A theory of origami world.pdf
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Carnegie Mellon University
Research Showcase @ CMU
Computer Science Department School of Computer Science
1978
A theory of origami worldTakeo KanadeCarnegie Mellon University
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Takeo Kanade
Department of Computer Science
Carnegie-Mellon University
Pit tsburgh, Pa. 15213, USA
and
Department of Information Science
Kyoto University
Kyoto, JAPAN
September 20, 1978
Abstract
The recovery of three-dim ensional configurations of a scene from Its Image Is one
the most Import ant steps In comput er vision. The Origami world Is a m odel funderstanding l ine drawings In term s of surfaces* and for f inding their 3-D configuration
It assumes t hat surfaces them selves can be stand-alone objects, unl ike conventional m odel
such as the trihedral world* which assume sol id objects. We have establ ished a label i
procedure for this Origami world* which can find the 3-D m eaning of a given l ine drawing
assigning one of the four labels* (convex edge)* - (concave edge)* * -* and -> (occludi
bounda ry) to each line. The procedur e u ses a filter ing procedure not only for junction Labe
as In the Waltz label ing for the trihedral world* but also for checking the consistency
surface or ientat ions. The theor y Includes the Huffm an-Clowes-Waltz labelings for the trih edr
solid-object world as a subset. It shows great potent ial for the applicat ion of recovering 3
configurations from region-segm ented Images; other Informat ion (such as spectr
Informat ion) avai lable from Im ages can also be Incorporated smoothly. This paper also reveaInterestin g relatio nships am ong previous research In polyhedral scene analysis.
This r esearch was support ed by the Defense Advanced Research Project Agency und
contract no. F33615-78-C-1551 and monitored by the Air Force Office of Scientific Research
UWVFHSITY LIBRARIES
pARHEGlE-WaiON
UNIVIRSITV ,
PITTSBURGH.PENNSYLVANIA 15213 i
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I Introduction
Origami is the Japanese traditional manual art of making various shaped objects (e.
animals) by f oldi ng a sheet of paper. Figure 1 is a typical example of Origami. It is easy
see that Figu re 1 is an Origami crane. This process of seeing and understandi ng may b
di vi ded into two processes: one is to determine the possible three-dimensional conf igurat ion
from the picture, and the other is to match them with some known concepts (such as "crane"
This paper deals wit h the fi rst process. Thus the problem is: how do we underst and th
possible three-dimensional configurations from a collection of lines?
One solu t ion i s: f ir st , model a worl d (I will call it the "Origami" wor ld ), where sur face
them selves can be stand-alone objects, rather than the conventional tr ihedral sol id -obje
wo r l d; secondl y, establ ish a procedure which can assign a 3-D meaning to each line. Th
pr ocedur e developed uses a fi ltering method both for finding consistent combinati ons
labels and for testing the consistency of surface orientations based on the gradient spac
repr esen t at ion. Not only does this surface-oriented Origami world include the cases of th
sol i d-ob ject wor ld, studi ed by Huffman [Huffman, 1971], Clowes [Clowes, 197 1], and Wal
[Waltz, 1972], as a subset, but It also demonstrates various features that have the potenti
to be used in image understanding tasks of real-world images.
Figure 1 Origami crane.
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Figure 2 Photograph of a carton paper box.
Figure 3 Line drawing of a carton paper box.
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I I Key Ideas and Related Work
Illustrative Examples
Let us have several illustrative examples of simple line drawings for the following
di scussions. Suppose that an image of a box case made of carton paper (Figur e 2) is gi ven .
How do we recognize that the object in the image has a shape of "box" (i.e., an open-faced
cube)? A line drawi ng der ived from the image such as Figure 3 has long been an import ant
product of t he initial feature extraction process. In fact, we can imagine the t h ree -
dim ensional shape of " box" from Figure 3. As other examples, the drawings in Figur e 4
usually convey to the viewer the meaning intended by the artist: (a) a cube, (b) a W-folded
pa per, and (c) two coordinate planes int ersected. However, take Figure 4(a) for example:
other possible configurations, such as those in Figure 5, are imaginable.
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L ARROW FORK T
Figure 8 Junct ion types t reated in this paper.
Figure 10 Examples of configurat ions at vert ices: (a) one quadrant plane whi ch
generates an L junct ion; (b) two quadrant planes which generate an
ARROW junct ion; (c) three quadrant planes whi ch generate a T
junct ion; (d) four quadrant planes which generate a T junction.
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I I I The Theor y of Origami World
The presentation of the theory of the Origami world consists of seven subsections:
(1) Terminology;
(2 ) The enumerat ion of legal combinations of line labels at juncti ons;
(3 ) The links bet ween regions;
(4) The probl em concerning consistency of surface orienta t ions;
(5 ) A test method for consistency of surface orienta ti ons;
(6 ) The actual labeling procedure;
(7 ) Examples of labeling.
Il l1 Origami World and Terminology
The wor ld is assumed to be made of a collection of surfaces. A line drawi ng is
picture (ort hograph ic projecti on) of such a composite in the scene. For the time being th
surface in our Origami world are assumed to be planar; i.e.. the orientation is constan
t hrough out a surf ace (actual ly, the restrict ion to plane surfaces will be relaxed a lit tl e in t h
la ter secti ons). In this respect it is not t he paper-surface (i.e., developable surf ace) w or l
investigated in [Huffman, 1976].
The terminology we will use for the Origami world parallels that for the Waltz labelintheory for the trihedral world [Waltz, 1972]. An edge is a straight boundary of a plan
surface. A vertex is a point wher e edges of the surf ace(s) meet. A l ine is an orthographi
pr oj ect ion of an edge to the picture plane. A junction is a point in the picture where li ne
meet . A junct ion can be the projecti on of a vertex or the point where an edge is i nt err upt e
by an occluding surface. A region is an area in the picture surrounded by li nes, and
corresponds to (a visible part of) a surface.
An edge can be classified according to its three-dimensional physical meaning in the
scen e. We wil l use the fol lowing terms and labels:
convex + : edge along which two surfaces meet and form a con vexi t yconcave - : edge along which two surf aces meet and form a concavi ty
occluding - or -> : edge along which one surface occludes another
The di rect ions of ar rows of occluding boundaries are given in such a way that the occluding
sur face is on the ir right hand side. A line can therefore be labeled wi th one of the f ou
labels (+, -,
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labels to t he lines in the drawing is to give a three-dimensional meaning to the drawi ng. A
set of assignments of line labels to the lines in the drawing is called an interpretation of the
dr aw ing. For example, the labeling shown in Figure 6 is an int erpret at ion of Figure 4 (a).
Junctions are classified according to the number of lines meeting at the junctions and
their geometrical configurations in the picture. In this paper we will confine ourselves to L
ARROW, FORK and T junctions shown in Figure 8.
111*2 The Enumerat ion of Legal Junction Labels
The physical world imposes constraints on the labels that lines can take at a particula
t yp e of j unct ion. A combination of line labels for one juncti on t ype is ref er re d to as junct io n la be L The crucial observat ion which was made by Huffman and Clowes, and whi c
was exploited to a great extent by Waltz, is that not all the combinatorial^ possible junction
labels can appear (are legal) in the picture. For example, for the ARROW junct ion, onl y t hree
jun ct ion label s out of the 4x4x4 possible combinations can occur in the t ri hedral wor ld
Needl ess t o say, unless we assume a certain restr ict ion on the three-dimensiona
configurations allowable at the vertices, the resultant constraints on junction labels will be
too weak to be useful. We need to confine ourselves to a reasonably limited world which
corresponds well to the real world images.
The confinement we adopt in the Origami world is that surfaces meet edge to edge
that no more than three surfaces of dif ferent orientati ons can meet at a ver t ex, and that thcombi nat ion of t he three orientat ions is "general" , in the sense that t hey span the t hr ee
dimensional space ( i .e. , each o r i en ta t i on has a vector component perpendicular to the ot he
t wo). Thus, no more than three edges of dif ferent directions are involved at a ver tex. Le
us call such vertices up-to-3-surface verti ces. This restri ction corresponds to the t ri hedra
vert i ces in the soli d-object worl d. Note, however, that the up-t o-3-sur f ace verti ce
generate a richer world than the world generated by the trihedral vertices, since the forme
can include 1- and 2 - surface vert ices; that is, it allows f ree extending surf aces as st an d
alone objects.
Possible junction labels for the up-to-3-surface vertices in the Origami world can be
enum era t ed in the fol lowing way. The planes of three general orientat ions int ersect andi vide each ot her i nto 12 partial planes. Thus we can think of 12 quadrant pl ane surf ace
around the vertex point as shown in Figure 9.
Let us fix our eye position in one of the eight octants separated by the quadran
pl anes, say, the octant bounded by the quadrants 0,4, and 7. Next, we generat e one by on
all the possible (4096) combinations by setting each quadrant plane to be either occupied o
vacant, and check how the vertex formed at the origin appears when viewed from the ey
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pos i t ion f i xed as above. Then we can give a label to each line at the junct ion based on i
meaning, and obtain a legal junction label. Figures 10(a) through 10(d) show examples of th
ve r t ex conf igurat ions and their derived junct ion labels. As previously stated, we conside
on l y t he combinat ions which result in the juncti on types shown in Figure 8. The number o
j unct ion labels thus obt ained is: 8 for L, 15 for ARROW, 9 for FORK, and 12 for T.
For t he junct ion type T, the four additional junct ion labels shown in Figure 11 ar
included as legal. They do not correspond to actual vertices, but to the cases in which th
jun ct ion is caused because the upper half plane is in the front and occludes the edge behi nd
Table 1 compares the number of legal up-to -3-sur face junction labels thus obtai ned wit h t ha
of legal t ri hedra l junct ion labels. It gives an idea of the degree of constraint imposed by t h
up -t o-3 -su r f ac e Origami world compared with the Huffman-Clowes t rihedral-juncti on w orl d
The appendix gives a complete list of legal juncti on labels in the Origami wor l d; for eacjunct ion label , it includes an il lustrat ive f igure of the conf igurat ion which the labe
represents, and the l inks which will be explained next.
Figure 11 Legal T juncti on labels not corresponding to vert ices.
Tabl e 1. Compar ison of the size of the Origami junct ion dict ionary wi th the
Huffman-Clowes dictionary.
Junction
Type
Huffman-Clowes
Dictionary
Origami World
Dictionary
L 6 8
ARROW 3 15
FORK 3 9
T 4 16
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I I I -3 The Links
Each junct ion label deri ved in the previous subsection implies which sur faces a
con nec ted at which edge in ord er to form that juncti on label. For later use, this inform at io
is also stored explicitly in the dictionary by means of a l ink, which links a pair of connec te
regions and points to the line at which they intersect. For example, the link in a legal FOR
jun ct ion label shown in Figure 12 represent s that the regions Rj and R2 are conn ected
the convex line L. Note that since the region R3 is totally occluded by the other two region
(in other words, it is the background), it has no relationship to others at this junction.
In the case of junction labels which involve partially occluded regions, a subt
si tuat ion occurs. Take the ARROW juncti on label shown in Figure 13(a) as an example. Th
junct ion label was original ly derived from the conf igurat ion shown in Figure 13(b); th
sur faces Sj and $2 connect at the edge BC. However, note that the junct ion label i tself ca
mean ot he r cases such as those shown in Figure 13(c) and 13(d): (c) is the case wh er e
and S2 intersect wi thin the angle ABC, and (d) is the case where Sj and S2 wil l int erse
outside of the angle ABC, when they are extended.
In the Origam i wor ld we will assume that the situat ion shown in Figure 13(c) is what
happening near the vert ex. This assumption allows more configurat ions than assumin
merely the case of Figure 13(b). It seems reasonable to exclude those situati ons lik e Figu
13 (d), because t hey are accidental cases caused by a particular relat ionship b et ween th
view direction and the vertex. In fact, if we move our view direction a little to the left, thve r t ex of Figure 13(d) may appear like Figure 13(e), even though we are looking at the sam
sides of the same surfaces.
Therefore, the link for the junction label of Figure 13(a) is given as shown in Figu
13(f); it represents that the region R^ and R 2 are connected at an "occluded intersectio
li ne L' (it s label is >), whi ch is located with in the angle ABC. Note that t he line L' can over la
Fig ur e 12 Link bet ween regions.
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wi t h BC, but not wi th AB, because if it did, the label of AB wil l change from
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D
(a) (b) (c)
Figure 14 Interpretation inconsistent with respect to surface orientations.
actually the same surface and therefore have aunique surface orienta ti on in the scene. Th
example demonstrates that we need a provision to check such global consistency of surfac
orientations.
It should be noted that the kinds of anomalies illustrated above, which are caused b
rel yi ng solely upon the* junct ion dicti onary, have also occurred in the Huff man-Clowes-Wal-
label ing for the tri hedral solid-object world. But because they occured "less fr equent ly
t hey did not show up as a very serious problem. Figure 15 is an example of such a
anoma ly show n in [Mackwort h, 1977]. All the junction labels in it are legal in the t ri hedr
world, but it can be seen that the configuration is not realizable in that world.
1 R 2 + RJ +RA
Figure 15 Anomaly in the Huffman-Clowes-Waltz labeling.
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viewer
Figure 16 Geomet ry involving the viewer , picture plane, and object .
A Tool
In order to carry out consistency checks of surface orientations it is necessary to
repr esen t surf ace orientat ions in the scene in connection wit h thei r proper t ies proj ect e
onto the pict ure. The gradient space introduced by Mackworth [Mackworth, 1973] provides
good tool for it.
Let Figu re 16 be the geometry involving the viewer , the picture plane, and the obj ec
in the scene. A plane in the scene whose surface is visible from the viewer can b
expressed as
-z ax + by + c. (1)
The two-dimensional space made of the ordered pairs (a,b) is called the gradient space G
Let us assume for our convenience that we align the directions of the coordinates of (x,y> i
the pi ct ur e wi th those of (a,b). All planes in the scene which have the same values of a an
b a re m apped into the point (a,b), called the gradient, in G.
The values of (a,b) represent how the planes are slanting relative to the view line (z
axis). For example, the origin 0Q* (0,0) of G corresponds to those planes (-zc
perpend icul ar to the view line. Pj (l ,0 ) corresponds to the planes (-z-x+ c) whi ch ar
slant ing hor izont al ly to the right. Mathematically,
a - d(-z)/ dx, b - d(-z)/ dy, (2)
whi ch is wh y (a,b) is called the gradient. Thus the l engt h-/ a 2 + b^ of the vector from OQKO,
to P(a,b) is the tangent of the angle between the picture plane and the plane
cor respondi ng to P; and the direction tan " l (b/ a) of the vector is the directi on of th
steepest change of -z (depth) on the plane.
Kanad
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One of the useful properties of the gradient space is the following [Mackworth, 1973].
Consi der two planes meeting at an edge and the orthographic pi cture made of regions Rj
and R 2 and a HneL, as shown in Figure 17, Then in the gradient space, the gradients Gj and
G2 of the two planes should be on a line which is perpendicular to the picture l ine L
Mo reo ver , if the edge is convex (+), Gj and G2 are ordered in the same directi on as are t he
cor respo nd ing regions in the pict ure. If the edge is concave (-), thei r order is rever sed .
In the Origami world, we additionally have the case of an occluded intersection (
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b
Figu re 19 Trace of gradients of the regions in Figure 14.
An Example of Nonconsistency of Surface Orientations
Now we are ready to test the example of Figure 14. Imagine a grad ient space an
ref er to Figure 19. Let Gj be the gradient of the region Rj . Rj and R 2 are connected at th
co nvex line AD in the picture. Thus, the gradient of R 2 should be somewhere on the half lin
G^a, which is perpendicular to AD and extends toward left, because of the property of the
grad ient space. Suppose it is at G2 . Again because R 2 is connected with Rj at a convex lin
BE, t he gradi ent of Rj should be somewhere on the half line G2 b as shown. Since we hav
f i xed t he gradi ent of Rj at Gj , this half line G2 b should pass Gj , whi ch is impossibl
wherever we select G 2 on the half line Gja. This means that t here is no combinat ion o
gradients for the regions Rj and R2
which results in the configuration of Figure 14(btherefore the configuration is inconsistent.
I IX-5 The Test Procedure in the Origami World
The above example has demonstrated the necessity of and the method for globa
con si st ency checks of surface orientati ons for a set of regions. This sect ion wil l present a
algorithm which indicates on what sets of regions the consistency checks are to b
per f or med and which tells whet her they can have consistent surface orientat ions. Given a
interpretation, the method consists of first constructing a labeled graph called a Surfac
Connection Graph, and then performing a type of filt ering operat ion on the constraint s in thgrad ient space. The test procedure to be presented is closely related to the idea of the dua
gr ap h of Huff man [Huffman, 1971] and the POLY program of Mackworth [Mackwort h, 1973
In fact, the Surface Connection Graph represents by and large the topological properties o
the dual graph, and the filtering procedure uses constraints in the gradient space in a more
t ho rou gh and systemat ic way than does POLY.
UNIVERSITY UBRARItS
CARNLGiE-MELLON UNIVERSITYPITTSBURGH. PENNSYLVANIA 152U
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Figu re 22 Spanning angle of a path.
In any event, the constraint which the SCG imposes between a node pair (p,q
connect ed b y a path 7=(p-q) is that if the gradient of one node is f ixed, the gradient of th
ot he r node should be within a fan-shaped area (or, in a special case, on a half l ine as
equat ion (3)), whi ch is spanned by a non-negat ive linear combination of vectors . Let us ca
the fan-shaped area the spanning angle S^ of the path (see Figure 22).
We ca n define the f o l l ow ing compu ta t i ons on spanning angles: inverse, union, an
int ersect ion. Suppose that a path (p-q) has a spanning angle S( p ^ q ) . If we t ra verse th
path* inversel y as (q->p), the corresponding spanning angle S( q_p ) is the angle spanned by
set of vectors obtained by inverting the vectors which define S(p_> q). Graphi call y, as show
in Figure 23(a), the spanning angle S(q
^p
) is the angle opposite to S(p
_*q
). This operation called the inverse of spanning angle, and is denoted by S q ^ p - " S p ^ q .
We can concatenate two arcs, or more generally, two paths (p-q) and (q-r) to form
Figu re 23 Computati on on spanning angles: (a) inverse; (b) union; (c) in tersect ion.
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longer path (p-q-r). The operation of spanning angles corresponding to this concatenation i
the union. Graphical ly, as shown in Figure 23(b), S( p ^ q ^ r ) is the angle of the area whic
eit her belongs to one of S^ ^ q j and ^(n^r) or belongs to the area which is separat ed b
S(p_>q) and S( q_> r) and which has an angle less than 180. Let us denot e this oper at ion b
S(p_>q_> r) ^(p->q) u ^(q-* r)- ^ y ^ e u n ' o n operation, the resultant spanning angle can
become 360: the whole 2-6 space. This happens when the set of vect ors includes mor
than t hree vect ors and the angles made by a neighboring pai r of vect ors are all less t ha
180. In such a case, the path does not give to the node pair any constraints rega rdi ng th
relative locations of their gradients.
Now, if there are two paths y and Y 2 ' R O M ^ E N 0 ( ^ E P * then t hey imposeconst ra int s on Gp and G q simultaneously; i.e.,
G q - G p - * kjx-Pji - 2 k u-Pifr (5)
wher e {P^ } is from y and {Pj 2 } from ? 2 . This means that G q should be within the
overlapping area of Sy and (Figure 23(c)). This overlapping area is the int ersect ion o
spann ing angles, and is denoted by n .
Loop-Free SCG and Elementary Paths
The operation of intersection of spanning angles suggests that we can reduce an SCG
in to a simpler f orm on which our test will be applied. First of ail, it is easily seen that onl ythose part s of the SCG which include a loop or circuit need to be actually consi dered. Thi
implies that if the SCG can be separated into two subgraphs by cutting a single arc, then
each subgr aph can be considered independently. In part icul ar, leaf nodes can be eli minated
f rom the considerat ion. Thus, we can "prune" and "cut " the SCG into a set of Leaf-fre
connected SCG (LF-SCG)% each of which is independently subject to the consistency check
In Figure 21, the leaf (R3, Rg) can be pruned, and the remainder is the LF-SCG.
Further, it is understood that the gradients of the nodes which are connected to
exact ly t wo ot her nodes (i.e., their node degree is two) in an LF-SCG, such as the nodes s
and t in Figure 24, are rela ti vely less important. They do not affect other nodes beyon d p o
q; it is only required that the relation (vector) between Gp and G q is kept the same. Thisimpli es that we can divide an LF-SCG into a collection of pat hs, each of which begi ns and
ends wi th nodes of degree more than two, and each of which contains only nodes of degree
t wo in bet ween. Let us call such a path an elementary path. In Figure 21, paths such as
(R2 ->R 3->R4) and (R2 ->Ri ) are elementary paths. We can associate a spanning angle wi th each
(di rect ed) element ary pat h. What we need now is a computational procedure on the spanning
angles of the elementary paths of an LF-SCG, to see whether the constraints on surface
orientations can be satisfied.
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T h e n , f i l t e r t h e S ^ n " l ) by the in tersec t ion o f S r . ( n ' l ) , and se t t he resu l t t o S ^ ( n )
i.e.,
S < n ) - ( n S r . * " - 1 * ) n S < n " 1 ) ,
If any S^ n ^ becomes null, then the test fails.
(3) If there exists an elementary path such that S^ n ^ ? S^ n ~ * \ then n-n+l and go to
(2). Otherwi se, the test terminates wit h success.
Four t hings should be noted about the test procedure. First , if an LF-SCG consist s o
a single circuit (i.e., the degree oi all the nodes is 2), then we can pick up any pai r of node
and regard the two paths connecting them as elementary paths. Or, al ternat ivel y, this i
equivalent to testing whether the spanning angle corresponding to the circuit is the entire
360.
Second, the i terat ion in the procedure always term inates, since all the S ^ n ^
monoton ical ly decrease in step (2 ) and the number of possible spanning angles which S ^ n ^ *
can take is finite. (It is bounded by the number of subsets of the vectors involved in the LF
SCG under the test.)
20
Filtering Operation on Spanning Angles of Elementary Paths
Now we are ready t o describe the fi lt ering operati on defined on the SCG. We assume
that the SCG for a given interpretation has been simplified and decomposed into LF-SCG's,
and that each LF-SCG is decomposed into elementary paths. If there is no LF-SCG, there is
no need for performing the filtering procedure, and the test trivially succeeds.
(1) For each elem entary path y, associate an init ial spanning angle which is
computed from a set of vectors defined for the arcs belonging to the path. Set n -l .
(2) For each elementary path y, let {Tj} be a set of all the paths that connect the same
node pair in the same direction as y connects. Since the LF-SCG is decomposed in to
elementary paths, each T\ is a concatenati on of several elementary paths Wj^};T j = 7 j 2 " * " The'spanning angle of T j , Sj \ ^ n ~^ , is computed from the union
of the spanning angles of the component paths, S^ .^ n -l ) ; i.e.,
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(a) (b)
Figure 24 Elementary pat h: (a) SCG; (b) Gradient space.
Third, the procedure is a necessary condition for surface orientations to satisfy all th
constraints represented in the SCG, but it is no t a sufficient condition for that . An exampl
of this will be given in IV-1.
Fourt h, the presented algorithm is a conceptuall y st rai ghtf orward one, but it
inef fi cient . Implementat ion of the algorithm can exploit several proper t ies of t he SCG t
increase efficiency.
If all the LF-SCG's pass the above test procedure, then the given interpretation is sai
to pass the test for t he surface orientat ion consistency. The above test pr ocedure, togethe
wi t h the up-t o-3 -sur f ace juncti on dict ionary, defines the nature of the Origami wor ld. A
interpretation of a line drawing is called plausible in the Origami wor ld, if ail the junct ions ar
given legal junction labels contained in the dictionary, and if its SCG passes the above test.
For Figure 20, it is easy to see that the SCG consisting of two nodes does not pass th
test . Let us next consider the SCG in Figure 21. The path Y(R2 -R3->R4) is an elementar
path, and the path I 1}-( R^ Rj ^ R/ i ) is one of the paths which connect R? and R^. Th
spanning angles sj"^ and S f , ^ are calculated as shown in Figure 2*5. Since theintersection is null, the spanning angle S ^ ^ will become null in step (2), and the t est f ail s.
On the other hand, if we change the concave line labels (-) between R 2 and R3 an
be t we en R4 and R3 in Figure 15 to occluding boundari es (- ) so that R3 occludes R 2 andRQ
t hen the cor respond in g SCG wil l consist of two subgraphs: one includes nodes {Ri ,R2 ,R4,R5
and the ot her ^ 3 ^ 5 } . It is easy to see that the latter subgraph has no LF-SCG and tha
the former passes the test ; therefore the test succeeds this time. As anot her example
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consi der Figure 26 (a) and one of its int erpretat ions Figure 26(b). It is a f igure obtained by
adding a convex line CF to Figure 14(b). It is a paper-sided, t runcated pyram id viewed f romabove. Its SCG consists of a single circuit (Figure 26(c)), and passes the test . Ther ef or e the
configuration of Figure 26(b) is plausible.
Two point s should be mentioned concerning the proper t ies of SCG. Suppose that i
Figure 21 we are filtering the spanning angle of the elementary path 7=(R4-Rg) against a
the paths that connect R4 and Rg. If we have fi lt ered 7 by the path Tj ^ R^ Rj -^ Rg ), the
we need not fi lt er by such paths which travel a component element ary path of I ^ , sa
7l=(R4-Ri), through a nonelementary path. For example, r 2=(R4-R3-* R2-Rj->Rg) travels y
t hrough the nonelementary path ( R^ -^ -^ R^ Rj ) . The reason is that since ^ ^ - ( R ^ R i ) i
itself an elementary path, the partial path ( R^ -^ -^ R^ Rj ) of r 2 has been or wil l be used i
f i l ter ing y , and th eref ore, r 2 does not add any constraint diff erent from T j .
Another point is that Tj ^ R^ Rj -^ Rg ) does surely have an overlapping spanning angl
with that of 7 = ^ 4 -^ 5 ) , for they form a circuit which surrounds a single junction. If we sto
and th ink , this is what the juncti on label means: it assures a local consistency around
junct ion. These two facts can, reduce the number of paths whi ch are act ual ly used i
fi lt ering. In our example, y needs to be filt ered only by ^ -( R^ -M ^ -^ R^ Rg ).
I I I -6 The Labeling Procedure in Origami World
In the previous four subsections we have first enumerated the legal junction labels i
the Origami world, and have stored them together with link information in the Origam
junct ion di ct ionary. Then , af ter introducing the concept of spanning angles, we have def ine
a test procedure on the SCG which can decide whether a given labeled interpretation of
line draw in g is consistent wit h respect to surface orientations. This subsect ion will present
labeling procedure which, given a Line draw ing% finds all its plausible in terpretat ions; that i
all the combinations of assignments of line labels which result in legal junction labels at a
the junctions in the drawing, and which pass the test procedure for surface orientations.
Strategy
The labeling procedure must include two tasks:
(1) Using the junct ion dict ionary of the Origami worl d, int erpretat ions are generat ed
whi ch the labels given to lines constit ute legal junct ion labels at all the junct ions. Th
could be done in two steps:
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Fig ur e 25 Spanning ang les o f paths in the SCG in F igure 21 .
(a) (b) (c)
Figure 2 6 Paper -sided, truncated pyramid: (a) line drawing-, (b) label ing; (c) SCG.
( l a ) Filt eri ng of juncti on labels (as in the Waltz labeling [Walt z, 1972 ]),
( l b ) Tree searching to obtain the individual consistent combinations of junct ion labels
(2) For each int erpr etat ion obtained in (1), the SCG is construct ed and the consi stency o
sur face orientat ions is tested by fi lt ering the spanning angles of elem entary pat hs.
However, the number of interpretations which are generated in (1) as being consisten
with respect to the junction dictionary is very large, and most of them are inconsistent with
resp ect to the surface orientat ions. Also, if a certain subconfigurati on is inconsistent wi t
respect to surface orientations, any interpretation which includes that subconfiguration is
never plausible. Theref ore, in the actual implementation of the labeling procedure the step
( l b ) and a par t of (2 ) are combined into one process: while assigning a junct ion label t o
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j unct ion in a dept h-f i rst manner, the process const ructs the SCG increm ental ly and check
the spanning angles as far as possible. When any inconsistency, either in the junction labeor in the surf ace orient at ions, is detected, the process backtracks the search for the ne
com binat ion. This combined process great ly increases eff iciency in the labeling p rocedure.
As a result, the labeling procedure consists of three phases:
(1) Filtering of junction labels,
(2) Tree searching combined with filtering of spanning angles on a partial SCG,
(3) Final filtering of spanning angles of elementary paths.
Phase (1): Filtering of Junction Labels
This process is exactly the same as the Waltz method [Waltz, 1972]. At first, eac
junct i on is gi ven a set of possible labels drawn from the dict ionary according to it s junct io
t yp e. Init ial constraints are that the outer boundaries should have the label ,
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const ruct ed. If t he link of the present junction label adds an arc and result s in the format io
of a new circuit in the SCG, the spanning angle of this arc is checked against all the existin
pat hs which connect the same node pair. If they have a null in tersect ion, the part i
interpretation is inconsistent.
If any inconsistency in the configuration is detected, either in the combination o
junct ion labels or in the surface orientat ions, the tree search process back track s one st e
and sear ches f or the next combination. If all the junctions are labeled consistent ly, th
resultant interpretation is handed to the final phase (3).
Let us see the example of Figure 27(a). Suppose that the junct ion is f i rst gi ven
junct ion label as shown in Figure 27 (b); the corresponding part ial SCG consi sts of a singl
arc ( R j - ^ ) . Then , J7 is given a juncti on label (see Figure 27(c)). Since the link it ha
bet we en Rj and R3 is the same as the existing one, two new arcs are added to the SCG: fi rs
( R^ R2 )i and then ^ 2 -^ 3 ) . A circuit is formed when ^ 2 -^ 3 ) is added (Figure 27(d)). It
spanni ng angle is checked against the existing path ^ - ^ R ^ - ^ ) . As shown in Figure 27(e
the intersection is not null, and therefore the search proceeds to Jg. Suppose that the f ir s
choi ce of j unct ion labels for it results in the interpretat ion and the correspondi ng SCG as
Figu re 2 7(f ). Since no new circuit is generated, the assignment of juncti on labels pr oceed
J3 is given a unique junct ion label determined by the line labels already given. As shown
Figu re 27 (g), when J3 is given the junct ion label, it adds an arc ^ - ^ 4 ) and the SCG has
new circuit. Thus, the arc ^ 2 ^ 4 ) is to be checked against paths ^ 2 ^ 3 - ^ 4 ) an
(R2-R 1 -^ 3 -^ 4 ). Since the spanning angle of ^ - ^ 4 ) does not have an overlapp ing are
wi th that of ^ 2 -^ 3 -^ 4 ) as shown in Figure 27(h), this int erpretati on turns out t o binconsi stent . The process winds back to Jg, and the next choice for Jg wil l be examined.
Consider another stage of the tree search in which the junction labels have been give
as shown in Figure 27(i). When the junction J3 is examined, it adds a new arc ^ 2 -^ 4 ) . Th
time, the spanning angle of ^ - ^ 4 ) is compatible with that of ^ 2 - ^ 1 - ^ 3 - ^ 4 ) 1 and therefor
t h e sea rch p ro ceed s . S ince the res t of junc t ions do not add new arcs to the SCG, th
interpretation shown in Figure 27(j) is handed to the final phase.
If we considered only junction labels, 90 interpretations would have been generate
for the line drawing of Figure 27(a). However, by means of checking spanning angles in th
course of tree searching, only 8 out of these 90 interpretations were passed to the fin
phase.
Phase (3 ): Final Fi lteri ng of Spanning Angles of Elementary Paths
The method of this phase is exactly the same as the test procedure described in III-
The reason for the necessity of this phase is that the SCG is not completely constructed un
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Figu re 27 (conti nued).
the tree search has completed the assignments of junction labels in an interpretation: w
could not identify the elementary paths, and therefore we have only checked the spannin
angle of newly added arcs against the existing paths which connect the same node pa
Partial duplication of this phase may appear inefficient, but usually most of th
inconsi stencies of surface orientati ons have been detected in the phase of t ree searchin
and only inconsistencies that involve very global relationships among regions remai
undet ect ed unt il this phase. This phase is also useful because it reveals the mutua
rel at ionshi ps among the gradients of regions in the SCG; this inform at ion is used i
reconstructing the 3-D shape of the scene.
In our example, all of the eight interpretations that are generated in the tree searc
pass this f inal phase. In the case of Figure 27(j), it is revealed that the gradients of the f ou
regi ons should be placed in the gradient space as shown in Figure 27(k). The diagram
represe nt s the fact that the four surfaces form a convex opening in the 3-D space, which i
probably a general description of a "box".
I I I -7 Examples of Labeling
A few examples of interpret at ions in the Origami wor ld foll ows. Figure 28 shows three
possi ble i nterpret at ions (wit hout counting rotations) that a line drawing of Figure 4 (a) ca
have: (a) a cube-like configuration; (b) a concave corner; and (c) a "roof" placed on a plane.
A "box" line drawing of Figure 3 has eight possible interpretations shown in Figure 29
(a) corresponds to an ordinary box; (b) is a "squashed" box whose front sides are pushe
backward; (c) is another "squashed" box whose rear sides are pulled forward; (d) is a box
with a triangular lid in the right corner; etc.
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Figure 30 shows 11 interpretati ons of Huffman's "im possible" pyram id: (a) a sol id
truncated pyramid; (b) a paper-sided, truncated pyramid; (c) a view of (a) from the bottom(d) a view of (b) from the bottom; (d) a triangular "dome" with an opening in the lowest side
etc.
Figur e 31 is another example of int erpret ing an "impossible" obj ect . It has 1
interpretations. The interpretation (a) corresponds to three twisted rectangular bars.
Figure 32 includes 16 possible interpretat ions of Figure 4(b). Int erpretat ion (a
corresponds to the W-folded paper.
IV Discussion
The discussion in this section is divided into three parts. The fi rst part discusses how
the test criteria employed for checking the surface orientations in the Origami world are
re la ted to plane surf aces. The second part reveals interesting relat ionships of the Origam
wor ld to other wor ld s dealt wi th in prior work on polyhedral scene, analysis. The t hir d par
discusses how knowledge is used in understanding line drawings.
IV-1 Plane Surfaces and Origami World
We have noted that the test procedure, described in III-5, to check the consistency osurface ori ent at ions in the Origami world is not a sufficient condition for the constra in ts i n
the SCG to be sat isf ied simul taneously. But it should be remembered that the const raint s in
the gradi ent space themselves are not a sufficient condition for the conf igurat ion to be
rea l ized by plane surfaces. Consider again the configurat ion of Figure 26 (b). The
configuration made of three regions Rj, R 2 , and R3 has passed the test . However, it is a
simple geometry problem to show that unless three lines AD, BE and CF meet at a single
point , the confi gura t ion is not realizable by the three planes corresponding to Rj , R 2 , and R3
The problem arises from the fact that the gradient space does not take into account
the value of c in equat ion (1): a consistent t race in the gradient space means that the
cor respon ding regi ons can take a consistent combination of (a,b) values, but it does nonecessari ly assure a consistent combination of (a,b,c) values. Huffman [Huff man, 1 97 7
presents a ^(^* )-point test as the necessary and sufficient conditi on for a "cut set " of li nes
(equ iva lent ly, a set of regions separated by those lines) to be reali zable b y plane sur faces
Consider again the example of a paper-sided, truncated pyramid shown in Figure 33(a), and
take the set of li nes AD, BE and CF cut by the dot ted loop. Each line belongi ng to the cut set
of lines is given an ori entat ion shown as a big arrow according to its label, eit her coming i nto
the loop (if t he label is +) or going out from it (if the label is -) (see Figure 33 (b)). Then the
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0(0')-point is a point that is to the right (left) of some line of the cut set and that is not to
the left (r ight ) of any other li nes. The (0' )-point test simply checks whet her ei t her a 0
point or a ^ ' -poi nt exists, and if either one exists, then the cut set is unreal izable. In fact
un less AD, BE and CF meet at a single point , or points exist, and t he ref or e the
conf igu rat ion of Figure 33(a) is unrealizable. Unfortunately, it can not be said that if all thecut set s in the int erpret at ion pass the (* )-point test, then the whole in terp ret at ion is
rea li zable by only plane surfaces. (Notice that the M' )-p oi n t test is the necessar y an
suf f ici ent conditi on for the readab i l i t y of a cut set of lines, not of the whole int erpretat ion).
To human eyes, the configuration of Figure 33(a) appears quite reasonable; sometimes
it takes time to persuade people that the configuration is not realizable by plane surfaces
Figu res 34(a) through 34(c) are other examples of interpretati ons of simple f igu res whi c
appear plausible but not actually realizable by only plane surfaces: they pass our test in the
Origami world, but not the ^(^')-point test.
The use of only the gradients (a,b) makes some sense mathematically w he n wconsider the manner in which we view a picture. Note that the gradient (a,b) of a plane i
invar iant to the x-y t ranslat ion of the picture plane (i.e. shift of eye position). In viewi ng a
or thogr aph ical l y project ed pict ure, we do not fix an absolute origi n in mind. When we see
line where two surfaces actually intersect, we tend to "place" the origin on that line, which
means we gi ve the two surf aces the same c value. Therefore, constrain ts about c ar
automat ical ly sat isf ied at that intersecti on. Also, when we see occlusions, we tend t
attribute it to the difference of the c value rather than to any relations among a, b, and c
As we shift our eye and move the origin in the picture, it is easy to keep the gradient of a
particular region in mind, but it seems difficult for us to "calculate" the value c which tha
reg ion should have in the new coordinates. These observations seem to expla in why th
objects in Figure 33(a), Figures 34(a) through 34(c) do not look impossible at first glance.
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Figu re 34 Examples of -plausible" int erpretati ons which can not be made of
plane surfaces (the unlabeled lines are occluding boundaries, - or
in the obvious direction).
Figu re 35 Solid t runcat ed pyramid. An example in which all the constraint s in the
gradient space cannot be satisfied simultaneously, but all the pairwise
constraints with respect to others can be satisfied.
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Another issue about the test procedure deserves comment: its insufficiency fo
assuring that all the constraints on the surface orientations represented in the SCG arsat isf ied simult aneously. The test procedure presented in II I-4 merely assures that for an
pair of regions which are connected by an elementary path (it means that they intersec
di rect l y or t hey are connected by a sequence of free-hanging surfaces between them), th
const ra int s on the surface gradients that relate the pair of regions are all sati sfi ed. In orde
to u nder stand the dif ference, consider the int erpretat ion of Figure 35(a), which is a soli
t runcat ed pyram id viewed from above. The corresponding SCG is shown in Figure 35 (b
The SCG passes our t est procedure, but it is not possible to fi nd a set of gradi ents for a
the r egi ons Rj through R4 so that all the constraints in the SCG are sati sfied sim ult aneousl
unl ess AD, BE, and CF meet at single point . This crucial dif ference stems from the fact t ha
when we pick up a pair of regions, say, Rj and R 2 , the gradients of regions R3 and R4 ar
not fi xed uniquely in checking whether the paths ( R ^ R ^ - ^ ) and ( Rj -^ 3 -^ 4 -^ 2 ) have aoverlapp ing spanning angle with that of ( R j - ^ ) .
I V-2 Origami World and Various Worlds
The theory of the Origami world and the labeling procedure for it have interestin
rel at ionsh ips wi th the work of Guzman [Guzman, 1968], Huffman [Huffman, 1971] , Clowe
[Clowes, 19 71 ], Waltz [Waltz, 1972], Mackworth [Mackworth, 1973] , and Huffman [Huffman
1977]. Thei r wor k all concerned the problem of recovering three-dimensional configurati on
f rom li ne drawings. (The historical development in polyhedral scene analysis has been wereviewed by Mackworth [Mackworth, 1977].)
Assume we consider only a set of line drawings which are reasonably "l ik ely" f igure
that is, it does not include f igures which show too anomalous behavior . We can consider
set of all the combinatorially possible interpretations (assignments of line labels) of those lin
dr aw ings. A subset exists containing those int erpretat ions which can be rea li zed by plan
sur faces. Let us denote that subset as the Plane Surface World, S p s w . We can also think o
a subset consisting of interpretations in which all the constraints on the gradients of surface
are com pletely sati sfied. Let us call it the Consistent Gradient World, S w . Obviousl
^cgw psw
We can view a labeling procedure as a method consisting of a generator and a tester
gi ven a line drawing, a generator generates interpretati ons in a certain manner, each one o
wh ich a t est er accepts or rejects based on a certai n method. Table 2 summarizes vari ou
label ing methods according to this taxonomy. Various subsets can be def ined which ar
genera t ed b y genera tors, or are determined as legal by testers. We wil l discuss the
relationships among those subsets, referring to Figure 36.
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Tabl e 2. Vari ous labeling schemes as a method of a generat or and a test er.
Method Generator
Subset
Tester
Subset
Huffman
Clowes
Waltz
Trihedral junction
dictionary
Trihedral junction
dictionary
with cracks, shadows,
etc
S t r i
Mackworth Sequential generation
of most connected
interpretations
Constructive test
on coherence rules
in the gradient spaceSpoly
Huffman ^(^') point test
for all the cut sets
in the line drawing
Origami
World
Up-to-3-surface
junct iondictionary
Sup3
Filtering of
spanning angleson the SCG
ongami
Huffman [Huffman, 1971], Clowes [Clowes, 1971] and Waltz [Walt z, 1972 ] used
t r ihedr al j unct ion dict ionary as the generator and did not use any tester . Let us denote th
subset of interpretations generated by the trihedral junctions as Sj rj (Waltz used crack
illuminations, shadows, etc., and the corresponding subset is different from Sj rj , but becaus
geom et ri cal ly his dict ionary is a tr ihedral one, it is included in this category). As we saw, S^
is larger than the subset of solid trihedral objects bounded by plane surfaces.
The Huffman's ^(^)-point test, when used on all the cut sets in the line drawing, ca
def ine a subset which is larger than Sp S W , but is the closest one to it articulated s
far . However , since there is no appropriate efficient generator , it would be dif fi cult , given
li ne drawi ng, to actually enumerate all the interpretat ions belonging to S^ * ) .
Mack worth' s POLY used a generator which generates combinations of line labels base
on some pr eferences. In particular, to achieve the most connected interpret at ions,
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generates first the interpretation in which all the edges are connected (i.e., all the lines are
given either + or -) , and then when such an interpretation fails to pass the test, it generates
an interpretation with all edges but one as connected edges, and so on. The consistency in
the gradient space is tested by a constructive method which actually tries to fix the positions
of gradients corresponding to the regions, step by step. In this way, POLY avoids the use ofpredetermined interpretations for particular categories of junctions (such as L, ARROW, FORK
etc.), and thus, theoretically, the subset Sp 0| y could be equal to S C g W . However , since it is
not practical to test all the interpretations, a certain selection criterion is needed to supply
the generator with advice or preferences concerning the order of generation. The
constructive test procedure also uses some heuristic rules, because the construction is not a
trivial process. As the result, the actual nature of Sp 0j y becomes a little unclear.
In the Origami world, the subset SUp3 generated by the Origami junct ion di ct ionary
properly includes Sj rj . The subset S o rj g a mj , consisting of plausible interpretations which
have passed the test procedure, is a little larger than the up-to-3-surface objects in the
S C g W , as shown in Figure 36. One feature of S o rj g a mj is that it has a clear definition of the
membership, which allows an efficient procedure to generate the member interpretations for
a given line drawing; i.e., filtering of junction labels and spanning angles on the SCG. The
locations of several example interpretations are indicated in the diagram, and they can serve
to illustrate relationships among the subsets.
origami poly
1: cube (Figure 6)
2: box (Figure 29(a))
3: Figure 34(b)
4: pap er-si ded , tr uncated pyramid(Figure 30(b))
W-folded paper (Figure 32(a))
Figure 34(a)(c)
5: solid tr uncated pyramid (Figure 30(a))
6: Figure 15
7: Figure 14
Figure 36 Relationship among various subsets of interpretations.
37
UNIVERSITY UBRARi tS
CABNEGIE-MEUCN UNIVERSITYfJTTSBURGH. PENNSYLVANIA 152
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It is interesting to review Guzman's work[Guzman, 1968] on object recognition at th
point . In its goal, his wor k is the forerunner of Huffman, Clowes and Waltz. He t ri ed to f in
"good" associations of regions into separate 3-D bodies based on heuristics concernin
jun ct ion t ypes. The idea of a "regi on" (which is a project ion of surface to the pi ct ure pl ane
is ve r y close to the idea of the surf ace-oriented wor ld . Also, the links he used represen t th
possible connections of regions like our links. However, since his links are def ined fo
jun ct ion t ypes rat her than for junct ion labels, his link information is a kind of "aver age" ove
var ious combinat ions of surfaces at a particular junct ion t ype. Because of its heur ist i
nature, Guzman's method could not explicit ly clari fy the wor ld for which it is in tended .
Table 2 suggests that we can employ different combinations of generator and teste
dependi ng on the wor ld in which we want to work. For instance, the tri hedral ju nct io
dictionary together with Huffman's ^(^')-point test is the closest to the plane-surfactrihedral solid object world; the up-to-3-surface Origami junction dictionary together wit
the $(
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In the case of the Origami wor ld, the precompiled knowledge in the Origami junct io
di ct ionary can reduce the possibi li ti es to Sypg by means of fi lt ering the juncti on labels. Th
filtering of spanning angles on the SCG can reduce the possibilities further to S o rj g a mj . Th
same type of f il tering procedure is used both for exploit ing the precompi led knowledge an
f or the dedicated computat ion; one is used symbolically, the other numericall y. It seems t ha
the difference SUp3 - S o rj g a mj is fairly large in the surface-object world, and thus the teste
is real ly needed. The fol lowing is to be noted. The junct ion labels hold inf orm at io
concerni ng a ve ry local consistency, as was pointed out in III5. They can propagat
in for mat ion t o the neighboring juncti ons only through line labels. In contrast , the links ca
globally transfer information to any junctions through regions.
V Appli cat ion of Origami Theory to Recover 3-D Configurations from Image
The theory of the Origami world has great potential in applications to image
understandi ng t asks that recover three-dimensional configurati ons. That it can deal wi t h
scenes which include free extending surfaces is very attractive, because real world images
incl ude obj ect s which are practicall y or conceptually flat. In fact, the Origami wo r l d
corresponds well to the way in which we would interpret a picture which has been
segm ent ed into regi ons. Suppose that Figures 37(a) and 37 (b) are obtained as the resul ts of
regi on segment at ion of "chai r" and "door" scenes, respect ively. They are sat isf ying to us
(a) (b)
Figu re 37 Region segmented pict ure of "chai r" and "door" scenes.
39
Minsky [Minsky, 1974] uses the side view of the "impossible" truncated pyramid as a
example to show how little humans rely on numerical three-dimensional geometrica
inf ormat ion i n seeing objects. However, since the int erpretat ion of Figure 15 does loo
unreasonable to us, a more precise statement would be that humans use the geometric
information to check certain consistencies in the gradients of regions, but not the tota
consistency among them.
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(a)
Figu re 38 (a) Moderat ely curved chair seat; (b) Possible line drawi ng
because we i nt erp ret them in terms of surfaces. Needless to say, the Origami worl d include
the solid-object world as its subset.
Just as we generalized from solid volume to surface, we can go further land say that
"pencil" has conceptually a line shape, thus we need a "wire-frame" world, and further,
"dot " wor l d. The more basic the unit of the worl d is, the broader class of pictures it can dea
with, but at the same time the less constraint s it provides. We feel that the Origami wor ld rich enough to accept a large class of line drawings, and at the same time it has enoug
structure to impose constraints on the possible label combinations.
Even though the Origami world is not intended for curved objects or imperfect lin
drawings, a certain class of line drawings including curved objects or imperfections can b
accommodated within it. As an illustrative example, suppose that a moderately curved cha
seat of Figure 38 (a) yields a line drawing of Figure 38(b). While it is an "impossible" f igur
in the t ri hedra l wor ld , it has an int erpretat ion in the Origami worl d corresponding to "
rect angul ar block wi th a flat sheet att ached". Once this is hypothesized, furt her processi n
pr ob ab l y i nvolving image data analysis, can discover the detailed shape and know whet her
is a square or curved, solid or flat object.
In real image understanding tasks, the number of lines is large, and ther ef ore th
num ber of possibl e interpret at ions is also large. Even the line drawing of a box (Figure 3
has eigh t i nt erpr etat ions, for instance. However, spectral (shading, color, etc.) an
geometrical (collinearity, parallelism, etc.) knowledge can be used here to reduce the numbe
of possi ble interpret at ions. There is knowledge that relates the nature of edges and the
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int ensi t y prof il es t aken across the edge [Horn , 1977 ]; for example, a peak -shaped edg
prof i le of t en suggests a convex edge. Another t ypical example which provides constrain ts o
li ne labels is a so-cal led matched T shown in Figure 39. If the edge prof il es of lines l j an
L 2 are simi lar (and, preferabl y, if the edge prof il es of L3 and L4 and those of Lg and Lg ar
also similar), then the labels of L.j and L 2 are likely the same, and L3 through Lg are l ikely t
be occluding boundar ies, wit h the middle region R obscuring Lj and L 2 . All these const rai n
conceivably in a probabilistic way, the possible combinations of line labels that a set of line
can take. Ther ef ore, "best " or "most plausible" interpretat ions can be defined and searched
For exam ple, in the case of Figure 27(a), if Lg and L^g are known to take the same label, th
num ber of possibi li t ies reduces from 8 to 3 (see Figure 29).
Furt her , heurist ics concerning surface orientati ons can be used to provide pref erence
f or i nt erpr etat ions. For example, if interpretat ions in Sj rj (i.e., all the junct ion label s gi ve
are trihedral ones) are to be chosen first, then in the cases of Figures 28, 30, and 31, the
int erp ret at io ns corresponding to a cube, a solid truncated pyramid, and three t wisted bar
are se lect ed, respect ivel y. Another heuristic is that parallel lines in the pict ure are als
pr ef er ab l y parallel in the scene; this would pref er the interpretati ons corresponding to a
or di na ry box and a W-folded paper over others in Figures 29 and 32 , resp ect ively. Thi
"parall el-in-the-picture/ paralIel-in-the-sceneM heuristic seems ver y powerful for pi cture
wh ich do not include strong perspect ive distort ions. The subject of applying the Origam
t heo ry t o real image understanding is furt her treated in [Kanade, 19 78 J
Figu re 39 Matched T configurat ion.
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VI Conclusion
The theory of the Origami worl d (up -to-3-sur face juncti ons) has been devel oped . Th
contributions of this paper might be the following:
(1) The concept of selecting surfaces as basic components of the worl d, ra ther t han th
conventional solid polyhedra;
(2) The enumerat ion of the up-t o-3-sur face junction labels;
(3) The use of links to capture the global relationships of regions in the form of
Surface Connection Graph;
(4) The f il tering procedure defined on the spanning angles;
(5) The discussion of relationships among various worlds dealt wi th in pr ior work o
polyhedral scene analysis.
It seems that the Origami wor ld defines a subset of int erpretat ions whi ch a r
interesting both from the standpoint of psychological perception of shapes and from that o
practical analysis of region segmented pictures.
Acknowledgements
I woul d l ike to thank John Kender, Allen Newell , Raj Reddy, and Steven Shafer for ve
stimulating discussions in the development of the theory presented in this paper.
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APPENDIX ORIGAMI JUNCTION DICTIONARYKanad
JUNCTION TYPE - L
L I * L2* L3* L4*
Al
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