Operations on Value Sets

The operations on and between elements of a given value set Å are the usual

elementary operations associated with Å . Thus, if ÅÛÚÐ6ÁÄ–Ê{Ä%Á  Ì

Ñ , then the binary

operations are the usual arithmetic and logic operations of addition, multiplication, and maximum, and the complementary operations of subtraction, division, and minimum. If

ż-Ë , then the binary operations are addition, subtraction, multiplication, and division.

Similarly, we allow the usual elementary unary operations associated with these sets such as the absolute value, conjugation, as well as trigonometric, logarithmic and exponential functions as these are available in all higher-level scientific programming languages.

For the set Ê

Ó

Î we need to extend the arithmetic and logic operations of Ê as

follows: Ü ÇݦÉvÈÞ\¼¶ÝzÉvÈÞ£Ç Ü ¼ÉvÈ Ü Ú(Ê‡Ò Î Ü ÇßÈà¼ÔÈQÇ Ü ¼È Ü ÚáÊ Î Ý¦ÉvÈßÞ%ÇÈh¼ÈÇÙÝzÉvÈÞX¼¶ÉvÈ Üâ ݦÉvÈÞX¼®Ý¦ÉvÈÞ âÜ ¼ Ü Ü ÚáÊ Ó Î

Note that the elementÉbÈ acts as a null element in the system ݞÊXÓ Î Ä

â

Ä Ç0Þ if we

view the operation + as multiplication and the operation

â

as addition. The same cannot be said about the elementÈ in the systemÝZÊ Ó Î3āã{Ä Ç0Þ sinceÝzÉvÈÞ.ǓÈä¼ÈˆÇLÝ«ÉvÈßÞ²¼®ÉvÈ .

In order to remedy this situation we define the dual structureÝZÊ Ó Î Ä ãåÄ Çbæ†Þ ofÝZÊ Ó Î Ä

â Ä Ç0Þ as follows: Ü Ç æJç ¼ Ü Ç ç Ü Ä ç ÚáÊ Ü Ç æ ÝzÉvÈލ¼®Ý¦ÉvÈßÞ£Ç æ Ü ¼¶ÉvÈ Ü Ú(Ê‡Ò Î Ü Ç æ Èh¼ÈQÇ æ Ü ¼È Ü Ú(ÊXΠݝÉvÈÞ–Ç æ Èà¼ÔÈÛÇ æ ÝzÉvÈލ¼ÔÈ Ü ã3Èh¼ˆÈã Ü ¼ Ü Ü ÚáÊ Ó Î

1. 3 Value Sets 11

Now the elementèœé acts as a null element in the systemêzëì–í3îï{î èðòñ|ó Observe, however,

that the dual additions è and èbð introduce an asymmetry between ôbé and è0éó The

resultant structure êžë ì£í î õ{î ï{î èLîvèð†ñ is known as a bounded lattice ordered group [1].

Dual structures provide for the notion of dual elements. For each ö¹÷dë

ì£í we

define its dual or conjugate öaø by ö6ø0ùàôåö , where ô0ê«ôbéßñ‡ùRé . The following duality

laws are a direct consequence of this definition:

ú.ûJü úýoþoüvþÿ ý ú ü úý|ü þ ÿ ý þ þ

and

.

Closely related to the additive bounded lattice ordered group described above is the multiplicative bounded lattice ordered group žë

í

î õ{î ïåî îð . Here the dual ð of

ordinary multiplication is defined as

ð ù î ÷áë ùÔë

with both multiplicative operations extended as follows:

3éhùé ùé ÷(ë í ð éhùé ð ùé ÷áë í !éàùÔé"#œù$ ! ð éhùé% ð Øùé

Hence, the element 0 acts as a null element in the system ë&('

í

î õ{î and the element è0é acts as a null element in the system ë)*'

í

î ï{î ð

. The conjugate

ö6ø of an element ö1÷ ë +'

í of this value set is defined by

ö ø-, . ö0/21 if ö&÷'ë if öùÔè0é èØé if öù3 ó

Another algebraic structure with duality which is of interest in image algebra is the value set 465%ø7oî õ{î ï{î98èLî-8è

ð;: , where 5%ø70ùê<5 7 ñ ì£í ù=5 7 >éî|ôåé?Øù@•îAîôbéî éB .

The logical operationsõ andï are the usual binary operations of max (or) and min (and),

respectively, while the dual additive operationsè8 and è8

ð are defined by the tables shown

in Figure 1.3.1. 8 è 0 1 ôbé é 8 è ð 0 1 ôbé é 0 1 0 é ôåé 0 1 0 é ôbé 1 0 1 é ôåé 1 0 1 é ôbé ôåé é é é é ôbé é é é ôbé é ôbé ôåé é ôåé é ôåé ôbé ôbé é

Figure 1.3.1. The dual additive operations è8 and è8 ð.

Note that the addition è8 (as well as è8

ð) restricted to

5

7

ùC0 îAD is the

complement of the exclusive-or operation, xor, and computes the values for the truth table of the biconditional statement EFG (i.e., p if and only if q).

12 CHAPTER 1. IMAGE ALGEBRA The operations on the value setHIJ can be easily generalized to its k-fold Cartesian

product H IJLKNM H IJ>O H IJPORQDQDQSO H IJ . Specifically, if T MVU TW6X2Y6YDY*X(T#Z0[]\^H IJLK and _ MU _ W XYDY6Y*X+_ Z [`\aH IJ K , where TcbLX+_*b\dH IJ for e Mgf

XYDY6YX(h , then Tji

k _ M l T W i k _ W XmYDY6Y*X(T Z i k _ ZDn . The addition i k

should not be confused with the usual addition To0prq

Z

on H J K .

In fact, for TPX+_`\sH J K TBi

k _ M l U T W k _ W [utvXY6YDY+X U T Z k _ Z [utn , where U T b k _ b[ t Mxwcy e{z U T b k _ b[|T}o0p~q Mf f e{z U T b k _ b[|T}o0p~q M y)€

Many point sets are also value sets. For example, the point set 

Mƒ‚m„ is a

metric space as well as a vector space with the usual operation of vector addition. Thus,

Uv‚„

X

k

[, where the symbol “

k

” denotes vector addition, will at various times be used both as a point set and as a value set. Confusion as to usage will not arise as usage should be clear from the discussion.