by Yang et.al [23] has the best performance to resst occluson, clutter and mesh boundary whch are exstent n UWAOR and UWA3M datasets (see Sect ), we use the z-axs n ths LRF to replace our LRA for generatng the SDASS descrptor, and call ths new ntegrated descrptor SDASS-Yang n the followng. The performances of all descrptors tested on UWAOR, UWAOR-IR, UWA3M and UWA3M-IR datasets are shown n Fg.10. Several observatons can be drawn from these results. Frst, the descrptors SDASS-Yang and SDASS acheve the best performance on all four datasets, and they outperform others by a large margn. In partcular, SDASS-Yang slghtly outperform SDASS on UWAOR and UWA3M datasets, and they have a smlar performance on UWAOR-IR and UWA3M-IR datasets. It s manly because that the performance of z-axs n the LRF proposed by Yng et.al [23] outperform our LRA when tested on UWAOR and UWA3M, and they have a smlar performance when tested on UWAOR-IR and UWA3M-IR. Second, the performances of all descrptors are unversally mproved from tested on UWAOR and UWA3M to UWAOR-IR and UWA3M- IR. It mples that mesh boundary has a sgnfcant mpact to the performance of the descrptors. Thrd, smlar to the results on BR dataset, the performances of SI-LPA, SHOT-LPA and LFSH- LPA are sgnfcantly mproved by combnng our LPA wth ther orgnal descrptors SI, SHOT and LFSH. It mples that the proposed LPA has a hgher robustness to clutter, occluson and mesh boundary compared to the normals (.e. TN, RN) Tme Effcency In ths secton, the effcences of these descrptors (ncludng SI, SHOT, RoPS, TrSI, LFSH, TOLDI and SDASS) are tested. Snce the effcency s manly correlate to the number of ponts n local regon, we just use the BR dataset to test these descrptors. Frst, we randomly select 1000 ponts n each model of BR dataset (6000 ponts n total). Second, the total tme costs of each descrptor mplemented on these ponts wth respect to dfferent support radus R are counted. Here, the values of R are the same wth the set n Sect The tested results of these descrptors are shown n Fg.11. Fg.11. The computatonal effcency of the seven feature descrptors wth respect to varyng support rad. The Y axs s shown logarthmcally for clarty. For the results shown n Fg.11, the effcences of LFSH, SI, SDASS, TOLDI and SHOT are comparable. Specfcally, LFSH and SI rank the frst compared to all the other descrptors, and SDASS and TOLDI acheve the secondary calculaton effcency, and SHOT has the worst performance among these fve descrptors. TrSI and RoPS have a smlar computatonal performance, and ther computatonal performance s nferor than the others by one order magntude. In concluson, our SDASS acheve a better computatonal performance although t s slghtly nferor to the computatonal performance of LFSH and SI. It s worth notng that RoPS and TrSI descrptors are the two most tme-consumng, and ther man tme consumptons are produced by the tme-consumng LRF constructon process (as verfed n Sect.4.2.1). 5 APPLICATIONS To further verfy the effectveness of the proposed SDASS descrptor, 3D regstraton s performed based on the SDASS descrptor. Consderng that each local feature descrptor can be applcable to a partcular ppelne of 3D regstraton (e.g. the LRF based descrptors are approprate to use 1P-RANSAC [41] for searchng consstency, whle the LRA based descrptors are approprate to use 2SAC-GC [3]), we next only ntroduce the applcaton of our SDASS descrptor, and the comparson wth other descrptors sn t nvolved. The UWA3M s a popular dataset used for 3D regstraton [2]. Here, we also smply use the UWA3M dataset for verfyng the applcaton of our SDASS descrptor n 3D regstraton. We frst use the SDASS descrptor to preform parwse regstraton. Transformaton estmaton s an mportant process n parwse regstraton, and there are many methods (ncludng RANSAC [42], 1P-RANSAC [41], 2SAC-GC [31]) for estmatng transformaton. Consderng that our SDASS descrptor s generated on LRA, the 2SAC-GC method, whch use two feature ponts and correspondng LRAs for estmatng transformaton, s approprate for our descrptor to perform parwse regstraton. For specfc procedure of performng par regstraton, the scene and model nvolvng n a parwse regstraton are frst unformly sampled to reduce computatonal burden. Then, the key ponts on scene and model are extracted also by unform samplng. In general, the key ponts extracted on scene s dense than that extracted on model for guaranteeng the key ponts on model havng correspondng key ponts on scene. Next, the SDASS descrptor s generated on these key ponts. For each SDASS feature on model, ts closest SDASS feature s searched on scene. The correspondng SDASS features between model and scene are sorted based on the ascendng order of ther Eucldean dstance. The frst n correspondences are selected to estmate the transformaton usng the 2SAC-GC method (n s set to 50 here). An llustraton of parwse regstraton wth respect to each object n the UWA3M dataset s presented n Fg.12. In the fgure, the percentage of correct correspondences (PCC) [4] n the 50 selected correspondences are counted. We can observe that our SDASS acheve a very hgh PCC. Based on the parwse regstraton ntroduced above, we further present the applcaton of our SDASS descrptor on multvew regstraton. We use the method ntroduced n [2] for performng mult-vew regstraton. The process of mult-vew regstraton manly dvded nto three steps: Frst, feature descrptor s generated on the key ponts of all partal vews; Second, shape s updated by usng parwse regstraton for gradually addng new partal vew; Thrd, fne regstraton algorthm [43] and surface reconstructon algorthm [36] are appled to construct a contnuous and seamless 3D model. The results of mult-regstraton for the four objects n UWA3M dataset are presented n Fg.13. It can be seen that our SDASS ntegrated nto the multvew regstraton method proposed n [2] acheve a better performance.

13 13 a) Chef, PCC=98% (b) Chcken, PCC=96% (c) Parasaurolophus, PCC=88% (d) T-rex, PCC=84% Fg.12. Four cases of parwse regstraton wth respect to each object n the UWA3M dataset. The PCC denote the percentage of correct correspondences n the selected 50 correspondences. Fg.13. A result of mult-vews regstraton for the four objects (Chef, Chcken, Parasaurolophus, T-rex) n the UWA3M dataset. 6 CONCLUSIONS AND FUTURE WORK In ths paper, we proposed a novel SDASS feature descrptor for 3D local surface descrpton. The promnent advantage of our SDASS descrptor s ts hgh descrptveness and strong robustness. Moreover, our SDASS has a superor performance n terms of compactness and effcency. For generatng our SDASS descrptor, we also proposed an mproved LRA and a robust LPA. Our LRA s developed from the LRF proposed by Yang et.al. [22]. Our LRA acheved a hgh repeatablty n the presence of nose and varyng mesh resolutons compared to exstng technques. Our proposed LPA acheved a good performance for resstng varous nusances (e.g. nose, varyng mesh resolutons) compared to the two methods of constructng normals (TN and RN). Based on the LPA, the devaton angles between LRA and LPA for encodng the geometrcal nformaton of local surface presented a hgh descrptveness and strong robustness. Our SDASS descrptor encode the combnaton of spatal and geometrcal nformaton of a local surface. The spatal nformaton s fully encoded by two features (heght and projected radal nformaton) on LRA. The geometrcal nformaton s encoded by usng the devaton angle between LPA and LRA. Owng to the strong robustness and hgh descrptveness of LPA, the geometrcal nformaton of local surface encoded n our SDASS descrptor presented a superor performance. We performed a set of experments to assess our SDASS descrptor wth respect to a set of dfferent nusances ncludng nose, varyng mesh resoluton and mesh boundary, etc. The expermental results showed that our SDASS descrptor outperforms the state-of-the-art methods by a large margn, and obtans hgh descrptveness and strong robustness to resstng nose, varyng mesh resoluton and other varatons. At last, our SDASS s appled n 3D regstraton presented a good performance of ts applcaton ablty. There are several nterestng research drectons for further research. Snce local feature descrpton s a fundamental task n 3D computer vson, further explorng the effcent method of constructng superor descrptor s valuable. Consderng that the texture of 3D objects s easy obtaned by scanner devces (e.g. the Mcrosoft Knect devce, stereo sensors, etc.), ntegratng RGB nformaton to the SDASS descrptor would be benefcal when the 3D models exhbt poor geometrc features but rch photometrc cues. In addton, ntegratng our SDASS descrptor to specfc applcatons (e.g. 3D object recognton and surface regstraton) also need to be further researched n the future. ACKNOWLEDGMENT The authors would lke to acknowledge the Stanford 3D Scannng Repostory, the Unversty of Western Australa (UWA) for makng ther datasets avalable to us. We also thank the help of Dr. Guo, Dr. Yang and Dr. Tombar for provdng ther codes to us. Ths work was partly supported by the Natonal Natural Scence Foundaton of Chna (Project No ), and the Shangha Muncpal Scence and Technology project (Project No ). REFERENCES [1] A. Petrell and L. D Stefano, "Parwse Regstraton by Local Orentaton Cues," Computer Graphcs Forum, vol. 35, pp , [2] Y. Guo, F. Sohel, M. Bennamoun, J. Wan, and M. Lu, "An Accurate and Robust Range Image Regstraton Algorthm for 3D Object Modelng," IEEE Transactons on Multmeda, vol. 16, pp , [3] J. Yang, Z. Cao and Q. 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