2.5D View models of nonconvex polyhedron on view sphere with perspective M. Frydler , W.S. Mokrzycki Institute of Mathematical Machines, 02-798, Warsaw, Krzywickiego 34,
[email protected] Institute of Computer Science PAS, 01-237 Warsaw, Ordona 21,
[email protected] Abstract: This article concerns generating 2.5 D models of nonconvex polyhedral that are a complete representation of this polyhedral, according to viewing sphere with perspective concept. Those models are going to be used for visual identification based on them and a scene depth map. We give a new conception and an algorithm for facedepended generation of multi-face views. It does not require any preprocessing or auxiliary mechanisms nor complex calculations connected with them. Key words: Object visual identification, depth map, 2,5 D precise models, viewing sphere with perspective, viewing points, models completion state of viewing representation.
1 Introduction Method of generating 2.5 D viewing representation of nonconvex polyhedral for object visual identification described in [10] is based on the following idea: centrally generate views relative object features chosen for identification, calculate one-view areas on viewing sphere which correspond with earlier generated views, check if whole viewing sphere is covered with one-view areas. If this cover is complete, generation of viewing representation is finished. If not generate additional views corresponding with uncovered areas of viewing sphere and again check if this cover is complete. Continue until complete viewing sphere cover is generated. Complete viewing sphere cover with one-view areas means that generated representation is complete. Methods from [16] and [18] are better. To achieve complete representation we don’t need to work in a loop. Complete representation is obtained by strict covering viewing sphere by one-view areas and controlling “edge” register (of no covered area). When register is set to “empty” generation of model viewing representation is done. Generated representation is complete which follows from the generation method. However to achieve complete representation we have to calculate one-view areas on viewing sphere and operate them in a given order. Without their help it is not possible to get a complete set of views of virtual polyhedron model. On the top of that described methods are for convex polyhedral only. In this article we present a method for generating a complete viewing representation of nonconvex polyhedral. It’s more calculation efficient then described above.
2 Research assumptions This research focuses on developing of a method and of an algorithm for generation of multi-view, nonconvex polyhedral representation. For representation generation we use viewing sphere with perspective concept. For this following conditions have to be met: 1. 2. 3.
Models are accurate - every model is equivalent to b-rep model. 2.5 D models are used – this model has enough information to identify 3D object Models are viewing models – it is possible to identify object from any view
Use of a viewing sphere (Fig. 1) with perspective as a projection space allows simple view standardization. Uses: Recognition of objects not bigger then a few meters and distant (from the system) not more then 10 – 20 meters. Mentioned above uses allow to make certain assumptions about recognition system strategies. We assume following steps of recognition processes 1. 2. 3. 4. 5. 6. 7.
Determining recognizable object types. Definition of identification task(choose an object’s shell feature used for object identification) Generation of viewing models for each object system should identify Creation of database containing all views of all models Acquisition of scene space data and visual data Isolation of scene elements and their transformation to model structures stored in the database Identification of objects by comparing them with database models.
3 View generation space – viewing sphere with perspective Let object be a non convex, non transparent polyhedral without holes or pits. Let’s consider its faces si as features areas, those areas will be used as a foundation for accurate viewing model determining. This model is a set of 2.5D accurate views, acquired through perspective projection from viewing sphere, according to the model from [10] (Fig 1). This model is best for 3D scene data acquisition and gives identification system reliability.
Fig. 1 Concept of view sphere and “one view” areas.
The concept from [10] of generating 2.5 D view representation based on assumed generation space model is as follows: - Create a viewing sphere. Each object has its own sphere, for all views of this particular object (fig 1). - Circumscribe a sphere on a polyhedron. Sphere is small (radius r) and its center is at the polyhedron center. - On this sphere place a space cone with angle of flare 2α. This is the viewing cone. Vertex of this cone is a model viewing point PW. Distance between polyhedron’s center and model viewing point is R. Viewing axis always goes through the sphere’s center. - Unconstrained movement of cone vertex, where cone is tangent to the small sphere creates large sphere with radius R. This sphere is called viewing sphere. - Dependencies between values α , r and R and polyhedron vertices coordinates (Xvi,Yvi,Zvi) are:
r = max i =1,. ,k R=
xvi 2 + yvi 2 + zvi 2 r sin α
- Generate views, taking into account only object features selected for identification i.e. faces. Faces visible in the viewing cone create a view, external edges from this view create view’s contour. - Calculate one-view areas. Those areas correspond with particular views. Complete set of views for a given polyhedron is obtained by covering of viewing sphere with one-view areas. Views are generated in such a way that corresponding one-view areas completely cover viewing sphere. Algorithm makes this approach complete.
Fig. 2 Covering viewing sphere with “one-view” areas. Changing one view to the other is a visual event. This event occurs as a result of point PW movement. This event is manifested by appearance of a new feature in a view, disappearance of a feature or both.
4 Method’s basic concepts and assumptions Consider nontransparent, nonconvex polyhedron. Its faces si are identification features. As a viewing space take a viewing sphere with perspective fig 1. View generation idea requires introduction of polyhedral natural representation concept. 4.1 Polyhedral natural representation Let’s consider a polyhedron as a set of pyramids with top vertices in polyhedron geometrical center and with polyhedron faces as bases. Pyramid top vertex is described by all polyhedron vertices and its base by a vertex
sequence of a corresponding face. This natural representation may be used for convex and nonconvex polyhedral alike. In described here idea of generation of polyhedral view representation this natural representation leads to acquiring of a complete set of views. This set I acquired through “views from above a face” mechanism. If we are able to get all polyhedron views from above all of its faces then we are able to create a complete set of views by adding together views from above all faces. 4.2 Basic concepts and definitions Viewing face SW is one of the polyhedron faces. During view generation each polyhedron face is a viewing face. Over viewing face we take and move current viewing point PW to generate all possible views from above this face View is created by faces that are visible in the viewing cone at a certain viewing point PW position. External edges of a view create a viewing contour. Viewing face pyramid is a pyramid described by the center of a viewing sphere and current viewing face. Face SW Viewing potential PWSW is a sum of faces visible from above face SW. Edges between faces from PWSW and other edges are called Face SW viewing potential boundary. Viewing potential pyramid OPW has a common vertex with viewing face pyramid but has wider angle of flare. Complementary viewing cone SDW is a cone defined by current viewing axis (it’s collinear with its height) and has an opposite direction of flare then the viewing cone. It intersects viewing cone with angle Π/2, so its angle of flare is Π/2-α. Complementary, boundary cone is a cone that complements cone DSW. Its angle of flare is twice wider than in cone DSW. Viewing representation is created for the polyhedral with following features: nonconvex polyhedral with no pits, if a face is intersected by viewing ax it’s visible.
5 Visual problem and its solution 5.1 Problem analysis Lets define our problem: Generate a 2.5 D complete view model of a nonconvex polyhedron. Number of polyhedron faces m >=6. Generation is conducted based on viewing sphere with perspective. Divide this problem into as many sub problems as the number of our polyhedron faces. Each sub problem is generating all possible polyhedron views from above current face SW. Lets take tangent vector for each face and translate it to center of view sphere. Sub “view task” is about to finding versors (for current face) that each of them lay inside complementary viewing cone. This cone certainly consist current face versor. So let’s take face SW for which we want to generate complete set of 2.5D views. We need to recognize its pyramid OSW, its viewing potential and the pyramid of this potential OPWsw. Next we need to recognize normal versors of polyhedron faces that lay inside of pyramids OSWsw and OPWsw. During our further task we will be interested only in such faces which tangent versors lay inside of pyramids OSWsw and OPWsw. It is proper because if some versor lay outside OSWsw its face is invisible only if view axis lay inside OSWsw. However, when vector lay inside OPW, it doesn’t mean that sw face (which defines pyramids) is visible. It depends on its surroundings. This face can be covered by other visible face (which belongs to viewing potential). Information, about presents (inside contour potential view) of faces witch tangent vectors lay outside potential view pyramid, is very important. Existing of such faces mean that inside OPWsw lay sk face which is invisible in every “view above sw face”. This information helps us to define visible condition for hollow polyhedron. 5.2 Generation of views from above the face – creation of complementary cones Let’s take a nonconvex polyhedron for which we won’t to generate views. We have to calculate normal versors vni of all faces and anchor them in polyhedron center. Next we need to circumscribe a sphere on this polyhedron and place both in a viewing sphere with an appropriate radius. Then we should choose one polyhedron face from which we will start generating views. Next we have to do the following steps:
•
Identify pyramid OSW and calculate pyramid OPWsw
•
Find normal versors
•
Find normal versors
OSW
∑ vn
j
in
k sw faces inside OSW cone
in
k p s w faces inside OPWsw cone
OPWsw
∑ vn
i
Further calculations we will constrain to normal versors from cone OPWSW. Idea of generating views from above a face which we will use here relays on finding all views that we can get with each versor from cone OSW and removing duplicate views. OPWsw
Let’s go to execution of the above idea. First we have specify which versors from the set
∑ vn
i
may go into
OSW
views with one vector from the set
∑ vn
j
. The most inaccurate activity mentioned here is reciprocal existence
of faces in one view i.e. faces sj (with normal versor vnj ) with rest of the faces si from the viewing potential of this (sj) face. One way of doing this is “rolling” of complementary cone around normal versor vnj and recognition of “entering” and “leaving” this cone by other normal versors vni. Lets go through possible configurations of complementary, boundary cone, versors and faces: 1. if complementary cone is empty there are no other faces accept for face sj 2. if in complementary cone of versor vnj there are normal versors of all neighboring (with sj) faces, sj can not appear in a view on its own 3. if we place around versor vn j complementary, boundary cone then in this cone we will find only versors that go into view with versor will get subsets:
∑
vn j . Doing this for all versors corresponding with viewing face pyramid we
SDG ( vn1)
in a view with versor
,. , ∑
SDG ( vnj )
,. , ∑
SDG ( ksw )
contains faces that coexist
vn j
OSW
4. 5.
versors from set
∑ vn
i
that have angle between them smaller than Π/2-α can not exist in the same view
All possible views with versor vnj we can achieve by “rolling” complementary cone. Executing this task for each versor vnj we will get complete set of combinations of views with versors in OSW.
Fig. 3 Scanning tangent vectors.
So the algorithm for finding set of views for viewing face could look like this.
From
∑
wid
∑
OSWsw
∑
OPWsw
take one versor
vni i.e. vn j and create (from set
vn j of versors that can exist in a view with versor vn j . Let number of versors in
Next calculate angular distances between versors
∑
) a set wid
vn j be k j .
vni and versor vn j and order them in an ascending manner:
rd r1 (vn1 ),. , rd (vnk ) . Versor v i which is most distant is going to be used for definition of scanning
direction KOS. For this direction scanning cone will start turning around versor vn j . Now let’s start calculating the views for versor vn j . 5.3 Algorithm loop PA
In set
∑
SDGvnj
find a versor closest to generator of the scanning cone ( vni 2 ) from the front side (in the
direction of cone movement). Do the same from the back side (opposite to the direction of cone movement). During cone movement versor that is closer to the generator will trigger visual event first. When this happens stop cone movement and register new view wid i +1 (vn j ).
∑ OSW we can move to the next versor from the set ∑
SDGvnj
If there are no more versors in set
we are done generating views for versor sw
beginning of the algorithm loop. As you can see scanning round versors from set
. If there are still versors in
∑
∑
vn j . Now
SDGvnj
go to the
SDGvnj
is based on executing a loop around
each versor with scanning cone. During this task we check if any visual events were triggered . .
6
Algorithms
6. 1 Algorithm of generating views. We assume following -View face lay inside View (because of type of polyhedron). 1.
Algorithm of computing view potential of sj face
1 Designate vectors of OSWsw and OPWsw. 2 For each sj versor, which belong to OSWsw, build complementary cone. 3 Designate versors that lay inside complementary cone.
2.
Algorithm of computing views
1 Rotate complementary cone around first versor which lay inside OSWsw and note every visual event. This event is manifested by appearance of a new versor inside OSWsw or by disappearance of versor which previously lay inside OSWsw. 2 Repeat previous step on every versor which lay inside OSWsw. 3 Remove events that repeat itself (events are considered the same when they consist of same set of vectors).
9 Summary This approach to generating 2.5 D models of nonconvex polyhedral requires farther studies and researches. Our next task is to implement this set of algorithms. Result will be presented.
8 REFERENCES 1987 [1] Connell J.H., Brady M.: Generating and generalizing models of visual objects. AI, 31, 159-183. 1990 [2] Bowyer K.W., Dyer Ch.R.: Aspect Graphs: An Introduction and Survey of Recent Results. SPIE, 1395. 1991 [3] Gigus Z., Canny J., Seidel R.: Efficiently computing and representing aspect graph of polyhedral object. IEEE Trans. PAMI, 13(6), 542-551. 1993 [4] Zhank S., Sullivan G.D., Baker K.D.: The automatic construction of a view-independent relational model for 3D object recognition. IEEE Trans. PAMI, 15(6). 531-544. 1995 [5] Leonardis A., Kovacic S., Pernus F.: Recognition and pose determination of 3D objects using multiple views. Proc. CAIP'95, LNCS 970, Springer-verlag, Berlin, 778-783. [6] Suk T., Flusser J.: The projective invariants for polygon. Proc. CAIP'95, LNCS 970, Springer-Verlag, 729-734. 1996 [7] Arbel T., Ferrie F.P.: Informative views and sequential recognition. Proc. ECCV'96, Cambridge, UK, April, 469-481. [8] Hlavac V., Leonardis A., Werner T.: Automatic selection of reference views for image-based scene representation. Proc. ECCV'96, Cambridge, UK, April, 526-535. 1997 [9] Arbel T., Ferrie F.P.: Informative views and sequential recognition. Proc. ECCV'96, Cambridge, UK, April, 469-481.
>@ 'ENRZVND 0 0RNU]\FNL :6 0XOWLYLHZ PRGHOV RI FRQYH[ SRO\KHGUDO 0* 9
[11] Madsen C.B., Christensen H.I.: A viewpoint planning strategy for determining true angles on polyhedral objects by camera alignment. IEEE Trans. PAMI, 19(2), 158-163. [12] Shimshoni I., Ponce J.: Finite-resolution aspect graphs of polyhedral objects. 1998
>@ 'ENRZVND 0 0RNU]\FNL :6 $ QHZ YLHZ PRGHO RI FRQYH[ SRO\KHGUDO ZLWK IHDWXUH GHSHQGHQW YLHZ MG&V, 7(1//2), (Proc. GKPO'98, Borki, Poland, 18-22 May), 325-334. >@ 'ENRZVND 0 0RNU]\FNL :6 &RQGLWLRQV RQ PRGHOV IRU REMHFW YLVXDO LGHQWLILFDWLRQ 3URF $&6
Szczecin, 19-20 Nov., [15] Kovacic S., Leonardis A.: Planning sequences of views for 3D object recognition and pose determination. PR, 31(10), 1407- 1417. 1999 [16] Kowalczyk M., Mokrzycki W.S.: Determining complete object's view model by joining one-view areas. Proc. ACS'99, Szczecin, 18-19 Nov., 68-72. 2001 [17] Kowalczyk M., Mokrzycki W.S.: Obtaining complete 21 2D view representation of polyhedral using concept of seedling single-view area. Submitted to CV &IU. 2002 [18] Kowalczyk M., Mokrzycki W.S.: A new method of finding one-view areas and tight view sphere covering. Proc. ICCVG'02, Zakopane, Poland, Sept. 25-29, 443-449.
