Bruce A. Draper & J. Ross Beveridge, January 25, Geometric Image Manipulation. Lecture #1 January 25, 2013
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1 Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Image Manplaton Lectre # Janar 5,
2 Brce A. Draper & J. Ross Beerdge, Janar 5, Image Manplaton: Contet To start wth the obos, an mage s a D arra of pels The pel locatons represent ponts on the mage plane The pel ales represent measrements of lght at those locatons Color mages : energes b freqenc ranges (RGB: three oerlappng ranges) Intenst mages : aerage energ across the sble range Yor CS ra tracers shold hae taght o abot mage formaton To drectl compare two mages, the shold be regstered Geometrcall : to presere the spatal pattern, whch pel n mage lnes p wth each pel n mage? Photometrcall : f measres a certan amont of energ n mage, t shold mpl the same amont of energ n mage
3 Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Regstraton It s not enogh for two matchng mages to hae the same set of pel ales The hae to be n the same relate postons Image from CalTech56 data set Otherwse, these two mages match
4 Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Regstraton (II) Geometrc regstraton fnds a mappng that maps one mage onto the other We wll lmt orseles to lnear transformaton We shold be able to regster these
5 Brce A. Draper & J. Ross Beerdge, Janar 5, Regstraton formalsm We denote an mage as a D fncton: I, ( ) Or, n homogeneos coordnates: I,, w ( ) The goal s to fnd the transformaton matr G sch that: I ( * (,, w) I j * G * ) # + '- '- '- &,
6 Brce A. Draper & J. Ross Beerdge, Janar 5, Interpolaton (foreshadow ) Seldom get nteger-tonteger mappng. Geometr part comptes real-aled postons of pel centers. We wll worr abot how to nterpolate ales later.
7 Brce A. Draper & J. Ross Beerdge, Janar 5, Image Transformatons The smplest set of transformatons are translaton, rotaton, and scale Together these are called the smlart transform. Smlart transforms hae degrees of freedom. In matr form these are: ' s # & # s& ' # & ' & # & cosφ snφ# & # snφ cosφ scale and... rotaton
8 Brce A. Draper & J. Ross Beerdge, Janar 5, Image Transformatons : Translaton # & # & # & t t Translaton (note the D homogeneos coordnates)
9 Brce A. Draper & J. Ross Beerdge, Janar 5, Translaton Appled to Images Translate n ' ' # &' Translate - n # & ( ( '(
10 Brce A. Draper & J. Ross Beerdge, Janar 5, Scale Appled to Images Note the orgn Scale Unforml b Scale Unforml b.5.5 ' ' '.5 ' # & ' # & '
11 Brce A. Draper & J. Ross Beerdge, Janar 5, Rotaton Appled to Images Rotate b 5 Rotate b -5 Note that a poste rotaton rotates the poste X as toward the poste Y as
12 Brce A. Draper & J. Ross Beerdge, Janar 5, Remember Composton of Matrces To rotate b θ arond a pont (,): ( ) sn( θ) cos θ ' ' ' ' sn( θ) cos( θ) ' ' # & '# & '# &' ( ) sn( θ) sn( θ) cos( θ) ( ) cos( θ) cos θ ' ' sn( θ) cos( θ) sn θ # & '# ( ) sn( θ) sn( θ) cos( θ) + ( ) cos( θ) + cos θ sn( θ) cos( θ) sn θ # ' ' &' ' ' &'
13 Brce A. Draper & J. Ross Beerdge, Janar 5, Smlart & Affne Transformatons All the smlart transforms can be combned nto one generc matr: & # & a b c #& # Hnt: dagonal d e f terms are not eqal, and b -d. Bt Ths matr does more. What? hnt: two more transformaton tpes nclded. hnt: 6 degrees of freedom (DOF) How can o specf ths matr? Ths s eqalent to addng two shear parameters (or neqal scalng & one shear).
14 Brce A. Draper & J. Ross Beerdge, Janar 5, Affne Eamples.5 ' ' # & ' ' ' # & '
15 Brce A. Draper & J. Ross Beerdge, Janar 5, Specfng Affne Transformatons There are s nknowns n the matr (a throgh f) If o specf one pont n the sorce mage and a correspondng pont n the target mage, that elds two eqatons: a + b + c d + e So prodng three pont-to-pont correspondences specfes an affne matr + f
16 Brce A. Draper & J. Ross Beerdge, Janar 5, Affne Specfcaton: Eample There s one affne transformaton that wll map the green pont on the rght to the green pont on the left, and algn the red and ble ponts too.
17 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng Affne Transformatons These lnear eqatons can be easl soled: WLOG, assme then c and f so: ( ) ( ) ( ) ( ) ( ) b b b b b a b a b a # & ' Calclaton of a, b & c s ndependent of calclaton of e, f & g.
18 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng Affne (cont.) Ths can be sbsttted n to sole for a The same process wth s soles for d,e,f Abot the WLOG: It was tre becase o can translate the orgnal coordnate sstem b (-, - ) So what do o do to compensate? Alternatel, set p a sstem of lnear eqatons and sole... Wll show ths for a harder case shortl.
19 Brce A. Draper & J. Ross Beerdge, Janar 5, Perspecte Transformatons We can go beond jst affne transformatons. We can do an perspecte transformaton of a plane to a plane. Therefore, we can model an mage as a plane n space, and project t onto an other mage. How does ths dffer from the perspecte projecton ppelne n CS?
20 Brce A. Draper & J. Ross Beerdge, Janar 5, Perspecte Matr ' ' ' ' # w& ' a b c ' ' d e f ' ' # g h & '# & ' ' w, ' w Wh does element [,]? How man ponts are needed to specf ths matr?
21 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng for Perspecte For correspondng ponts prodce eght eqatons, eght nknowns --- bt we can t obsere w ' ' w w a g d g + b + h + e + h + c + + f +
22 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng (cont.) Mltpl to get rd of the fracton ( g + h + ) + b ( g + h + ) d + e + f Now, remember that the s, s, s & s are known; grop the nknown terms a d + b + e + + c f a g g + c h h
23 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng (III) And epress the reslt as a sstem of lnear eqatons # & # & # & h g f e d c b a
24 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng (IV) Fnall, nert the constant matr and sole # & # & # & h g f e d c b a
25 Brce A. Draper & J. Ross Beerdge, Janar 5, Solng (V) : Qestons Is there alwas a solton? Is the solton alwas nqe? Under what condtons s the matr nertble?
26 Brce A. Draper & J. Ross Beerdge, Janar 5, Perspecte Image Transforms (Intton) What does the followng matr do? & #
27 Brce A. Draper & J. Ross Beerdge, Janar 5, Matr Decomposton Note that sch decompostons are: not nqe (wh?) dffclt to ntt # & # & # & orgnal Scale b Rotaton b 5
28 Brce A. Draper & J. Ross Beerdge, Janar 5, More Inttons What wll the followng matr do? & # More specfcall, what wll t do to the graffe mage?
29 Brce A. Draper & J. Ross Beerdge, Janar 5, Check Yor Inttons What s gong on here?
30 Brce A. Draper & J. Ross Beerdge, Janar 5, More Intton Checkng Part of what o are seeng s a scale effect poste terms n the bottom row create larger w ales, and therefore smaller, ales Somethng mch werder s also gong on: What happens when +? How do o nterpret ths geometrcall? Isn t the perspecte transform lnear? So how do o select transform matrces?
31 Brce A. Draper & J. Ross Beerdge, Janar 5, Perspecte Transform of D Planes Contned. Recall the basc eqaton for the perspecte transform ' a b c ' ' ' ' ' d e f ' ' # w& ' # g h & '# & ' ' w, ' w The onl practcal wa to specf an mage transform s b prodng for pont correspondences
32 Brce A. Draper & J. Ross Beerdge, Janar 5, Comptng Transformatons Remember how to bld a transformaton from for pont correspondences. # & # & # & h g f e d c b a
33 Brce A. Draper & J. Ross Beerdge, Janar 5, Comptng... So f we want the followng mappng: (,) (,), (,) (,), (5,) (5,5), (5,) (5,9) # &
34 Brce A. Draper & J. Ross Beerdge, Janar 5, More Comptng... What does Ths sa abot? & # & # & a # &.7 # b c.7 d.77 5 e 5 f. 5 g.97 9 h Remember the earler WLOG? c, M - & ector How does Ths alter t?
35 Brce A. Draper & J. Ross Beerdge, Janar 5, elds & #
36 Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Transformaton : Reew Goal: I (,,w) I j (G [,,] T ) Transformaton classes: Smlart ( DoF : translaton, rotaton, scale) Affne (6 DoF) Perspecte (8 DoF) Specfed a pont correspondences
37 Brce A. Draper & J. Ross Beerdge, Janar 5, Stll To Do. Photometrc regstraton Interpolaton Transformatons are not nteger to nteger Then we can tackle matchng nder aros transformaton classes
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