Optics Using MATLAB [TT111] 9781510608313

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Chapter 3

Curve Fitting and Statistics

One of the big differences between working on purely numerical solutions and working with data from experiments is that the experimental signals are often more complex than expected. This is not improved by the additional noise usually found in measured signals. Analyzing this data often requires that trends in the data be extracted and ideally expressed as a simple mathematical function. Determining an appropriate mathematical func­ tion is the domain of the many curve-fitting techniques that are commonly used. Complex curves can also be created from many simpler mathematical functions by adding them together. The individual components of these synthesized curves can be explored separately. We have already shown that there are ways to introduce noise into numerical functions and that the noise in data can be filtered to reduce its impact. Adding noise to a synthesized curve can make the curve more like an experimental data set. The process of determining whether a curve fits a data set requires the development of metrics to use for comparison. These metrics provide a measure of the quality of the fit between the curve and the data. One simple metric that we will develop will provide a "goodness of the fit" test. This simple test relies on the curve through the data being, on average, equidistant from the curve so that the sum is zero. In this chapter we will introduce sorne simple tools for working with complex signals contaminated with unwanted information. The appropriate statistics such as the mean and the standard deviation will be used, and we will go farther by showing that we can find the trend in the data and remove this bias so that we can look in more detail at the noise signa!.

3.1 y>olynomial Synthesis and Curve Fitting Mathematical functions can be summed over a common range of inputs' to generate or synthesize a curve. Polynomial functions are composed of a number of terms with a variable of interest raised to sorne power and a coefficient 37

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Chapter 5

in bytes (B) or megabytes (MB). ln order to work u'ith graphics files, it

is

important to understand that each color element, or pixel, is just a nuntber oL a sct ol numbers, and a picture is r.rothing more than a matrix of numbers.2 lmagc files can be separated into two classes, raster and vector graphics. Raster graphics arc typically a rectangular array of numbcrs that represent the various colors in the image, with the fir'st pixel located in one comer olthe itnage and then lollowing a row colun.m otder until the final pixel is reached. Raster graphics will be thc prirnary focus ol this chapter. Vector graphics, as the namc implies, alc based on vectors used to represent positions of objects in the irnage plane. Compalcd to raster graphics, vector graphics make t¿rsks such as scaling, rotation, and translation ol objects considerably easier; however, vector graphics are not as effcctive lor representing everyday images taken with current digital carrcras. Most display and printing dcvices work with raster infonnation. Fortunately. conversior.r from vector graphics to raster graphics is straightforn'ard. although raster images usually rcquire larger file sizcs thau vector images. Iu a rastcr image the overall color of each pixel is defined by a specific intensit¡r levcl lor each of the colors red (R), glccn (G), and blue (B). The intensity level is a numbcr typically between 0 and 255.2 Using this RGB model, tlie color blue would be rcpresenled by the number triplet (0, 0. 255), red by (255, 0, 0) and green by (0, 255, 0). Other colors, such as purplc (128, 128, 255), make use of intensity levels in each plane. Onc of the easiest rvirys to explore how colors are madc rip in images is io use a graphics program such as Microsof'tt Painl to make a small, single-color image and then decompose the imagc into numbers. Some of the more common image or graphics hles that MATLAB is able to work with arc BMP. GIF, JPEG, PNG, TIFF, and thcre are many more. In this chapter we u'ill work with just two of these file typcs to introduce some ol the basics of wolking with images. In most cases, the MATLAB Help will let you take the basic skills that wc introduce here and adapt them to work with other file fom.rats. 4 The MATLAB built-in function to read in an image file is rmread 1¡ .r'3 This function reads in an image file of a defincd lormat and places the values into a matrix. Example 5.1 shows how to use amr:ead O to load a bitmap file and store it in a matrix as a set of numbers that can be uscd lor calculations. Several new idcas are included here that will be explained further. Example 5.1 Use imread

(

) to load

a file and separate out color planes.

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fife to

a

matrix

& command l',/indow

lmaqes and Image

;

fname:'rec.: - imreao ( :. ::-

M

te¡ Separ.ale

::

redPt ane - i: -:

greenP Lane - :-: bfue Pl ane : ::: !¡

? Dcne

The r¡r,::.-:, file in this ca s; semicolon is r¡: returned in thitt ,n" a*""r1i6r.i rr: v¿rriables can b: panel.

The

cottr er.

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type double. :i

mathematical .: occur when inte Should ther. values in each.. using the :-:reconstructcd iil next section.

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Chapter 5

5.2 lmage Commands Working with irnage files is an integral part of MATLAB.' and numerous built-in functions support thc rcading, writing. and display ol irnagcs. To derronstl-ate hon' these commands can bc used. Examplc 5.2 changes a red image to blue ancl saves tl.rc file to a new name. More detailed leatures of the commands are avaiiable in thc MATLAB Help files. Example 5.2 Use ¡,4ATLAB image commands to turn a red image to blue.

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r.r; U

);

:::. SeparaLe cof

or pfanes .eaP-ane:M(:, |, !) ; greerPfane --14(:, : ,2) ; bfuePlane:M(:,:,3); coior pfane order N(:/:,1)-greenPlane; N(:,:,2):bfuePiane; N (:, :,3) : redPfane; rl-¿ Sr,rap

.ó

e. Shor,¡

image s

figure ; lmage (M) figure (2) ; rmagesc (N) ; (1 )

!¿:; Save lmag;e f

iLe

imw¡ i le (N, 'biue.bmp', 'bnp' ) ;

The conversion to a blue in.rage is failly simple: the order in whicl.r the inrage planes were placed back into the new matrix N was switched hom the original order so that the blue plane now has values and the red plane has zclos. Sincc wc are not doing any real mathematical operalions on the values in the matrix, thcrc is no necd to cor.lvclt thcm from their original uintS data typc. The lmage O and imagesc o functions were used to display thc in.rages. Thc dilftrence between the

lnages and lmag.

in o function. : ¡he image ploi The appr..-, one image t¡ p: bitmap: rr e c,-..-r cbange both th: savedl

The reirl pr image tc' n. an reconstructe¡ ::

5.3 lmage

Sir

Working u ith .: files can be r ei. or 3ooo x lt)l) 3 numbcrs li¡r: the file. tlie m;:: header that c¡:complession i.' inlorrnation c..: reduction in ::compressed di-: infomation L.. compression

r'--:

tion process i: : Iarge pictur. j: ijlagc, not all . the image b¡ re:

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Chapter 5

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ar

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originai

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.'r'ery tl.titd :r. gre¿1tl) redui: atvcragc ol n::irnf gesa r :-larger pirel:. climensions.

Thc net: nealcst nclgir:

Example 5.3 Reduce an image file size by using super-pixels.

,: Ii 5 3.nL i SIir 2-25 16 !a Averag= paxers to a smaffer m¿tr1¡ .a::r

tiousekeepr:lg

clea¡ aLf; cfc; : cfears the '!or{space ! c¡mm¿¡o ,,]i¡do-;!: Ccnf lgiure mat¡ices 14:rand(10tC); lrMax, cMaxl : slze (M) ; r: !; Scan ancl lep.ace sum: C; sr 1; sc !; Ícr r:2:3: rMar 'at c:2:3:cl'llax -of s:1 ior t: -1:1 sum: s,J.n + M (r+ s, c+t

:cr O loop: thc M arra\. a sun]l]l¿ltiolr

I

figurcs using

t:.

grayscale

..

nr.'.

by using thc gr in the 'Shou r: two images .Li value of the t::

This pr.oc.' tl.re desir-ed le:

);

end e

€:

1ci

(sr, sc) - sun/ 9,' sum: C; sc:sc', i; !óincr counte r

x.-

end

sr:sr+1; sc I; -ond

1:

Figure :;

5.2

l

A ::

with pixelation ¿informat on iLra :

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'pe^erqc€ sr cZIs elu.to uorlnlosoJ peJrsep eql

Illun 3lU o8urur u9.ro xulpur l? uo sourrl IE.raAos unlel oq uuJ sseJold srqJ 'pepoeu raqurnu paxu e le 1es rq plnoJ lr lnq 'e8eurr lsrq orll.lo enle^ Jr lexrd runurxeu aqt sr ¡cxrd eJuaJoJeJ srrlt o-raH (1'1) laxrd aql lu se8eurr o,n1 eql uoo,r\laq ecuere-¡o.r flrsua¡ur uoutrüoJ e sepr,r.o:d uollJes ,sllnsei,roqS, orll ur eurJ lslu eqJ ' (Ierb) deuroToc uorlcun3 Surdduu ro¡oc alecs,{er8 eql Bulsn,{q lre1q o1 seuol ,(e:3 qinorql elq,r ruor3 aSuu s.ro1o: 1eql Surueeru e1els,{e:8 ll ol pJus^uoc uaaq e^pq saiuur cq¿ pul?rxtrro3 O csobeurr er¡l Sursn sa:n8g crqde:8 se pe,{e¡dsrp eJÉ slJs ptep o,\\t oqJ slcxrd euru eql Jo uortpruruns e cpl,r.o.td sacrpur 1puú s rrJqt r¡lr,r ,sdoo¡ loJ, o,\\t lsul sq1 pue ,,{e:re ¡41 erp

q8nolp dels s¡Jrpur o pue I rrrqt qlr,u ,sdoo1 roJ, o,rt ls.lu eqJ .sdool g -¡ol patsau -rnoJ qinolql p:qsqdurorce sr lsxrd relueJ rqt pue s:oqqSreu lsaleau lqira rqt 8ur8e:e¡¿ pue ,{u-r:u cq1 qSno:q1 Surddels Jo poqteru eq.l. 'suorsueurrp

llsie,\o Jurps eql 8ui^pq s¿ u,\\oqs oru so8urur qloq oJurs .s¡exrd r:F.re1 :,req o1 s-ruodde 1r¡3r.r sqt uo ,{e:le ,nau eql ¿ g 3rg ur uorlcun; ( ) c s abeur ¡ql Sursn pa1lo1d e:e s,{¿llp qtog ,{u.l:e ler ur eq1 ur slexrd euru 3o a8urc,ru eql sr ,{prre ,lrau eql ul slcxrd aql 3o qJue álaq,r ,ozrs ,{utla pecnpe.t ,{11ee.ri sloqqireu tselpáu áqt Sur8e:c¡.¿ pue lurod pJrqt ,,{.re^a

e sr tlnsei

leur6r.io

eql ]o

=

:r¡1

t¿ ,{ei-re oql q?no.rqt Surddals ,(q tos etpp oqt Jo ezls eql sáJnpái pue ,{¿llp a,Ienbs p ur sJequnu tuopul?l Jo ,{er.te yerlrur up selpl g.g aldruexa

3UOC:

I (.{ef 6) deu.roTor j (N)csabeulr 1 (¿) arntrr__ I (,{e.rb ) deu¡roIoc I (r{) c s abEu.rr i ( 'i ) ernb r_ 1(I,I)r^r: (I,I)N j((:)N)rpur- (I /T)¡, slTnssf Moqs ?i: g )etdeLtC

Chapter 5

70

5.4 Golor Models and Conversions There are many \\ays to present color inlormation to the viewer.

Results of rr Thcsc

nrethods include using truc color and using false color images. where the colors are altered to highlight particular inlormation in an imagc. There are also dillerent ways to represent the colors that arc being used. One of thcse mcthods. RGB. lias already been introduccd, Others include HSV and CMYK color nodcls, or tables that provide dillerer]t ways to var)' color. These colol models pt'ovidc different ways to model color, but sincc the range of color lcally doesn't changc very much to tlre obscrver. thcsc other Ícprcsentations are just transforms of other models. The transformatiol] between RGB and CMYK proceeds as fbllou's:

K-l

/R G B\ nrrxl^-,^-.^-1. \ /\\ /\\ /\\/

lmages and lm¿.

(5.1)

c:0 M:0.846 Y K

- 0.64 - 0.368

,-

This ls

a

determine u hi¡ the tht'ee coltrl: be used to del: defined as a c¡ . either bc ll an::

MATL.IB

:

color schente.

IK

(5.2)

intensity 1e,,.-1 r imagc, as thc i:-.: lront whitc to i.

lor a figule is; coior map ker

.

(s.3)

Color rlaps u r,.

5.5 Spatial Fi (5.4)

These equatior]s can bc constructed into a fturction that will return the CMYK values lor an RGB input. This function is shown in Example 5.4. Example 5.4 Use the conversion function for RGB to C|\4YK.

functron lC, M, Y, K I : -a;:¡ I'{ 5 4.rn !i SI\'T 2 26-16

RGB2CYMK (R, G, B)

!¡ RGR -)> CI4YI{ !ó 'i Conve r s Lon

RGB: fR/255; G/255; B/2551 ; K-1 rnax(RGB) ; c: (1-RGB(1) K) / (1-K) ; M: (1-RGB(2) K) /(1 K); Y- (1 RGB (3 ) - K) / (1-K); E i.i

One of the mo:r Thesc filters ¿i rc 1o smooth

or sl-. ap¡rly these tú. introduccd in Sr-, size of the tll.-. :: Spatial trlre: i.e.. a spatiai da:. a point. In Sec::, smooth nois\ d¡: filter did reduce :: The simplest f ur,. sharpening. Sntt r fliters tcnd ¡o cn: tion of the filter : shows tlre o¡-ser:: applications.: Th: lliter.2 In the h:-i sharpening Lapla.-

oql tnq 'peJunouoJd ssol sr lcol]o Suruedftqs eql 'l < V loJ luerceldel Suruedreqs e Jo rruoJ er{l uI sr rettu eq} ¡I:v uoq/\\'rellu lsooq-q8rq 3q1 uI ¿ ro}lu lsooq-q8rq € Jo ru:oJ eql ur pesserdxe sI xuleru Surued:eqs eq1 .'suorluoqdde Suruad:uqs pue Surqloorus roJ xulerü Jollg eql 3o uoqezrue8ro eq] s,troqs arnSg er¡1 3o eprs pueq-lqSrr eql pue 'los €l€p oql uo sseco.¡d rotrlg oql Jo uoq -ezrue8ro eql s,{\oqs t S 3!C Jo eprs pueq-Uel aq¿ se8ps e¡u¿que 01 puot s.rállg 's:e¡g 8uu-rnlq se u,trou{ osle are s:e11g Surqloous Suruod:¿qs

I # i

Surued-reqs e¡q,,lt

pue Surqloours eJe 1ás ¿lep eql 01 perTdde oq uec lpql suortounJ tse¡durrs sq¿ 'suorsuárürp o,ul ur pssn 0q UEJ srallu r¿lrurs esrouJo s130]J0 eql aJnpel prp rá1lg eql 'Álalolduroc peJeAoJeJ 1ou se,r pu8rs purSuo eql eIq¡A etep,{srou qloorus ol pesn sp,lr lpql pecnpoJlur s",^\ rálfu Sur8e¡el¿ u¿ '€'¿ uorloas uI lurod e Jo enle^ eql ol¿¡nclmar o1 slurod elep Surpunorms áql sesn lás elep ¡erleds e ' e r lslexrd ,{e¡¡e ue 01 uorllunJ leJq€ureqleru e ssqdde 3uue11g 1e4ed5 3o 'e8erur er{l. Jo {oo[ eql ,(Jrpoiu o] s¡ esod:nd sql 'elu oqt Jo azrs eql SuronpeJ Jo asodrnd aql 8ur^eq ueq] ráqleJ tnq '€'S uorlJes ur pecnporlur qceo.rdde eq1 ur pe^lolur ssoql 01 ru¡nurs Á:e,t ele suoDrunJ oseql ,{[dd€ o1 Sursn aq [[r,r e,r leql s¡decuoc crs¿q eqJ 'e3¿ur ue usd:eqs :o qtoorus ol pasn oq u¿J l¿ql suoflounJ lsorl¿uieqleru uuo3red o1 peuSrsep eJe sJellu eseql ler¡eds3o osn oql sr Surssaco:d a8erurlo spoqlaru l3oJrp lsoru eglJo euO

'¡ 9 e¡duer3 eql uJnleJ IJr.\\ (t

's.ra11g

buue¡¡rg

E)

le¡eds 9'9 (s's)

'suorlcunJ pele¡er-e8eur Sursn s:e no,( :e^.eueq,\\ á[qe¡e^e eq 1¡t,,lr sdeu.r rolo3 's:aqlo Suoruu ',Jerürüns, pue '.1e[ '.reddoc, ere,{¡t o] sp:o,,n,{e¡ deru:o1oc Ieuonrppu euos uorlcunJ O d€xrroloc eql Sursn pellojluoJ sr ernSg e :o3 duu roloc aq¿ sroloc fer3 3urue1:ep -¡o seues e q8norql lculq ol álrq,lt uro{ seSueqr :oloc pa,(eldsrp orll ' l ot 0 uor3 se8uuqo an1e.t ,{1rsue1ur oql se 'o8elur o¡ers,(e:8 e ur'qceo:dde pasn,{¡uouuroc stql.¡o elduexe ue sy'1e,r.e1 ,{1rsua1ur eql uo pes€q laxrd e 3o rolo3 aql a8uuqr sdeu Joloo aseqJ 'sauráqcs ioloc luareJ1rp ,{eldsrp tpqt suorldo deru roloc Jo raqunu e sepr,rord SV'IIV¡¡ (: l'pesn eq plnoc )atu LUJOJ oql Jo 'pesodsu€rl aq rorllrs plnoc gDU Joqlre 'lolco^,\\or e áJo,{\ lrJr 'Je^e,^Aoq :rolma urunloJ e se peuUap sr gCl{ l€ql ac4o¡1 a1du1 eq1 Jo onle^ runrurxeur eqt ourruJstep ol pesn áq u€c uorlJunJ O xÉu¡ aql 1eq1 os xrrleur e¡8urs e olur peceld a:e sJoloo aelrlt aql 'eldurrs srql e{eur oJ 'lseq8rq sr g ro 'C '1¡ Jo senle^ eql Jo qcrq,tr ourruJotep peou op no,{ q8noql¡e 'uorsrá^uoJ pre,tr.ro31q3re:ls ,ften e sr srq¿

(¿ s)

(t

s)

uollBrIIloJsu¿Jl J Jáqlo oseql .l¡ u á8ueJ eql aJurs ln ':oJoc ,{¡¿,r o1 s i

ol

pue ASH epnlru sseql Jo euo .p¡s s:e e:eq¿ .o8eur

88t'0 - t Lb9'A:L 9tB'0-h

oql

0:a) (S9, tZ, 9ET ) X¡{CZsgli

e.raq,ü 'sa;eLur

3SárlJ le,{or,\ ¡r{l

: [x, I,r{, C] 6qruun¡ Jo sllnseE I

dussaxu¿ a6eal pue sadewl

,L

reldeqC

Chapter 5

72

Images and lmag..

column of the starting at ( 1.1 the image u ith either of these r

lrnaoe Dáta

r

11'1 111

111

111

111

111

1111

11 11

111

111

111

111

11

11

1

11

11

1

111

1'l

11

1

111

5.3

1t9 1t9

1t9

in.rage.

1ts 1t9

1t9

1t9 1t9

1t9

location is cali summation ol t:

Each time

the data set tha:

to the final

0 -1

0

-1 A+4

-1

0-1

0

half at a valu.Example 5.5 P:

5 5.::. 5-2- ' ? Tunabfe ::. ?i?

M

',: SIriT

(left) A matrix populated with 1s showing how a 3 x 3 filter would be positioned

on the matrx. The dark center shows the pixel location where the result of the filter mult plication is placed in a new matrix. (right) Two filters used in spatial filtering.

inragc u ill appear bdghter. The opposite of the sharpening filter is the snoothing filter. uüich reduces the high-f'r'equency variations and blurs the image. Thc smoothing fllter shown in Fig. 5.3 has the 3 x 3 matrix populated with ls, so it musL bc divided by 9 so that thc brightncss ol thc filter does not inctcrse.

The proccss of spatial filtering that is being described is also knowt.t ¿ts discrete convolution and involves the gene¡ation of a ncw matrix based on applying a spatial filter matlix to the larger image data set. The flltcr matrix passes over the data matrix one pixel at a time, and the element-by-element product of the two matrices is summed. This result is placed into thc resulting matrix. This process is shown mathematically as

c(r,r)

r alLi

showsa5xi.

Sharpening

111 F¡gure

111

Smoothing

- | lt-

ti

I

\) lll,.l,l+M(r-/r.t t: t"

.lt),

(5.5 )

where C(r, c) is an element of thc nen' matrix. / is a square filter matrix of size N-2nt- 1, and M holds the elemcnts being included ln the original irnage

matdx. The application of the 3 x 3 filter matrix to the data is shown in Fig. 5.3. rvhcre the first calculation has the filter centered on the in-rage data point located at row - 2, colunn:2 or (2,2). By starting at this location rathcr than at the (I,1) location, all of rl.re filter elements are within the inage. Had the proccss started at the point (l.l) in the imagc. the top row and lafJcft

? Hou se ke a. r: cfear a--; :

?t Matr.r- :-

siZe:25; .: . M-zeIoSl:::: M(l1:size , lrM,cMl:- .

-

; Fif ter l.:--:: F- | 1 1 - , 11

11 1_, lrF,cFl -.-

noRm- s.r::

{=

ri

noRrn

¿¿

Scan. ..:

-

-

i

:

:

- ilocr (.i , r:R : slart, . for¡-s]-:r,: iorc-:. T:.', s

]/l

:r)l,il-I O 1{rllEluqns alEuj 1 I (s+il:s-c's+l:s S-lr"¡¡: lf elS - c .iol S-l/il.l:llelS:J lo-: s.tr3Junoc :las !¡ jJ-reJS - Ec jl-rEls : u.r ls+ i:f:rets I (z/Jf ),looTf -s aceTdel Pue a^To^uo¡ 'ueos 1;i 1pué iT:ur8Qu l0 :> u¡Uou -lJ ur ssnTe^ llns ? l((:)J)LLIns-uuoLl xef^lc /xpl^tl 01 szrs u6rssec'j I (J) ezrs - IJJ /Jrl ]I T T ] I I IIT iI IIIII TTIII I I I I t I :J a.Íenbs aq Jsnul xr,rletr{:rell-!J

I ot

:i

xE,l,,lir 'xet\lr oJ azrs ubrsse e,: I (r{) azrs - [I^]3'tilf, 'lszrs:Tl)I,,1 fas != 1I - (llZ/azts),rooTl:Il jo xr:rfPl/il I i (aZrs)soraz:U j (azrs)so]3z =t¡

U3[-JeJ puP \(rj pe¡1 aSurur :qr leqlEr uorlll¡ol ! lurod PtEp ¡;rL::

't g 8lC uI u \1!,t

aSeur Jeut-r-ro :.. xlnetlt l..t ozls Jo

(E E)

8ul1¡rse: tr¡]

p 1o 6uua1¡r1 ¡eueds

urolod

99

crl'.::

luerue a-,{q - I u : r::. YrJll?m ,r3tlrl ¡ql ¡

uo pSsPq xutlrri: u,{\ou{ o.!lf r:

sl?

'JSee.IJUt 1a:

-rTPLI

eo:raz

xilrJElll 3-renbs Jo ¿zas i !92-azts dn las xr.trler.il i-..

¡opura puEllnxotr ! ailedst{,roM or{f s:reeT¡ q: ic , il I lTe -reaTa 6urdaelasnoH:.

:reflrl Terleds aTqeunl ' 9I_LZ g JMS . u|g g I^l :: !: 'rallu9x9eqlrMlas elep

'lr os 's qtr.\\ p¡t:. I cq1 e8uur aqr .

Sutqloorus arp

sr:

'6uuollU

rellrl oql Jo llnsa,

?

:

p€uorlrsod aq p nc',

.L

aldutexf

'0 1B J[¿q puP I Jo anle^ e lu JlPtl qlr,{\ ',{llslle'}ul ut ¡tlds st l¿ql les lr}?p e uo 8ullJE ra11¡ lerleds E S n s,toLl\ g E e¡durexg tq8re,,lr pu¿ uirs lextd lrllU aqt,{q pa¡o.t1uoc sI anle^ [PuU otll o] lexrd las ulep qcueJo uortnqriluo¡ eql Jellu eql ,{q pere^o. sl luql ios elup otll Jo s,{¿Jreqns eqt ueá,{\l¡g uortuJlldtllnLu }u:ura¡a-,(q-luaua1a 3I.l1Jo uolll?tllttlns eql sr uoqPlnslÉJ oqf xllletu ,t\au P uI paJols puP p¡18lnJI€J sI uolleJol loxrd eql JoJ anle^ 'reu e'1extd P tlo pJuolltsod sI Jollg áql ,.t,t,'"l.tn_,

66,,

oql olul lqanoiq Sutaq st uotluuuojul IPIcUl e 'sotlJ€olddP o,\\] eseql Jo rJtlllé ul senle^ Iexrd e3p3 3q1 alerrldnp o1 Jo '(SuIppPd olez) solsz qll{\ oáPrul rt{1

eL-

( 3o s:8pc cr¡1 puetxa .rerltre ot Jf,ltl¿.Id uouttuoc P sI 1I 'perlsep sl l' [ ) 1¿ SullJels 3-rerl,\\ esec 3rl1 UI los elBp erll Jo lJo rq plnor\ xtJl€tu lellu 3q1 Jo ulunloJ

Duissaioj.J o6eq pue sabeal

g )atdeqC

lmages and lmage

Chapter 5

74

*F;

if the sum is 0. a , in the next sectio The third sec coding, where n

fiftered natrix R (rR, cR) :sum(N(:)) /noRm; co-cR+ l; ?lnc courLó"

N:

T.

Z

end

rR:

rR

+1; cR: slart;

?

incr rR reset

accounting of inc

cR

end ?g Show re ligure (1) figure (2)

*

su 1Ls

colornap (gray) ;imagesc(R) ;colorrnap (gray) ; imagesc (M) ;

; ;

Done

This script is fairly complex and shows many interesting and useful concepts. Functionally, the script is in four sections, with each section identiirable by the double percent symbols 'o/'|%' . The first section is the header, where the only active code is for clearing variables and workspace'

The fourth section shows the two images on the scleen with only a command to set the colors used to display the images. The second section %% Matrix set up does exactly as that first line indicates: the matrices to be used are set up. Even though we are talking about images in this script, the starting point will be numerical matrices' The diménsion of the data matrix is defined using the 'siZe' variable MATLAB has a si ze ( ) function, but by capitalizing just one letter in the function name, we can use the function as a variable. Two matrices are created next: M, which will be used to hold the original data, and R, which will be used to hold the processed data. Both M and R are initially set up to be 25 x 25 element matiices with the elements set to zero. Approximately half of the values in the M matrix are set to 1, creating a sharp transition between the values. This line is sufficiently interesting to look at in detail:

for the new spati F matrix. The n spaces, their pu¡ The T matrix is a while the F mat column position i step, in which rl element. The \ r value. The locari controlled b1 rF appropriate.

MATLAB

result of runnin_s t

the righthand im the left. Because t calculation, thel around the right-l

f1: sizel , [1:

ffoor (síZe/2) ] ) :1;

2 set

haff to

:

Example 5.6 Use Z% %

¿

M_5_6.m

sWT 5-27-1€ Tunable spa:

_

?3 House keec _:.: c

M(

h¿

convolution ol rh

lear al1; c-:;

1

The commands in brackets identify regions in the matrix; [1: size] identifies rows l through 25' and l1: ¡1oo r \size/2) I identifies the columns beginning with the first and running through half the size value, 12.5, then rounded downward using the floor O function to 12. The size g function passes the matrix size to two variables that hold the maximum size information. Each of these variables is tracked individually so that the data set matrix is not required to be square. The firlter matrix F that will be used is lequired in this case to be square, but it can be any dimension provided it is smaller than the data matrix. The sum of the values in the matrix is calculated as an 'if statement' to ensure that

Matrix se-_ _! siZe-25;: s r:: r"l - zeros ( st _- ; M([1:saze]/ -_:¿*

= size (M) ; lrM, cMl : sizs

R

? Fi

'1.

Lter Mat:_:.:

F: I 111 _ 111,_ 1111_

ra]lLr.

elenbs aq

JSnur

xel^I. ?xptil.i oJ ozts u6rSse !: I

III i, IIITi I T T I I ] :J -r,rleh] falT lJ :l

(t¡f )

azTs

:

ll^lc

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I (t^l)32rs -

Iof fTeqtes I jI = \[(¿/3zrs),rooT]::Tllazrs:Il)t^i

U

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l¿ql alnsus o1 . t eql xl.tl¿ru U1:; 'c:enbs aq ot:., elep aqt lPrll AZIS UINUII\T?

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e.

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The script in Example 5.6 includes the conv2 g function approach, and the function results of applying the hlter are shown in Fig. 5.5. The conv2 supports several keyvords that allow the user to tune the way the function is applied. The keywords include 'full', 'same', and'valid'and change the resulting shape of the matlix. The 'valid' key,vord returns the convolution without using zero padding.

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The usefulness of spatial filtering becomes apparent when working directly with images, as the filtering effects can be masked by the viewing resolution. The effect of many hlters is difficult to show except in electronic images, where it is possible to zoom in on features of interest. An imagefiltering techniq te lhat does have clearly evidelt effects uses sharpening masks, which enhance regions of abrupt change.

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(right) Or¡ginal image and (left) the sharpened image.

enhancing images. Being able to construct and apply custom image filtels is a powerlul skill and can also be interesting.

5.6 Practice Problems What 3 x 3 flrlter would return the same image that you started with? 2. What is the ellect of the filter 0.5*[0 0 0; 0 I 0; 0 0 0]? 1.

Answers

1.[000; 010;000] 2. It reduces the brightness by half.

References 1. D. Hanselman and B. Littlefield, Masrering MATLAB@ 7, Prentice-Hall, Upper Saddle River, New Jersey (2011). 2. R. C. Gonzales and R. E. Woods, Digital hnage Processing, Second Edition, Prentice-Hall, Upper Saddle River, New Jersey (2002). 3. H. Moore, MATLAB@ Jor Engineers, 4th Edition, Prentice-Hall, Upper Saddle River, New Jersey (2014). 4. R. Pratap, Gexing Started vvitlt MATLAB 7: A Quick Introduction for Scienrisrs and Engineers, Oxford University Press, Oxford (2006).

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Chapter 7

Spectrometers Spectroscopy can be thought of as working with rainbows, i.e., the separation of white light into its component colors, which range from red to violet. Seeing a rainbow, or even better, double rainbows, has great aesthetic value but is also part of the science behind analyzing light. On the beauty side, cut glass crystals such as those that might be used in a chandelier will also show the colors of the rainbow. Refining the shape of these crystals for use in science has led to new insight in many fields. The light intensity as a function of wavelength is known as a spectrum and is used in many analytical systems to determine the composition of materials. lt is well known that the sun produces a wide range of wavelengths and that the water vapor in the Earth's atmosphere blocks many wavelengths from reaching the surface. The areas of the spectrum that can penetrate the atmosphere are known as atmospheric windows. 1-5 Metallurgy has benefitted from the use of spectrometers for determining temperatures of highly heated metals to ensure that the metal processing happens at the right temperatures. Optical spectrometers can be made usi�g prisms, which rely on refraction and gratings. Gratings use diffraction to separate out the various wavelengths. Hybrid devices known as grisms are made from combinations of gratings and prisms and have the advantage of making very compact instruments. In this chapter we will model the sorne of the key equations used m spectrometers and how they relate to the final spectrum.

7.1 Dispersion in a Material The index of refraction of a material is wavelength dependent and typically will have a higher index of refraction at shorter or blue wavelengths than will redder or longer wavelengths of light. The results is that if a beam of white light-one that contains ali wavelengths-pa.sses through a thickness of glass at a non-normal angle of incidence, the emerging light will have separated into its component colors. This separation of colors is known as dispersion and is shown in Fig. 7 .1. 95

Chapter 11

Polarization

An object viewed through certain crystals will appear as a double image. Such crystals demonstrate double refraction; i.e., the two directionally dependent refractive indices through the material result in one image being displaced from the other. l-4 The explanation of this effect led to our current understanding of polarization effects. A wide range of devices incorporate polarization; one of the most common of these in use today is the liquid crystal display. Liquid crystals were discovered in the late l 800s and have only recently become the primary display technology.4 More traditional devices based on polarization include Pockels cells, Nicol prism, Glan-Thompson prism, and more.3 Polarization, which can be induced by reflection, is found in nature, an example of which is sunlight glancing off of a body of water. A similar effect can be created by light interacting with a collection of thin glass plates. It is important to appreciate that there is a difference between simple reflection and a polarization effect on reflected light. Polarization of light is an important topic and often does not receive the consideration that it should be accorded in many optic courses. Here, we are interested in the mathematics involved in calculating polarization using matrix methods. The two formalisms, the Jones calculus and the Mueller calculus provide an effective means of determining the influence of a polarizer on a beam of light. In this chapter we will see how two different yet related matrix approaches are applied in polarization calculations and how to set up the calculations in MATLAB®. Interestingly, under certain conditions these two matrix approaches can be transformed from one to the other.

11.1 Polarized Light The double image seen in a calcite crystal placed on a typed page from a book is one of the more visually striking images associated with the phenomenon of polarization. Double images were first reported in 1669 by Erasmus 155