1 - CS, Technion
TECHNION - Israel Institute of Technology
Technion - Computer Science Department - Tehnical Report CS0616 - 1990
Computer Science Department
SELF-SIMILARITY PROPERTIES OF DIGITIZED STRAIGHT LINES by
A.M. Bruckstein A.M.
Technical Report #616 Technical March 1990
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
, SELF-SIMILARITY PROPERTIES OF DIGITIZED STRAIGHT LINES
Alfred M. Bruckstein
Department of Computer Science Science
Technion - Israel Institute of Technology Technology
Haifa 32000. Israel Israel
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March
19~9 .
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~~ ABSTRACT
A digitized straight line is the boundary of a linearly separable dichotomy of the lattice points in the plane. This paper presents several interesting self-similarity properties of chain-codes of digitized straight lines. These properties are recognized to express the reg invariance of the digital straightness property under re-encoding of digitized lines on regular subgrids embedded in the integer lattice.
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1. INTRODUCTION A digitized straight line is defined as the boundary of a linearly separable dichotomy of the set of points with integer coordinates, Z2
={(i,j)
I i,j E Z}, in the plane. The
boundary points of the dichotomy induced by a line with slope m and intercept n, y =m x +n, are
L(m,n)
={
(i,hi) lie lie Z, hi
=Lmi+nJ}
(1.1)
nonde Without loss of generality let us assume that m > 0, so that the sequence hi is a nondecreasing sequence of integers. Associate to the set of boundary points L (m, n) a string of two symbols, 0 and 1, coding the sequence of differences hi+l-hi' as follows
• (1.2)
where
o
if hi+l-hi =0
Ci(m,n) = Ci(m,n)
(1.3)
if hi+l-hi = k and 1k means 11 ... 1 with k 1'so C (m,n) is called the chain code of the line L (m,n). Note that the sequence C (m,n) can be uniquely parsed into its components Ci(m,n), since a separator must precede every 01 string and follow every run of 1's and each of the remaining O's must be a single component, as well.
Cle~ly,
given some hio value and
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C (m,n), the entire sequence hi can be recovered. The graphical meaning of the chain
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code associated to L(m,n) is depicted in Figure 1. The set of points L(m,n) uniquely detennines the slope of the line m. Indeed, L(m,n) ==L(m',n') implies m =m', since otherwise the vertical distance between y
=mx+n and y =m'x+n' would become unbounded at x ~ ±
00
and their hi sequences
would differ starting at some large enough i's. Furthennore, if m is irrational we have, by a classical result, that the vertical intercepts of y
=nix +n modulo
1 are dense in [0, I].
Therefore for every E > 0 there exist i 0 and j 0 such that
and changing n by
E
mio+n -Lmio+nJ < £
(1.4a)
mjo+n -Lmjo+nJ > 1-£
(l.4b)
would result in a change in L (m,n). Therefore for irrational m,
L(m,n) uniquely detennines both m and n. If m is rational there exist oniy a finite set of
distinct vertical intercepts of y = mx +n modulo 1, therefore n is determined only up to an interval and the length of the worst interval of uncertainly for n depends on the minimal
p/q representation of m. This discussion also shows that the chain code C(m,n) deterdeter mines m uniquely, and if m is irrational, n is also determined modulo 1, since we clearly have C(m,n) == C(m,n+l). This paper deals with some interesting general properties of the sequences C(m,n), chain-codes of linear dichotomies. The paper is organized as follows: in the next section we overview some elementary properties of chain-codes of digitized straight edges and Sections 3 and 4 present a series of striking, self-similarity properties and their proofs. Section 5 discusses some previously known results of this type and highlights the ~stori cal developments in the research of digitized straight lines.
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2. THE BASIC PROPERTIES OF CHAIN CODES 2.
Technion - Computer Science Department - Tehnical Report CS0616 - 1990
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From the definition of chain-codes C (m,n) we obtain several immediate and basic properties a sequences of 0/1 symbols must have in order to correspond to a straight edge.
In the case of m < 1 the difference
hi+l-hi =lm(i+l)+nJ - lmi+nJ
(2.1)
can only be either 0 or 1. In this case the ch$-code of a digitized line has runs of O's separated by single 1's, and the O's occur in runs with length detennined by the number on integer coordinates that fall within the intervals determined on the x-axis by the points· defined by
m Xi +n = ieZ,
i.e.,
ii n ··=--=--m m m
XJ
••
(2.2)
The intervals Xi+I-xi have a constant length of 11m and therefore the number of integer coordinates covered can be (sec Figure 2a) either Pi =lllmJ or Pi =lllmJ + 1. Therefore, if m < 1, C (m, n) looks like
C (m,n) = ... 1 efl where Pi
E
l(f:1 l(f3
1 ... ...
(2.3a)
flllmJ, LlImJ + 1 }.
In case m > 1, the difference hi+I-hi
= lm (i +1) + nJ - llni+n J is always greater
than 1, and the chain-code C(m,n) therefore has runs of 1'5 separated by single 0'5. Since lm+mi+nJ - Lmi+nJ equals the number of integer coordinates between the
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
integral values in consecutive intervals of length m, see Figure 2b. This shows that the run-lengths Pi in this case, will be either Pi
=Lmj =LmJ
or
Lmj +1. LmJ +1.
Therefore if m > 1, the chain-code C (m,n) has the form form
C (m,n) = .. , 0 I P1 0 I P2 0 I P, 0 ... with Pi
E
(2.3b)
{Lmj, Lmj +1}. {LmJ, LmJ
The question that immediately arises is the following: is there any order or pattern in the appearance of the two values for the run length of the symbols 0 or 1, i.e. in the . sequences {Pi} that arise from "chain coding" digitized straight lines? The results of the next section address .and settle this and other questions concerning the patterns (and patpat terns of 'patterns, and patterns of patterns of patterns, etc.) that appear in the chain codes
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of digitized lines.
3. TRANSFORMATION RULES AND SELF·SIMH..ARITY RESULTS Suppose we are given the chain code of a digitized straight boundary C (m,n). We know that C (m,n) is a sequenc~ composed of two symbols,
°
and 1, and that it looks
either like (2.3a) or (2.3b), thus it has the following general form (3.1)
where Pi e {p,p +1 }, P
E
Z, and A, 0 stand for either 0,1 or 1,0, respectively.
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We next define several transformation rules on two symbol, or AID, sequences of the type (3.1), transformations transformationS that yield new AID sequences.
RULE X: interchange the symbols A and D (i.e. A ~ D and D ~ A). Application of X to a chain code C(m,n) yields a
ne~
sequence of symbols, with
O's replacing the l's and 1•s replacing the O's of the original sequence.
RULE S: replace every D A sequence by A. Application
. .. AD P, A 0
PI+l
of
the
S-transformation
to
a
chain
code
of
the
form
A "', yields a sequence of the same type with the run length Pi
replaced by Pi-I. Applying S, p times yields the next transformation rule.
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RULE SP: replace DP A by A and DP+l A by D A. Notice that, in contrast to the transformation rules X and S, this rule depends on the
{Pi! sequence, Le. it is adapted to the given pattern (3.1). Indeed we can apply the SS transformation successively at most p times where p is the minimal value of Pi'S. Mter that, we need to do an X transformation in order to bring the sequence of symbols to the form (3.1).
RULE R: replace DP A by A and DP+l A by 0. We may view the action of Rasa result of applying SP first then replacing 0 A by
o . This rule too is adapted to the sequence on which it operates. "
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The next transformation role is a bit different, since it replaces symbols in a way
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that depends on the neighborhood or the "context".
RULE T: replace D 11 by D and the D's followed by a D by 011. Application of role T has the effect of putting a 11 between every consecutive pair of D's and removing all the l1's appearing in the original sequence. For example the sequence ... 01100001100011000110 ... will be mapped, under T to ... 00110110110011011001101100 ...
Up to this point the transfonnation rules were completely specified by rather simple local symbol replacement laws. The next two transformations, or rather classes of transformation rules, also require the setting of an initial position and a bilateral parsing for the generation of the transformed sequences.
V-RULES: given the sequence of 11 0 choose a 11 symbol as an initial position, then to the right and to the left of the chosen 11 delete batches of Q-l consecutive
A's. This transformation has the effect of joining together (from the starting position) Q consecutive 0 -runs. The sequence 11 0
••
PI-1l •••
A0
Pi-1
~ 0 PI
11 0
PI+1
11 . .. 0
PI-t/l-1
A
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
will be mapped to
if the !:J. preceding 0 PI is chosen as the initial position. Therefore if a A I 0 -chain code
sequence of type (3.1) is specified by the 0 -run length sequence {p;}
ieN
a V-
transformation as defined above will produce a sequence of type (3.1) specified by
{
for a given io and a given integer Q
Pio+nQ+Pio+nQ+l +...+Pio+{n+l>a-1} neN
~ 1.
H-RULES: given the sequence of !:J. 0 symbols choose a starting point between
two consecutive symbols and p~e the sequence to the right and left of the starting subse point, counting the number of D's seen. After seeing PO's, replace the subsequence by one 0 followed by the number of !:J.'s encountered while accumulating the P, 0 ·s. In counting the A's encountered, apply the following rules: 1) when parsing to the right: if the p-th 0 symbol is followed by a A count this A too and start accumulating the next batch of P symbols after it and 2) when parsing to the left: if the P-th 0 symbol is preceded by
~
do not count this A and start accumulataccumulat
ing the next batch of P 0's right away. As an example, consider applying an H-transformation to the sequence below, with the indicated initial position,
...
D~DDDf!,.DDDAtDDDD~DDDADDDADOO~O...
and parameter P=7. We obtain the parsing
... tD~DDD~DDD~tOOOO~DDO~tOOD~DOO~Dt
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
that yields the output
..
The same initial conditions with parameter P = 3 provide the parsing
...
D~i OOO~i OOD~i DOO~i DOOr O~OOr D~DOr O~OOr D~OOr O~OOr D~OOr O~O O~D
...
and an output sequence
...
iD~i D~i
Dr O~i
O~i O~t
We have" defined seven rules for transforming ~/O sequences into new ~/D sequences. The first five of them were uniquely specified in tenns of local string replacereplace ment rules, the last two being classes of transformations that depend on the choice of an initial positions for parsing and have to be further specified by an arbitrarily chosen integer (Q or P). The main results that we prove in this paper are the following:
RESULT I: Given a ~ 0 sequence of type (3.1), the new sequence produced by applying to it any of the transformations X, S , SP, R or T, is the chain code of a digitized straight line if and only if the original sequence was the chain code of a digitized straight line.
RESULT 2: If a ~ 0 sequence is the chain code of a digitized straight line, then the H sequences obtained from it by applying any transformation according to the Hrules, or V-rules, are also chain codes of digitized straight lines.
Notice that, for the X, S, SP, R, and T-transformation rules we have stronger results than
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the corresponding results for the class of HN rules. The reason for this will become ,j,.
obvious from the proofs, in the next section. The properties stated above will be called
self-similarity results, since what we have is that a given chain code pattern generates, under a given set of transformation rules, new
pattern~
of the same type: chain codes of
digitized straight lines.
4. PROOFS VIA EMBEDDED LATTICES 4. We shall argue that the chain code transformations defined in the previous section are re-encodings of digitized straight lines on regular lattices of points, embedded into the integer lattice Z2. This observation, combined with the fact that the embedded latlat tices are generated by affine coordinate transformations, readily yield the results stated in Section 3. Indeed, choose any two linearly independent basis vectors Bland B 2 with integer entries and a lattice point (io,jo) for the origin
00. Define a regular
embedded
lattice of points as follows
E' The straight line y
={
(io.Jo) + is l+jB, I (i.j)
E
zl
(4.1)
=mx+n defines a dichotomy of the points of Z2, and also of the
points of E 2 c Z2. By definition, there exists an affine transformation that maps lattice E2 the embedding back into Z2, Le. the point (io,jo) + iB 1+jB 2
E
E2 into (i,j)
E
Z2,
and the same transformation maps the line y = mx+n into some new line Y =MX+N, on the transformed plane. Since the points (i o,j 0) + iB 1+jB 2 from the original (x,y)-plane
•z
map into (i,j) the transformation from (X,Y) into (x,y) is
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(4.2) and therefore the inverse mapping from (x,y) to (X,Y) is
[~)
=[
Bf BI
r [[~) -[~:) ].
(4.3)
From these transformations the mapping of the line parameters (m,n) into (M,N) can also be readily computed, in terms of BIB 2 and
no.
After performing the transformation (4.3) the line Y =MX+N can be chain-coded with respect to the lattice Z2 (that now is the image of E 2) and the resulting chain-code will somehow be related to the chain-code of y
=mx+n defined on the original grid Z 2 •
The key observation, proving all the results stated in Section 3, is that the transformations introduced in the previous section represent rather straightforward re-encodings of digitized lines with respect to suitably chosen embedded lattices E 2 • Let us next show the choices of basis vectors that lead to each of the sequence transformations discussed in Section 3 (see Figures 3 and 4).
1) The X-transformation rule, the interchange of A and 0 symbols is clearly accomaccom plished by the coordinate-change mapping that takes (i,j) into U,i). Here B 1 = [0,1] and
B2
=[1,0] and we have thaty =mx+n maps into Y =(lIm)X-(nlm) under the transfortransfor
[0 1]
. matnx . Mx = 1 0 . mation
2) The S-role which reduces every integer of the (Pi) sequence by 1 is induced by
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the mapping that considers a 0 step as ~ step in the B 1
=[1,0] direction, but a combined combined
o A-step as the unit step in the B2 = [1,1]-direction (see Figure 3a). Therefore, the SS-
transformation
matrix
is
M - [1 1] -1 s- 01
= [10 -1] r
and
y =mx+n
maps
into
Y =X m/(l-m) + nl(l-m).
3) The adaptive SP-transformation rule which replaces DP A by A and DP+l A by
o A, A. corresponds to choosing B 1 =[1,0] and B 2 =[p, 1] (see Figure 3b). The transfortransfor
y =mx+n
is
transfonned
into
Y =Xm/(l-pm) + nl(l-pm).
Note
that,
if'
m < 1, I. P =LlImJ and we denote the fractional part of 11m by a( =11m - LlImJ ), ). we have m/(l-mp) = lIa > 1. This shows that one SP-transfofI!l8tion, that is adapted to the run-length of the 0 -symbols replaces the'slope m with (11m - LlImJ ••
r
l
.
Therefore,
repeated application of this adapted transformation followed by an X-transformation will produce a sequence of slopes recursively given by mt = lImt-l - LlImt-d •, mo =m. Hence. the sequence of adapted "exponents" of the corresponding SP-transformations Hence,
Pt =L 1/mtJ are the sequence of integers of the continued fraction representation of mo, mo. 1
1
(4.4)
mo=---~---
1
Po+---- 1
4) The transformation rule R maps B1
=[P+ 1,1]
and
B 2 = [P, [Po 1]
oP A into A and
(Figure
3c).
The
Op+IA into O. Therefore
transformation
matrix
is
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
- 1313 MR
= [P+l 1
p] -1 1
=[1-1
-p] p+l
and an original line y =mx+n is mapped into
Y =X[(p+l)m-l]/(l-pm) + m/(I-pm). Note here that in terms of a = 11m -lllmJ the new slope is
(1~)/a,
when m < 1.
5) The last of this class of transfonnations, rule T, replaces D!!. by 0 's and 0 's followed by a 0 by D!!. 'so We may view this transformation as a sequence of two maps: the first one replacing DP+l !!. by 0 and OP !!. by!!., by the adapted rule R, the second replacing 0 by D!!. 0 !!. 0 ... !!. 0 with (P +1) 0 's and !!. by D!!. 0 !!. 0 ... !!. 0 with pO's. This would imply that we first do an R-transformation via the matrix
transformations we obtain
(4.5) and this is not surprising. Indeed, D!!. is mapped by B 1 = [1 1] into one 0 step, but a 0 followed by another 0 will have to be mapped into a sequence of two steps, B 2 B 1, the first one being B 2 =[0, -1] (see ·Figure 3d). We readily see from the MT transformation that y = mx+n maps into Y = (l-m)X -no Therefore the slopes of the two lines add to 1. Indeed, "summing up" the two sequences in the sense of placing a !!. whenever there exists a !!. in either the original, or the T-transformed chain code, we get the sequence ...0 !!. 0 !!. 0 !!. 0 ... , which represents the lines of the type y
=x +n.
Up to this point, all the transformation matrices, whether adapted to the chain code parameter p or not, were matrices with integer entries and had the property that
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
det (M)
=±1. This implied that the matrices had inyerses with integer entries and, as a
consequence, the embedded lattice. E 2 was simply'a "reorganization" or "relabeling" of the entire integer lattice Z2. In mathematical terms, unimodular lattice transformations deter are isomorphisms of the two dimensional lattice. The 2x2 integer matrices with determinant ±1 (called unimodular matrices) form a well known group called GL(2,Z) and this group
i~ finitely generated by the matrices
[21 AJ [A n[ f
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
Computer Science Department
SELF-SIMILARITY PROPERTIES OF DIGITIZED STRAIGHT LINES by
A.M. Bruckstein A.M.
Technical Report #616 Technical March 1990
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
, SELF-SIMILARITY PROPERTIES OF DIGITIZED STRAIGHT LINES
Alfred M. Bruckstein
Department of Computer Science Science
Technion - Israel Institute of Technology Technology
Haifa 32000. Israel Israel
~
March
19~9 .
~
~~ ABSTRACT
A digitized straight line is the boundary of a linearly separable dichotomy of the lattice points in the plane. This paper presents several interesting self-similarity properties of chain-codes of digitized straight lines. These properties are recognized to express the reg invariance of the digital straightness property under re-encoding of digitized lines on regular subgrids embedded in the integer lattice.
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
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1. INTRODUCTION A digitized straight line is defined as the boundary of a linearly separable dichotomy of the set of points with integer coordinates, Z2
={(i,j)
I i,j E Z}, in the plane. The
boundary points of the dichotomy induced by a line with slope m and intercept n, y =m x +n, are
L(m,n)
={
(i,hi) lie lie Z, hi
=Lmi+nJ}
(1.1)
nonde Without loss of generality let us assume that m > 0, so that the sequence hi is a nondecreasing sequence of integers. Associate to the set of boundary points L (m, n) a string of two symbols, 0 and 1, coding the sequence of differences hi+l-hi' as follows
• (1.2)
where
o
if hi+l-hi =0
Ci(m,n) = Ci(m,n)
(1.3)
if hi+l-hi = k and 1k means 11 ... 1 with k 1'so C (m,n) is called the chain code of the line L (m,n). Note that the sequence C (m,n) can be uniquely parsed into its components Ci(m,n), since a separator must precede every 01 string and follow every run of 1's and each of the remaining O's must be a single component, as well.
Cle~ly,
given some hio value and
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
C (m,n), the entire sequence hi can be recovered. The graphical meaning of the chain
•
code associated to L(m,n) is depicted in Figure 1. The set of points L(m,n) uniquely detennines the slope of the line m. Indeed, L(m,n) ==L(m',n') implies m =m', since otherwise the vertical distance between y
=mx+n and y =m'x+n' would become unbounded at x ~ ±
00
and their hi sequences
would differ starting at some large enough i's. Furthennore, if m is irrational we have, by a classical result, that the vertical intercepts of y
=nix +n modulo
1 are dense in [0, I].
Therefore for every E > 0 there exist i 0 and j 0 such that
and changing n by
E
mio+n -Lmio+nJ < £
(1.4a)
mjo+n -Lmjo+nJ > 1-£
(l.4b)
would result in a change in L (m,n). Therefore for irrational m,
L(m,n) uniquely detennines both m and n. If m is rational there exist oniy a finite set of
distinct vertical intercepts of y = mx +n modulo 1, therefore n is determined only up to an interval and the length of the worst interval of uncertainly for n depends on the minimal
p/q representation of m. This discussion also shows that the chain code C(m,n) deterdeter mines m uniquely, and if m is irrational, n is also determined modulo 1, since we clearly have C(m,n) == C(m,n+l). This paper deals with some interesting general properties of the sequences C(m,n), chain-codes of linear dichotomies. The paper is organized as follows: in the next section we overview some elementary properties of chain-codes of digitized straight edges and Sections 3 and 4 present a series of striking, self-similarity properties and their proofs. Section 5 discusses some previously known results of this type and highlights the ~stori cal developments in the research of digitized straight lines.
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2. THE BASIC PROPERTIES OF CHAIN CODES 2.
Technion - Computer Science Department - Tehnical Report CS0616 - 1990
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From the definition of chain-codes C (m,n) we obtain several immediate and basic properties a sequences of 0/1 symbols must have in order to correspond to a straight edge.
In the case of m < 1 the difference
hi+l-hi =lm(i+l)+nJ - lmi+nJ
(2.1)
can only be either 0 or 1. In this case the ch$-code of a digitized line has runs of O's separated by single 1's, and the O's occur in runs with length detennined by the number on integer coordinates that fall within the intervals determined on the x-axis by the points· defined by
m Xi +n = ieZ,
i.e.,
ii n ··=--=--m m m
XJ
••
(2.2)
The intervals Xi+I-xi have a constant length of 11m and therefore the number of integer coordinates covered can be (sec Figure 2a) either Pi =lllmJ or Pi =lllmJ + 1. Therefore, if m < 1, C (m, n) looks like
C (m,n) = ... 1 efl where Pi
E
l(f:1 l(f3
1 ... ...
(2.3a)
flllmJ, LlImJ + 1 }.
In case m > 1, the difference hi+I-hi
= lm (i +1) + nJ - llni+n J is always greater
than 1, and the chain-code C(m,n) therefore has runs of 1'5 separated by single 0'5. Since lm+mi+nJ - Lmi+nJ equals the number of integer coordinates between the
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
integral values in consecutive intervals of length m, see Figure 2b. This shows that the run-lengths Pi in this case, will be either Pi
=Lmj =LmJ
or
Lmj +1. LmJ +1.
Therefore if m > 1, the chain-code C (m,n) has the form form
C (m,n) = .. , 0 I P1 0 I P2 0 I P, 0 ... with Pi
E
(2.3b)
{Lmj, Lmj +1}. {LmJ, LmJ
The question that immediately arises is the following: is there any order or pattern in the appearance of the two values for the run length of the symbols 0 or 1, i.e. in the . sequences {Pi} that arise from "chain coding" digitized straight lines? The results of the next section address .and settle this and other questions concerning the patterns (and patpat terns of 'patterns, and patterns of patterns of patterns, etc.) that appear in the chain codes
•
of digitized lines.
3. TRANSFORMATION RULES AND SELF·SIMH..ARITY RESULTS Suppose we are given the chain code of a digitized straight boundary C (m,n). We know that C (m,n) is a sequenc~ composed of two symbols,
°
and 1, and that it looks
either like (2.3a) or (2.3b), thus it has the following general form (3.1)
where Pi e {p,p +1 }, P
E
Z, and A, 0 stand for either 0,1 or 1,0, respectively.
IN
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
We next define several transformation rules on two symbol, or AID, sequences of the type (3.1), transformations transformationS that yield new AID sequences.
RULE X: interchange the symbols A and D (i.e. A ~ D and D ~ A). Application of X to a chain code C(m,n) yields a
ne~
sequence of symbols, with
O's replacing the l's and 1•s replacing the O's of the original sequence.
RULE S: replace every D A sequence by A. Application
. .. AD P, A 0
PI+l
of
the
S-transformation
to
a
chain
code
of
the
form
A "', yields a sequence of the same type with the run length Pi
replaced by Pi-I. Applying S, p times yields the next transformation rule.
•
RULE SP: replace DP A by A and DP+l A by D A. Notice that, in contrast to the transformation rules X and S, this rule depends on the
{Pi! sequence, Le. it is adapted to the given pattern (3.1). Indeed we can apply the SS transformation successively at most p times where p is the minimal value of Pi'S. Mter that, we need to do an X transformation in order to bring the sequence of symbols to the form (3.1).
RULE R: replace DP A by A and DP+l A by 0. We may view the action of Rasa result of applying SP first then replacing 0 A by
o . This rule too is adapted to the sequence on which it operates. "
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The next transformation role is a bit different, since it replaces symbols in a way
..
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that depends on the neighborhood or the "context".
RULE T: replace D 11 by D and the D's followed by a D by 011. Application of role T has the effect of putting a 11 between every consecutive pair of D's and removing all the l1's appearing in the original sequence. For example the sequence ... 01100001100011000110 ... will be mapped, under T to ... 00110110110011011001101100 ...
Up to this point the transfonnation rules were completely specified by rather simple local symbol replacement laws. The next two transformations, or rather classes of transformation rules, also require the setting of an initial position and a bilateral parsing for the generation of the transformed sequences.
V-RULES: given the sequence of 11 0 choose a 11 symbol as an initial position, then to the right and to the left of the chosen 11 delete batches of Q-l consecutive
A's. This transformation has the effect of joining together (from the starting position) Q consecutive 0 -runs. The sequence 11 0
••
PI-1l •••
A0
Pi-1
~ 0 PI
11 0
PI+1
11 . .. 0
PI-t/l-1
A
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
will be mapped to
if the !:J. preceding 0 PI is chosen as the initial position. Therefore if a A I 0 -chain code
sequence of type (3.1) is specified by the 0 -run length sequence {p;}
ieN
a V-
transformation as defined above will produce a sequence of type (3.1) specified by
{
for a given io and a given integer Q
Pio+nQ+Pio+nQ+l +...+Pio+{n+l>a-1} neN
~ 1.
H-RULES: given the sequence of !:J. 0 symbols choose a starting point between
two consecutive symbols and p~e the sequence to the right and left of the starting subse point, counting the number of D's seen. After seeing PO's, replace the subsequence by one 0 followed by the number of !:J.'s encountered while accumulating the P, 0 ·s. In counting the A's encountered, apply the following rules: 1) when parsing to the right: if the p-th 0 symbol is followed by a A count this A too and start accumulating the next batch of P symbols after it and 2) when parsing to the left: if the P-th 0 symbol is preceded by
~
do not count this A and start accumulataccumulat
ing the next batch of P 0's right away. As an example, consider applying an H-transformation to the sequence below, with the indicated initial position,
...
D~DDDf!,.DDDAtDDDD~DDDADDDADOO~O...
and parameter P=7. We obtain the parsing
... tD~DDD~DDD~tOOOO~DDO~tOOD~DOO~Dt
.
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
that yields the output
..
The same initial conditions with parameter P = 3 provide the parsing
...
D~i OOO~i OOD~i DOO~i DOOr O~OOr D~DOr O~OOr D~OOr O~OOr D~OOr O~O O~D
...
and an output sequence
...
iD~i D~i
Dr O~i
O~i O~t
We have" defined seven rules for transforming ~/O sequences into new ~/D sequences. The first five of them were uniquely specified in tenns of local string replacereplace ment rules, the last two being classes of transformations that depend on the choice of an initial positions for parsing and have to be further specified by an arbitrarily chosen integer (Q or P). The main results that we prove in this paper are the following:
RESULT I: Given a ~ 0 sequence of type (3.1), the new sequence produced by applying to it any of the transformations X, S , SP, R or T, is the chain code of a digitized straight line if and only if the original sequence was the chain code of a digitized straight line.
RESULT 2: If a ~ 0 sequence is the chain code of a digitized straight line, then the H sequences obtained from it by applying any transformation according to the Hrules, or V-rules, are also chain codes of digitized straight lines.
Notice that, for the X, S, SP, R, and T-transformation rules we have stronger results than
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
the corresponding results for the class of HN rules. The reason for this will become ,j,.
obvious from the proofs, in the next section. The properties stated above will be called
self-similarity results, since what we have is that a given chain code pattern generates, under a given set of transformation rules, new
pattern~
of the same type: chain codes of
digitized straight lines.
4. PROOFS VIA EMBEDDED LATTICES 4. We shall argue that the chain code transformations defined in the previous section are re-encodings of digitized straight lines on regular lattices of points, embedded into the integer lattice Z2. This observation, combined with the fact that the embedded latlat tices are generated by affine coordinate transformations, readily yield the results stated in Section 3. Indeed, choose any two linearly independent basis vectors Bland B 2 with integer entries and a lattice point (io,jo) for the origin
00. Define a regular
embedded
lattice of points as follows
E' The straight line y
={
(io.Jo) + is l+jB, I (i.j)
E
zl
(4.1)
=mx+n defines a dichotomy of the points of Z2, and also of the
points of E 2 c Z2. By definition, there exists an affine transformation that maps lattice E2 the embedding back into Z2, Le. the point (io,jo) + iB 1+jB 2
E
E2 into (i,j)
E
Z2,
and the same transformation maps the line y = mx+n into some new line Y =MX+N, on the transformed plane. Since the points (i o,j 0) + iB 1+jB 2 from the original (x,y)-plane
•z
map into (i,j) the transformation from (X,Y) into (x,y) is
.
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(4.2) and therefore the inverse mapping from (x,y) to (X,Y) is
[~)
=[
Bf BI
r [[~) -[~:) ].
(4.3)
From these transformations the mapping of the line parameters (m,n) into (M,N) can also be readily computed, in terms of BIB 2 and
no.
After performing the transformation (4.3) the line Y =MX+N can be chain-coded with respect to the lattice Z2 (that now is the image of E 2) and the resulting chain-code will somehow be related to the chain-code of y
=mx+n defined on the original grid Z 2 •
The key observation, proving all the results stated in Section 3, is that the transformations introduced in the previous section represent rather straightforward re-encodings of digitized lines with respect to suitably chosen embedded lattices E 2 • Let us next show the choices of basis vectors that lead to each of the sequence transformations discussed in Section 3 (see Figures 3 and 4).
1) The X-transformation rule, the interchange of A and 0 symbols is clearly accomaccom plished by the coordinate-change mapping that takes (i,j) into U,i). Here B 1 = [0,1] and
B2
=[1,0] and we have thaty =mx+n maps into Y =(lIm)X-(nlm) under the transfortransfor
[0 1]
. matnx . Mx = 1 0 . mation
2) The S-role which reduces every integer of the (Pi) sequence by 1 is induced by
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the mapping that considers a 0 step as ~ step in the B 1
=[1,0] direction, but a combined combined
o A-step as the unit step in the B2 = [1,1]-direction (see Figure 3a). Therefore, the SS-
transformation
matrix
is
M - [1 1] -1 s- 01
= [10 -1] r
and
y =mx+n
maps
into
Y =X m/(l-m) + nl(l-m).
3) The adaptive SP-transformation rule which replaces DP A by A and DP+l A by
o A, A. corresponds to choosing B 1 =[1,0] and B 2 =[p, 1] (see Figure 3b). The transfortransfor
y =mx+n
is
transfonned
into
Y =Xm/(l-pm) + nl(l-pm).
Note
that,
if'
m < 1, I. P =LlImJ and we denote the fractional part of 11m by a( =11m - LlImJ ), ). we have m/(l-mp) = lIa > 1. This shows that one SP-transfofI!l8tion, that is adapted to the run-length of the 0 -symbols replaces the'slope m with (11m - LlImJ ••
r
l
.
Therefore,
repeated application of this adapted transformation followed by an X-transformation will produce a sequence of slopes recursively given by mt = lImt-l - LlImt-d •, mo =m. Hence. the sequence of adapted "exponents" of the corresponding SP-transformations Hence,
Pt =L 1/mtJ are the sequence of integers of the continued fraction representation of mo, mo. 1
1
(4.4)
mo=---~---
1
Po+---- 1
4) The transformation rule R maps B1
=[P+ 1,1]
and
B 2 = [P, [Po 1]
oP A into A and
(Figure
3c).
The
Op+IA into O. Therefore
transformation
matrix
is
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-
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
- 1313 MR
= [P+l 1
p] -1 1
=[1-1
-p] p+l
and an original line y =mx+n is mapped into
Y =X[(p+l)m-l]/(l-pm) + m/(I-pm). Note here that in terms of a = 11m -lllmJ the new slope is
(1~)/a,
when m < 1.
5) The last of this class of transfonnations, rule T, replaces D!!. by 0 's and 0 's followed by a 0 by D!!. 'so We may view this transformation as a sequence of two maps: the first one replacing DP+l !!. by 0 and OP !!. by!!., by the adapted rule R, the second replacing 0 by D!!. 0 !!. 0 ... !!. 0 with (P +1) 0 's and !!. by D!!. 0 !!. 0 ... !!. 0 with pO's. This would imply that we first do an R-transformation via the matrix
transformations we obtain
(4.5) and this is not surprising. Indeed, D!!. is mapped by B 1 = [1 1] into one 0 step, but a 0 followed by another 0 will have to be mapped into a sequence of two steps, B 2 B 1, the first one being B 2 =[0, -1] (see ·Figure 3d). We readily see from the MT transformation that y = mx+n maps into Y = (l-m)X -no Therefore the slopes of the two lines add to 1. Indeed, "summing up" the two sequences in the sense of placing a !!. whenever there exists a !!. in either the original, or the T-transformed chain code, we get the sequence ...0 !!. 0 !!. 0 !!. 0 ... , which represents the lines of the type y
=x +n.
Up to this point, all the transformation matrices, whether adapted to the chain code parameter p or not, were matrices with integer entries and had the property that
---
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Technion - Computer Science Department - Tehnical Report CS0616 - 1990
det (M)
=±1. This implied that the matrices had inyerses with integer entries and, as a
consequence, the embedded lattice. E 2 was simply'a "reorganization" or "relabeling" of the entire integer lattice Z2. In mathematical terms, unimodular lattice transformations deter are isomorphisms of the two dimensional lattice. The 2x2 integer matrices with determinant ±1 (called unimodular matrices) form a well known group called GL(2,Z) and this group
i~ finitely generated by the matrices
[21 AJ [A n[ f
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