M y. Image Warping. Targil 7 : Image Warping. Image Warping. 2D Geometric Transformations. image filtering: change range of image g(x) = T(f(x))

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1 Hebrew Universi Image Processing - 6 Image Warping Hebrew Universi Image Processing - 6 argil 7 : Image Warping D Geomeric ransormaions hp:// Man slides rom Seve Seiz and Aleei Eros Image Warping Hebrew Universi Image Processing - 6 Image Warping Hebrew Universi Image Processing - 6 image ilering: change range o image g() (()) image ilering: change range o image g() h(()) g image warping: change domain o image g() (()) image warping: change domain o image g() (()) g 3 4 Hebrew Universi Image Processing - 6 Parameric (global) warping Eamples o parameric warps: Hebrew Universi Image Processing - 6 Parameric (global) warping ranslaion roaion aspec aine perspecive clindrical 5 p (,) p (, ) ransormaion is a coordinae-changing machine: p (p) Wha does i mean ha is global? Is he same or an poin p can be described b jus a ew numbers (parameers) Le s represen as a mari: p M*p M 6

2 Scaling Hebrew Universi Image Processing - 6 Scaling a coordinae means mulipling each o is componens b a scalar Uniorm scaling means his scalar is he same or all componens: Scaling Non-uniorm scaling: dieren scalars per componen: Hebrew Universi Image Processing - 6 X, Y Scaling Hebrew Universi Image Processing - 6 -D Roaion Hebrew Universi Image Processing - 6 Scaling operaion: Or, in mari orm: a b (, ) (, ) a b θ cos(θ) - sin(θ) sin(θ) cos(θ) scaling mari S Wha s inverse o S? 9 -D Roaion (, ) (, ) θ φ Hebrew Universi Image Processing - 6 r cos (φ) r sin (φ) r cos (φ θ) r sin (φ θ) rig Ideni r cos(φ) cos(θ) r sin(φ) sin(θ) r sin(φ) cos(θ) r cos(φ) sin(θ) Subsiue cos(θ) - sin(θ) sin(θ) cos(θ) -D Roaion his is eas o capure in mari orm: cos sin ( θ) sin( θ) ( θ) cos( θ) R Hebrew Universi Image Processing - 6 Even hough sin(θ) and cos(θ) are nonlinear uncions o θ, is a linear combinaion o and is a linear combinaion o and Wha is he inverse ransormaion? Roaion b θ For roaion marices, de(r) so R R

3 Marices Wha pes o ransormaions can be represened wih a mari? D Ideni? D Scale around (,)? s * s s * s s s Hebrew Universi Image Processing Marices Wha pes o ransormaions can be represened wih a mari? D Roae around (,)? cosθ* sinθ* sinθ * cosθ * D Shear? sh * sh * Hebrew Universi Image Processing - 6 cosθ sinθ sh Shear along ais sinθ cosθ sh 4 Marices Wha pes o ransormaions can be represened wih a mari? Hebrew Universi Image Processing - 6 Marices Wha pes o ransormaions can be represened wih a mari? Hebrew Universi Image Processing - 6 D Mirror abou Y ais? D ranslaion? NO! D Mirror over (,)? Onl linear D ransormaions can be represened wih a mari 5 6 Hebrew Universi Image Processing - 6 All D Linear ransormaions Linear ransormaions are combinaions o Scale, Roaion, Shear, and Mirror Properies o linear ransormaions: Origin maps o origin Lines map o lines Parallel lines remain parallel Raios are preserved Closed under composiion a c b d Hebrew Universi Image Processing - 6 Homogeneous Coordinaes Q: How can we represen ranslaion as a 33 mari? a c be d g i h k j l 7 8 3

4 4 9 Hebrew Universi Image Processing - 6 Homogeneous Coordinaes Homogeneous coordinaes represen coordinaes in dimensions wih a 3-vecor coords homogeneou s λ λ λ Mulipling b a scalar does no change our poin! are all equivalen o: Hebrew Universi Image Processing - 6 Homogeneous Coordinaes Q: How can we represen ranslaion as a 33 mari? A: Using he righmos column: ranslaion Hebrew Universi Image Processing - 6 ranslaion Eample o ranslaion Homogeneous Coordinaes Hebrew Universi Image Processing - 6 Homogeneous Coordinaes Add a 3rd coordinae o ever D poin (,, w) represens a poin a locaion (/w, /w) (,, ) represens a poin a inini (,, ) is no allowed Convenien coordinae ssem o represen man useul ransormaions (,,) or (4,,) or (6,3,3) 3 Hebrew Universi Image Processing - 6 Basic D ransormaions Basic D ransormaions as 33 marices Θ Θ Θ Θ cos sin sin cos sh sh ranslae Roae Shear s s Scale 4 Hebrew Universi Image Processing - 6 Aine ransormaions Aine ransormaions are combinaions o Linear ransormaions, and ranslaions Properies o aine ransormaions: Origin does no necessaril map o origin Lines map o lines Parallel lines remain parallel Closed under composiion Aine ransormaion beween images e d c b a e d c b a

5 Aine ransormaions Hebrew Universi Image Processing - 6 Hebrew Universi Image Processing - 6 Projecive ransormaions (homograph) When do we mee hem? When he camera is scanning a plane in parallel, combining roaion, zoom. Camera moving plane Projecive ransormaions a d Aine ransormaions, and Projecive warps w g Properies o projecive ransormaions: Origin does no necessaril map o origin Lines map o lines Parallel lines do no necessaril remain parallel Closed under composiion b e h c i w When muliple ransormaion are applied normalizaion is done onl in he las sage 5 6 Hebrew Universi Image Processing - 6 Projecive ransormaions Projecive ransormaions beween images a b c w g h d e w g h a d g b e h c Hebrew Universi Image Processing - 6 Projecive ransormaions Cenral projecion map poins on one plane o poins on anoher plane 7 8 Hebrew Universi Image Processing - 6 Projecive ransormaions Cenral projecion: lines are mapped o lines (represened b planes hrough he projecion cener) Hebrew Universi Image Processing - 6 Projecive ransormaions Eamples o projecive ransormaions : Camera roaion (same cener o projecion) / camera varing he ocal lengh 9 Image o a plane and i s shadow on anoher plane ransormaion beween images induced b a world plane 3 5

6 Hebrew Universi Image Processing - 6 Removing perspecive disorion Hebrew Universi Image Processing - 6 Compuing ransormaion Corresponding ses o poins enable compuing he ransormaion mari Original image Warped image each poin can conribue equaions ( g h ) a b c ( g h ) d e 3 3 Mari Composiion ransormaions can be combined b mari muliplicaion Hebrew Universi Image Processing - 6 Mari Composiion Hebrew Universi Image Processing - 6 Marices are a convenien and eicien wa o represen a sequence o ransormaions General purpose represenaion Hardware mari mulipl cosθ sinθ s sinθ cosθ s w w p (, ) R(Θ) S(s,s ) p p ( * (R * (S*p) ) ) p (*R*S) * p Mari Composiion Be aware: order o ransormaions maers Mari muliplicaion is no commuaive Hebrew Universi Image Processing - 6 Hebrew Universi Image Processing - 6 Roaing Abou An Arbirar Poin Wha happens when ou appl a roaion ransormaion o an objec ha is no a he origin? p * R * S * p?

7 Hebrew Universi Image Processing - 6 Roaing Abou An Arbirar Poin Wha happens when ou appl a roaion ransormaion o an objec ha is no a he origin? I ranslaes as well How Do We Fi i? Hebrew Universi Image Processing - 6 How do we roae an abou an arbirar poin? Hin: we know how o roae abou he origin o a coordinae ssem Hebrew Universi Image Processing - 6 Roaing Abou An Arbirar Poin Hebrew Universi Image Processing - 6 D image ransormaions p (R)*p Where should we add he scale? 39 hese ransormaions are a nesed se o groups Closed under composiion and inverse is a member 4 Hebrew Universi Image Processing - 6 D image ransormaions Hebrew Universi Image Processing - 6 Compuing ransormaion Corresponding ses o poins enable compuing he ransormaion mari hese ransormaions are a nesed se o groups each poin can conribue equaions How man poins are required or he dieren ransormaions (ranslaion, rigid, similari, aine, projecive)? Closed under composiion and inverse is a member 4 4 7

8 Image warping Hebrew Universi Image Processing - 6 Forward warping Hebrew Universi Image Processing - 6 (,) (,) g(, ) (,) (,) g(, ) Given a coordinae ransorm (, ) h(,) and a source image (,), how do we compue a ransormed image g(, ) ((,))? Send each piel (,) o is corresponding locaion (, ) (,) in he second image Q: wha i piel lands beween wo piels? Forward warping Hebrew Universi Image Processing - 6 Inverse warping Hebrew Universi Image Processing - 6 (,) (,) g(, ) - (,) (,) g(, ) Send each piel (,) o is corresponding locaion (, ) (,) in he second image Q: wha i piel lands beween wo piels? A: disribue color among neighboring piels (, ) Ge each piel g(, ) rom is corresponding locaion (,) - (, ) in he irs image Q: wha i piel comes rom beween wo piels? Known as splaing Inverse warping Hebrew Universi Image Processing - 6 Bilinear inerpolaion Hebrew Universi Image Processing - 6 Sampling a (,): - (,) (,) g(, ) Ge each piel g(, ) rom is corresponding locaion (,) - (, ) in he irs image Q: wha i piel comes rom beween wo piels? A: Inerpolae color value rom neighbors neares neighbor, bilinear, Gaussian, bicubic

9 Hebrew Universi Image Processing - 6 Forward vs. inverse warping Q: which is beer? A: usuall inverse eliminaes holes however, i requires an inverible warp uncion no alwas possible... Warping in Malab How do we avoid loops? - Use meshgrid o represen image indees Hebrew Universi Image Processing - 6 Use malab inerp o perorm he inerpolaion Find holes (NaN) and ou o range inerpolaion values