FORMACIÓN E INFORMACIÓN A LOS TRABAJADORES.

The basic concepts of elementary function theory provide the underlying founda- tion of a functional specification of image processing techniques. This is a direct conse- quence of viewing images as functions. The most elementary concepts of function theory are the notions of domain, range, restriction, and extension of a function.

Image restrictions and extensions are used to restrict images to regions of par- ticular interest and to embed images into larger images, respectively. Employing standard mathematical notation, the restriction of Ú_ú0ûü

to a subset Z of X is denoted by ÚÌýþ , and defined by ÚÌýþ0ÿ ÚÙbûÛkÜ]ÝnÙ5ÞkßLÚ+Ù4ÞÛLÛPá™Þ&ú8ð–ù Thus, ÚÌýþEúÓû þ

. In practice, the user may specify Z explicitly by providing bounds for the coordinates of the points of Z.

18 CHAPTER 1. IMAGE ALGEBRA There is nothing magical about restricting a to a subset Z of its domain X. We can just as well define restrictions of images to subsets of the range values. Specifically, if

and

, then the restriction of a to S is denoted by and defined as

"!

In terms of the pixel representation of we have #%$ &(')*+& )-, *+& /. . The double-bar notation is used to focus attention on the fact that the restriction is applied to the second coordinate of

0

.

Image restrictions in terms of subsets of the value set

is an extremely useful concept in computer vision as many image processing tasks are restricted to image domains over which the image values satisfy certain properties. Of course, one can always write this type of restriction in terms of a first coordinate restriction by setting 1 # $ &- , "2& /.

so that 3# 45 . However, writing a program statement such

as 6

,

# 45 is of little value since Z is implicitly specified in terms of S; i.e., Z

must be determined in terms of the property “*+&

.” Thus, Z would have to be precomputed, adding to the computational overhead as well as increased code. In contrast, direct restriction of the second coordinate values to an explicitly specified set S avoids these problems and provides for easier implementation.

As mentioned, restrictions to the range set provide a useful tool for expressing various algorithmic procedures. For instance, if-87

and S is the interval 29"'): /

7 ,

where k denotes some given threshold value, then <;>=@?ACB denotes the image a restricted

to all those points of X where a(x) exceeds the value k. In order to reduce notation, we define ED = F ;G=H?ACB. Similarly, @I"= @J =H?ACB 'KEL"= ;MNA?=OB 'P@= @Q =R '8S@TVUHWN= ;M"A?=YX !

As in the case of characteristic functions, a more general form of range restriction is given when S corresponds to a set-valued image

[Z]\_^H` ; i.e., 2& acb &d .

In this case we define

@ #[$ +&('Y"+& Y-, "+& 2& .e!

For example, for 'f6g[7

we define F Wih $ +&('Y"+& Y-, "+& kj 6/G& O. '- LVh $ &(')"& )-, "2& l 6/+& m. ' F Iih $ +&('Y"+& Y-, "+& kn 6/G& O. '- D h $ &(')"& )-, "2& o 6/+& m. ' F h $ &(')*+& P, *+& # 6/G& . '-F_p q h $ &(')*+& P, *+& sr # 6/G& .t!

Combining the concepts of first and second coordinate (domain and range) restrictions provides the general definition of an image restriction. If-

, 1

, and uv

, then the restriction of a to Z and S is defined as

4;5 ? B # Y1w ("! It follows that 4 ;5 ? B # $ &(')"& )-, &x1ySHTiUd*]& . , 4 ; ? B # , and 4 ;5 ? ^ B # 45 . The extension of d to 6 {z

on Y, where X and Y are subsets of the same topological space, is denoted by E4

h and defined by E4 h +& #v| "& ~}€ &- 6/G& }€ &-‚8ƒE !

1. 4 Images 19 Two of the most important concepts associated with a function are its domain and range. In the field of image understanding, it is convenient to view these concepts as functions that map images to sets associated with certain image properties. Specifically, we view the concept of range as a function

„E…t†V‡‰ˆ‹Š_ŒŽH‘

defined by„E…t†V‡‰ˆH’+“"”–•˜—E„s™8ŒšŠC„–•›“"’2œ”(VžE„ Ÿ<žE¡Kˆ œ™-¢8£

. Similarly, the concept of domain is viewed as the function

¤

žE¡K…t¥+†Š_Œ §¦¨G©2ª/«i©­¬¯® Ž° s±

where

Œ{ ¦¨²© ª «i© ¬® •y³H´šŠs´x•“ ¦¨>µV¶·t® ± “-™¸Œ ±/¹ ™xH ±(º ™8H‘N»

and domain is defined by

¤

žE¡P…t¥†F’+´(”•y³Hœ™¼¢½Š¾“ ¦ ¨GµV¶·t®

’+œ” •´/’Gœ”(•u„–VžE„ Ÿ<žE¡Kˆ-„s™xŒ »À¿

These mappings can be used to extract point sets and value sets from regions of images of particular interest. For example, the statement

ŸCŠÁ•

¤

žE¡P…t¥†F’2“ÂEÃ"ÄE”

yields the set of all points (pixel locations) where a(x) exceeds k, namely ŸÅ• —œ™-¢½Š–“"’2œ”/ÆÇV£

. The statement

Ÿ‹Š•›„E…‰†V‡tˆE’2“ ÃNÄ ”

on the other hand, results in a subset of È instead of X.

Closely related to spatial transformations and functional composition is the notion of image concatenation. Concatenation serves as a tool for simplifying algorithm code, adding translucency to code, and to provide a link to the usual block notion used in matrix algebra. Given“-™xŒFÉEÊ

«

ÉmË

and´d™¸Œ(ÉÌÊ

«

É@Í

, then the row-order concatenation of a with

b is denoted by ’“ ¦ ´(” and is defined as ’+“ ¦ ´(”Γ ¦ÏtÐ ¨GÑO¶ Ä ® ¿

Note that ’+“ ¦ ´”-™[Œ{ÉEÊ

«

ÉEÍY҉Ó

.

Assuming the correct dimensionality in the first coordinate, concatenation of any number of images is defined inductively using the formula ÔÕcÖ ×dÖØ_٠ژÔYÔ+՚Ö/×Ù@ÖØ_Ù so

that in general we have

ÔÕ"Û_ÖÕÀÜtÖVÝ<Ý@ÝÖÕÀÞGÙ(Ú%Ô)Ô+Õ*ÛHÖÕiÜ_ÖVÝ@Ý@ÝÖÕVހßÛÙ<ÖÕiÞ²Ù"à

Column-order concatenation can be defined in a similar manner or by simple transposition;

i.e., áâ â â â â â â âã Õ"Û ä Õ Ü ä .. . ä ÕVÞ åæ æ æ æ æ æ æ æ ç Ú[ÔÕ"ÛHÖÕiÜtÖVÝ@Ý<Ý"ÖÕiÞGÙ2èéà

20 CHAPTER 1. IMAGE ALGEBRA

Multi-Valued Image Operations

Although general image operations described in the previous sections apply to both single and multi-valued images as long as there is no specific value type associated with the generic value set ê , there exist a large number of multi-valued image operations

that are quite distinct from single-valued image operations. As the general theory of multi- valued image operations is beyond the scope of this treatise, we shall restrict our attention to some specific operations on vector-valued images while referring the reader interested in more intricate details to Ritter [1]. However, it is important to realize that vector-valued images are a special case of multi-valued images.

Ifêxëuìí andîPïxêFð , then a(x) is a vector of formî*ñ+òóë%ñ+î"ô@ñ+òó@õ"ö@ö<öVõÀî

í

ñ2òóYó

where for each ÷xëùø_õ"ö@ö<öVõú , îiû­ñ+òóïì . Thus, an image îüïñ]ì{íió

ð is of form îýë[ñî ô õ"ö<ö@ö"õtî

í

ó and with each vector value a(x) there are associated n real valuesî û ñ+òó.

Real-valued image operations generalize to the usual vector operations on ñ]ìíÀó ð . In particular, if îõ2þvïÿñ]ìíió ð , then î þxë%ñ+î ô þ ô õ"ö<ö@öiõYî í dþ í ó îOþxë%ñ+î"ôOþ/ô<õö@ö<öVõYî í Oþ í ó îKþxë%ñ+î*ôKþeô@õö@ö<öVõ)î í þ í ó îKþxë%ñ+î ô Kþ ô õö@ö<öVõ)î í þ í ó If 8ë3ñ ô õ"ö<ö@öVõ í

óï ìí , then we also have

îPëuñHô î"ô@õö@ö<öVõ í î í ó"õ îPë%ñ ô î ô õ"ö<ö@ö"õ í î í ó"õ

etc. In the special case where ë%ñEõEõVö<ö@öVõ@ó , we simply use the scalarsï8ì and define

î›îõEîEî , and so on.

As before, binary operations on multi-valued images are induced by the corre- sponding binary operation_ìí §ìí"!0ìí on the value setìí . It turns out to be useful

to generalize this concept by replacing the binary operation by a sequence of binary

operations $#%iì{í& ìí! ì , where'Pë ø_õ"ö@ö<öVõú , and defining

î(Àþ)[ñî(Àôþ õYî(+*Oþkõ"ö@ö@öiõYî(

í

þó,

For example, if -#.ì{í. xì{í/!½ì is defined by

ñ10 ô õ"ö@ö<ö"õ0 í ó2 # ñ3 ô õ"ö@ö@öiõ3 í óë54%67098$0Àû:%3 # ýø<;÷=;>'(?{õ then forîFõ)þdïdñ2ìíÀó ð and

@Cë›î(Àþ , the components of@tñ2òó(ëÿñ@ ô ñ2òóOõ"ö<ö@ö"õ@

í

ñ+òó)ó have

values

@A#éñ+òó(ë›î"ñ2òó-#Eþ/ñGòó(ë54B670C8@î û ñòóDKî+#éñ2òó"ýø;÷=;E'F?

for 'wë øéõ"ö<ö@ö"õ¯ú .

As another example, suppose ô and * are two binary operationsì

* ¼ì * !ì defined by ñ0 ô õ0 * ó ô ñ3 ô õ3 * ó(ëG0 ô 3 ôIH 0 * 3 * and ñ0 ô õ0 * ó * ñ3 ô õ3 * óëJ0 ô 3 * E0 * 3 ô õ

1. 4 Images 21 respectively. Now ifKDL2M)NOP=Q R-S represent two complex-valued images, then the product TBU

KWV(M represents pointwise complex multiplication, namely T7XZYD[IU\X KC] X^YD[ M_] XZYD[=` K Q XZYD[ M Q X^YD[ LWKC] X^YD[ M Q XZYD[Da K Q XZYD[ M] X^YD[[b

Basic operations on single and multi-valued images can be combined to form image processing operations of arbitrary complexity. Two such operations that have proven to be extremely useful in processing real vector-valued images are the winner take all jth- coordinate maximum and minimum of two images. Specifically, ifKDLM>N

X

P=c

[

S , then the jth-coordinate maximum of a and b is defined as

Kd)efgM U\h$XZY L TiX^YD[[BjT7X^YD[kU K XZYD[=lnm K+f X^YD[o MCf X^YD[ L=p+qrWsut2v lxw s T7XZYD[IU M XyYD[Az L

while the jth-coordinate minimum is defined as

K{)ef-M U\h$XZY L TiX^YD[[BjT7X^YD[kU K XZYD[=lnm K+f X^YD[| MCf X^YD[ L=p+qrWs tv l}w s T7XZYD[IU M XyYD[AzFb

Unary operations on vector-valued images are defined in a similar componentwise fashion. Given a function ~

j

PE€P , then f induces a function P=c%P=c , again denoted

by f, which is defined by ~ X‚ ] L ‚ Q LCƒuƒ-ƒCL ‚ c [I„…X ~ X‚ ] [ L(~ X‚ Q [ LDƒ-ƒ-ƒ:L(~ X‚ c [2[ b

These functions provide for one type of unary operations on vector-valued images. In particular, if K U†X K ] L7K Q LCƒ-ƒuƒCL7K c [ N X P=c [ S , then ~ X K [I„ ~‡_K U…X ~ X K ] [ L(~ X K Q [ L,ƒuƒ-ƒ:L(~ X K c [2[ b Thus, if ~ U‰ˆ-Š‹\j PŒP , then ˆ-Š^‹,X K [IUŽX1ˆ-ŠZ‹,X KC] [ L,ƒuƒ-ƒ:L ˆ-ŠZ‹,X K c [[Cb Similarly, if ~ U‘F’ , then  ‘F’ X K [IU O  ‘+’ X K ] [ L,ƒuƒ-ƒ9L  ‘+’ X K c [ R b Any function ~ j

P=c“P”c gives rise to a sequence of functions ~ f UG• f ‡~ j P=c%€P , where– U— LCƒuƒ-ƒCL ‹

. Conversely, given a sequence of functions ~f

j P=c%€P , where– U˜— LCƒ-ƒuƒCL ‹

, then we can define a function ~

j P”c™€P”c by ~ XYD[k„JX ~ ] XYD[ L(~ Q X^YD[ LCƒuƒ-ƒ9L:~ c XZYD[[ L whereYU\X‚ ] L,ƒuƒ-ƒ9L ‚ c [

NP=c . Such functions provide for a more complex type of unary

image operations since by definition

~ X K [IU\X ~ ] X K [ LCƒuƒ-ƒ9L~-š X K [[IU…h$X^Y L2M X^YD[[%j M X^YD[IU…X ~ ] X K X^YD[[ LCƒ-ƒuƒ9L~-š X K X^YD[2[[z L

which means that the construction of each new coordinate depends on all the original coordinates. To provide a specific example, define ~ ]

j PIQ“P by ~ ] X‚ L› [<UŒˆ-Š^‹,X‚:[=a œ$ˆ$žX › [ and ~ Q j PkQŸP by ~ Q X‚ L› [ Uœ $ˆgX‚:[=aJˆuŠZ‹CžCX › [

22 CHAPTER 1. IMAGE ALGEBRA

¡E¢_£¤=¥A¦u§©¨ª£¤I¥A¦-§

given by

¡)«˜¬¡®­$¯¡

¥A°. Applying f to an image ±>² £¤I¥A¦u§ results in the image ¡¬ ± ° «\³®¬^´I¯2µ¬y´ °° ¢¶µ¬Z´ ° «\¬·$¸^¹,¬ ± ­u¬Z´ °°Cº¼» ½ ·$¾¬ ± ¥ ¬^´ °2° ¯ » ½ ·$¬ ± ­A¬^´ °° º ·u¸Z¹C¾C¬ ± ¥ ¬^´ °°° ¯=´ ²%¿\À7Á

Thus, if we represent complex numbers as points in ¤I¥

and a denotes a complex-valued image, then ¡¬

±

° is a pointwise application of the complex sine function.

Global reduce operations are also applied componentwise. For example, if

±G² ¬¤” ° § , and à « »AÄFÅ$Æ ¬ ¿ °, then Ç ± «…¬Ç ± ­$¯CÈuÈ-È9¯Ç ± Â9° «ÊÉËÍÌ Î Ï-Ð ­ ± ­ ¬^´ Ï ° ¯CÈ-È-È:¯Ì Î Ï-Ð ­ ±  ¬^´ Ï °1ÑÒ ² ¤  Á

In contrast, the summation

Â Ó ÔyÐ ­ ± Ô « Â Ó ÔyÐ ­uÕ Ô ¬ ± ° ² ¤ § since each ± Ô ² ¤ §

. Note that the projection functionÕ Ô is a unary operation ¬¤” ° § ¨Ö¤ § . Similarly, × ± «…¬ × ± ­-¯DÈ-ÈuÈ9¯ × ± Â(° ¯ Ø ± «…¬ Ø ± ­ ¯DÈ-ÈuÈ9¯ Ø ±  ° ¯ and Ù ± «\¬ Ù ± ­$¯DÈ-È-È:¯ Ù ± Â(° Á

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