Robust Surface Reconstruction from Orthogonal Slices

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Robust Surface Reconstructon from Orthogonal Slces Radek Svták 1, Václav Skala Department of Computer Scence and Engneerng, Unversty of West ohema n Plsen, Unverztní 8, 306 14 Plzeň, Czech Republc

The fnal result of surface reconstructon s always based on the correct estmaton of the structure of the orgnal obect. Ths paper s a case study of the problem of the structure determnaton.

1 Robust Surface Reconstructon from Orthogonal Slces Radek Svták 1, Václav Skala Department of Computer Scence and Engneerng, Unversty of West ohema n Plsen, Unverztní 8, Plzeň, Czech Republc E-mal: rsvtak@kv.zcu.cz bstract The surface reconstructon problem from sets of planar parallel slces representng cross sectons through 3D obects s presented. The fnal result of surface reconstructon s always based on the correct estmaton of the structure of the orgnal obect. Ths paper s a case study of the problem of the structure determnaton. We present a new approach, whch s based on consderng mutually orthogonal sets of slces. new method for surface reconstructon from orthogonal slces s descrbed and the beneft of orthogonal slces s dscussed too. The propertes and sample results are presented as well. 1. Introducton The crucal task of the surface reconstructon from slces s a correct estmaton of the orgnal obect structure,.e. the soluton of the contour correspondence problem. Most of the exstng methods smply consder the overlap of contours n a par of consecutve parallel slces as the only correspondence crteron. Therefore, they produce unacceptable structure estmaton when the angle between the axs of the obect and the normal of the slces ncreases. Hgher densty of slces can help to solve ths problem, but t s not always possble because of the resoluton lmt of the scannng devce, etc. It s obvous that other slces n non-parallel planes offer an addtonal nformaton. In ths paper we wll concentrate on the beneft of orthogonal slces for the reconstructon process. In comparson to the exstng methods, our currently acheved results show, that for a set of obects the resultant surface s sgnfcantly more accurate wth respect to the smlarty to the orgnal surface. The concept of the new proposed method s presented and results of comparsons wth the exstng methods are dscussed as well. D Fgure 1: Problematc cases when solvng the contour correspondence problem. Expected problems usng the overlappng crteron:,, C, D; generalzed cylnders:, D; MST: C, D; Reeb graph based methods: D. C Ths work s was supported by the Mnstry of Educaton of the Czech Republc proects: 1 FRVŠ 1348/004/G1 MSM

2 . ref survey of exstng methods Several methods for surface reconstructon from slces have been developed snce about In ths secton we wll classfy them accordng to ther approach to solvng the contour correspondence problem. For more extensve study of the exstng methods from the other vewponts, see [, 4, 6, 7]. The smplest methods estmate the contour correspondence locally between each consecutve par of contours. Typcally, contours that overlap each other are consdered as correspondent. Ths works f the densty of slce s hgh,.e. the dstance between slces s low, and the axs of the nput obect s nearly perpendcular to the slces planes. more advanced method uses generalzed ellptcal cylnder to solve the correspondence problem [1, 11, 1]. Contours are frst classfed as ellptcal or complex by determnng how well the vertces of ther permeter can be ft by an ellpse. If the ft s too poor, a contour s classfed as complex, and can not be ncorporated nto an ellptcal cylnder. Then the ellpses are grouped to the cylnders. When as many contours as possble have been organzed nto cylnders, then the algorthm uses the geometrc relatonshp between cylnders to group them nto obects. Ths method s most useful for elongated smooth obects wth roughly ellptcal cross secton. pparently the best exstng approaches that have been publshed are two graph-based methods. The frst of them presented by Sknner [10] computes a mnmum spannng tree based on contour shape and poston. In the frst step a graph s constructed by representng each contour as a node and connectng each node to all nodes representng contours n adacent sectons. The best fttng ellpse s computed for each contour. The cost of an edge of the graph reles on the mutual poston and sze of two ellpses: c (,) (x x ) + (y y ) + (a a ) + (b b ) =, where (x, y, z ), (x, y, z ) represent the centers of the ellpses of contours,, respectvely, and a, b, a, b are ther maor and mnor axs lengths. The mnmum spannng tree computed for the graph represents the soluton to the correspondence problem. The method works well for naturally tree-structured obects, the man lmtaton s ts nablty to solve the correspondence problem correctly for general graph topologes, e.g. genus > 0. Fgure : ) Data set of slces of the cochlea. Usng the Reeb graph t s possble to detect and represent the rght contour correspondence. The advantage conssts n the possblty of consderng the correspondence among contours of one slce (). Taken from [8]. The second graph based method presented by Shnagawa [8, 9] uses surface codng based on Morse Theory to construct a Reeb graph [14] representng the contour connectvty. Each contour represents a node n the graph, edges of the graph represent the contour correspondence relaton. Edges are added to the graph n the manner to avod makng connectons that would result n a surface that s not a -manfold. For each par of contours that can be legally connected, a weght functon s evaluated, and ts value s used to establsh a prorty for connectng that par of contours. The algorthm proceeds by makng the hghest prorty connectons n regons where the number of contours n each secton does not change, and then adds connectons n order of decreasng prorty wth respect to the a pror knowledge of the number of connected components and the topologcal genus.

It s necessary to note that all the exstng solutons ust estmate the contours correspondence,.e. the structure of the orgnal obect, should be emphaszed.

We dstngush two knds of cells of M, the surface-crossng and the surfacepassng cells.

Orthogonal slces One set of parallel planar slces s one of the well-known boundary representatons of a 3D obect.

3 It s necessary to note that all the exstng solutons ust estmate the contours correspondence,.e. the structure of the orgnal obect, should be emphaszed. In Fgure 1 there are some typcal example data sets to llustrate capabltes of the approaches mentoned n ths secton. of grd M. We dstngush two knds of cells of M, the surface-crossng and the surfacepassng cells. There are parts of contours on some sdes of a surface-crossng cell, whch means that the resultant surface ntersects the cell, see Fgure Orthogonal slces One set of parallel planar slces s one of the well-known boundary representatons of a 3D obect. Usually the planes of such slces set are perpendcular to the z-axs, and thus called z- slces. If we slce an obect by more then one set of parallel slces and moreover when these sets are mutually orthogonal, we get orthogonal sets of slces. Consder now that we have z-slces, x-slces and y-slces of an obect, see Fgure 3. Note, that the contours are supposed to be polygonal, orented the way that when lookng from the postve drecton of the gven slces set axs, the contours have the nteror on ts left sde and the exteror on the rght sde, see Fgure 4. Fgure 3: n example of three orthogonal slces sets. Fgure 4: Correct contour orentaton. 3.1 Contour correspondence The man advantage of orthogonal slces conssts n the approach how the contour correspondence can be determned. It s mportant to emphasze that two orthogonal contours whch ntersect each other comes aparently from one and the same surface component of the nput obect. It means that the ntersecton of contours s very mportant snce t provdes accurate nformaton about the correct structure, see Fgure 5. It s obvous that f the slces n the orthogonal sets sample the obect suffcently, then the ntersectons of contours from the orthogonal slces dentfy the correspondence relaton accurately,.e. the correct structure of the orgnal obect. Fgure 5: The mutual crossngs of orthogonal contours defne the correspondence relaton. 4. The algorthm The planes of slces dvde space nto a set of spatal cells of a spatal grd M. In Fgure 3 can be seen three mutually orthogonal planes Fgure 6: surface-crossng cell. Parts of contours on the sdes of the cell together wth node ponts form spatal polygons. Node ponts are denoted as whte crcles. Each edge of G s adacent wth two cells of M.

The ntersecton of two orthogonal slces consstng of curvlnear contours s a set of ponts and we call them node ponts, see Fgure 7. Now we focus on surface-crossng cell.

4 The ntersecton of two orthogonal slces consstng of curvlnear contours s a set of ponts and we call them node ponts, see Fgure 7. Now we focus on surface-crossng cell. n mportant observaton s that parts of nput contours and the node ponts form spatal polygons. Each such polygon s enclosed n a surface-crossng cell, ts patch s part of the resultant surface, see Fgure 6. representaton of a contour correspondence relaton. It holds that each node pont must le on the edge of the grd M. Snce the contours are supposed as polygonal curves, we cannot compute the ntersectons of two orthogonal sectons drectly. We obtan them n two phases. 4.1 The correspondence problem t ths moment we suppose that the correspondence of contours s dentfed suffcently by the ntersectons of orthogonal contours as t has been dscussed n secton 3.1. Consder the ntersecton of two contours as the relaton of correspondence. Note that the number of components of a graph constructed of such a relaton corresponds to the number of dsont components of the resultant surface. 4. Node ponts computaton node pont s geometrcally the ntersecton of two contours. Topologcally t s the Fgure 7: ) n nput contour, the lattce represents postons of orthogonal slces planes. ) The contour formed by ts node ponts (black spots). In the frst step ntersectons of each contour and the grd M are computed. These ntersectons are added among the current contour vertces on the approprate poston. They are regstered on the correspondng edge of the grd M smultaneously. Our algorthm works on the same prncple as the Cohen- Sutherland s lne clppng algorthm [3]. n ntersecton of a slce plane and all other orthogonal slce planes forms a lattce frequency of occurence Fgure 8: Polygon sze (number of edges) hstogram.

wth cells, see Fgure 7. Each node pont arses as the ntersecton of a contour and a sde of a cell.

In the second step the node pont constructon s completed. The correspondent vertex, whch s a member of the orthogonal contour and also a member of the same edge of M, must be found.

Each two nearest ntersectons comng from orthogonal contours regstered on an edge of M are qualfed as correspondent vertces buldng together a node pont. polygon.

Requres the central pont C determnaton. 4.3.

Note that the geometrcal shape of the edges stll corresponds to the approprate parts of contours.

5 wth cells, see Fgure 7. Each node pont arses as the ntersecton of a contour and a sde of a cell. Snce the contour s supposed to be polygonal, a node pont s smply computed as an ntersecton of two segments. Sngular cases when a contour crosses a cell at ts corner are handled separately [13]. In the second step the node pont constructon s completed. The correspondent vertex, whch s a member of the orthogonal contour and also a member of the same edge of M, must be found. s t was sad before t s done very fast searchng the auxlary regstratons of contour ntersectons on the approprate edge of M. Each two nearest ntersectons comng from orthogonal contours regstered on an edge of M are qualfed as correspondent vertces buldng together a node pont. polygon. Then usng the center pont each polygon s dvded nto set of quadrlaterals, whch are easer to patch, see Fgure 9. Fgure 9: Partton of a generc polygon n the set of quadrlaterals. Requres the central pont C determnaton Constructng the surface Now suppose graph G, whose set of vertces conssts of a set of the node ponts and whose edges represent the parts of contours between two node vertces. Note that the geometrcal shape of the edges stll corresponds to the approprate parts of contours. Now the task s to fnd such cycles of graph G, whch have the property that ther geometrcal representaton les wthn one cell of M. Those cycles represents spatal polygons that le on the surface. We suppose each edge e of G s adacent wth cells 1 and, see Fgure 6. Each cell from { 1, } ncludes one cycle c of our nterest, whch s adacent wth e (that results from the consderaton of -manfold obects). The crcle c represents the spatal polygon beng searched. Thus for each e two e e cycles c 1, c must be searched and then polygons p c, p 1 c correspondent to those cycles are constructed. s soon as all polygons are obtaned, we can start to patch them. We can use any arbtrary patchng technque. Note that the number of sdes of such polygon can be hgh, but n cases of our data sets t s n range 10, see the graph n Fgure 8. The proposed method starts wth fndng a sutable pont n the center of each C D Fgure 10: Results of the surface reconstructon. ) n nput data set (courtesy of Martn Čermák), ) VTK surface reconstructon from slces class, C) common volume based method, D) Proposed method for surface reconstructon from orthogonal slces.

6 5. Results ll the problematc data sets mentoned n secton 1 and many more have been processed usng: - surface reconstructng from slces class from VTK, - a common volume based method; see [5] for more detals, - our proposed method for surface reconstructon from orthogonal slces. The results of the reconstructon of one data set are llustrated n Fgure 10, the complete documentaton and expermental results can be found at 6. Concluson and further research Our current research proves that the advantages of orthogonal slces n the process of surface reconstructon are sgnfcant. There s a set of obects for whch the orthogonal slces are almost the only way to reconstruct them correctly. The proposed method supposes that the obect s sampled well enough, so that the number of components of the correspondence graph G equals to the number of dsont components of the orgnal surface. The man pont of our further research s the soluton of problems caused by undersamplng,.e. to deal wth data sets that do not sample the nput obect suffcently. Furthermore we would lke to study the nfluence of contour naccuracy on the node pont computaton. References [1] resler, Y., Fessler, J.., Macovsk,.: ayesan approach to reconstructon from ncomplete proectons of a multple obect 3D doman. IEEE Trans. Pat. nal. Mach. Intell., 11(8): , ugust [] Cong, G., Parvn,.: Robust and effcent surface reconstructon from contours. The Vsual Computer, (17):199-08, 001 [3] Foley, J. D., van Dam,., Fener, S. K. and Hughes, J. F., Computer Graphcs: Prncples and Practce, ddson-wesley, [4] Jones, M., Chen, M.: new approach to the constructon of surfaces from contour data. Computer Graphcs Forum (13): 75-84, 1994 [5] Klen, R., Schllng,.: Fast Dstance Interpolaton for Reconstructon of Surfaces from Contours. In proceedngs of Eurographcs '99, Short Papers and Demos, September [6] Meyers, D.: Multresoluton tlng. In Proceedngs, Graphcs Interface '94, pages 5-3, anff, lberta, May [7] Meyers, D.: Reconstructon of Surfaces From Planar Contours. PhD thess, Unversty of Washngton, [8] Shnagawa, Y., Kun, T.L.: Constructng a Reeb graph automatcally from cross sectons. IEEE Comuter Graphcs and pplcatons, 11(6): 44-51, November [9] Shnagawa, Y., Kun, T.L., Kergosen, Y.L.: Surface codng based on Morse theory. IEEE Comuter Graphcs and pplcatons, 11(5): 66-78, September [10] Sknner, S.M.: The correspondence problem: Reconstructon of obects from contours n parallel sectons. Master s thess, Department of Computer Scence and Engneerng, Unversty of Washngton, [11] Soroka,.I.: Understandng Obects From Slces: Extractng Generalsed Cylnder Descrptons From Seral Sectons. PhD thess, Unversty of Kansas Dept of Computer Scence, March TR [1] Soroka,.I.: Generalzed cones from seral sectons. Computer Graphcs and Image Processng, (15): , [13] Svtak, R., Skala, V.: Surface Reconstructon from Orthogonal Slces, ICCVG 00, Zakopane, Poland, 00 [14] Wood, Z. J.: Computatonal Topology lgorthms For Dscrete -Manfolds. Calforna Insttute of Techology, PhD Thess, May 003

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