# Report on Image warping

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1 Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms. CONTENTS Algorthms Inverse dstance weghted nterpolaton methods Calculate weght at every pont Interpolaton Radal bass functon transform mage warpng Whole descrpton Choose the bass functons General method for fold-over free mage warpng Whole descrpton Implement steps: Results Dfferent results for dfferent algorthms Correspondng result under dfferent parameters Future work Trangle dssecton based mage warpng ALGORITHMS Image warpng s the process of deformng a dgtal mage geometrcally. The problem of mage warpng gven a set of scattered translaton vectors s essentally the problem of nterpolatng two functons smultaneously. One functon s for the translaton n the drecton of the x-axs and the other for the translaton n the drecton of the y-axs. Inverse dstance weghted nterpolaton Inverse dstance-weghted nterpolaton methods were orgnally proposed by Shepard and mproved by a number of other authors, notably Franke and Nelson. Calculate weght at every pont For each data pont P, a local approxmaton f ( P) : R 2 R wth ~1~

2 f ( P ) = y = 1,2,..., n s determned. The nterpolaton functon s a weghted average of these local approxmatons, wth weghts dependent on the dstance of the observed pont from the gven data ponts, f ( p) = n = 1 w ( p) f ( p) where 2 f ( p) = y, = 1,..., n. w : R R s the weght functon, whch must satsfy the condton: n w ( p ) = 1, w ( p) = 1, and w ( p) 0, = 1,..., n. = 1 These condtons guarantee the property of nterpolaton. Shepard proposed the followng smple weght functon: w ( p) = n σ ( p) 1 wth σ j = µ d( p, p ) σ ( p) j= 1 j Where d(p, p ) s the dstance between p and p. Radal bass functon transform mage warpng Transformatons based on radal bass functons have proven to be a powerful tool n mage warpng. Images are regarded as 2D objects. In ths respect, an mage s a fnte doman of a plane wth a grey level (or color) assocated wth each pont. A warpng of an mage s then prmarly a transformaton of the plane to tself, and the grey level values are transformed accordng to the transformaton of ther assocated coordnates. Our man concern s the constructon of a mappng of mages (planes) that are determned by the mappng of some anchor ponts - ponts whose mappng s predetermned. Ths requrement leads us to nterpolaton theory.. Whole descrpton In two-dmensonal nterpolaton theory we deal wth computng a functon T ( x ) R 2 2 = y : R satsfyng T ( xk ) yk ( k = 1,2,..., N) = n the anchor ponts. Radal bass functons have proven to be an effectve tool n multvarate nterpolaton problems of scattered data. Gven scattered ponts {x R } 2, and ~2~

3 2 correspondng data values{s } R, we look for an nterpolatory functon T(x) of the form: N T ( x) = a g( x x ) + Aq + α 3 = 1 T 2 x y where A = ( α, α ), α R,1 N, α = { α, α R Eucldean norm on 1 2 } 2 R and 2. Here. denotes the usual g : R + R. T( x ) = F, 1 N. A functon of ths form s usually referred to as radal sum wth a lnear tem. Thus S s determned by N+3 coeffcents n R 2. The computaton of those coeffcents nvolves the soluton of two square lnear systems of sze N+3 each, where N condtons are derved by the nterpolaton requrements. To solve the equaton, we can add an addtonal tem and solve the lnear system defned by followng. The system of equatons for the vectors of unknowns s: k k T k k k T k = ( a 1,..., an and vk = ( α 1, α 2, α 3 ), k = 1,2 u ). The lnear system s descrbed as: Gu H T k u + Hv k = 0 k = b k k k Here b = ( y 1,..., y ), k 1 2 { q, q,1},1 N. N N G { g( q q j )}, j= 1 =, and H s an N 3 matrx wth th row Choose the bass functon Well-known radal bass functons are mult-quadrcs, orgnally proposed by R.Hardy : 2 2 µ / 2 = wth > 0 R ( d) ( d + r ) r and µ 0 Here, µ = ± 1 has been used successfully. General method for fold-over free mage warpng In mage warpng, an addtonal problem to that of nterpolaton s that of mantanng the one-to-one property of the warpng transformaton. Each pont n the transformed mage should correspond to only one pont n the orgnal mage and vce-versa. If the one-to one property s not satsfed the warped functon wll have the appearance of beng folded, rather than only stretched.. ~3~

4 Whole descrpton The one-to-one property of the change of varables can be preserved usng two facts. Frst, the concatenaton of two one-to-one maps must be one-to-one,.e. gven two one-to-one mappngs: (x, y) ( f, g) Wth Jacoban J 1 > 0 and ( f, g) (u, v) Wth Jacoban J 2 > 0, then the combned mappng: (x, y) (u, v) wll have Jacoban = J J 0 (by the chan rule of dfferentaton),and so be one-to-one. J > Second, an overlappng functon of the form (x, y) (x + f, y + g) can be made one-to-one by scalng the functons f and g by an approprate constant α satsfyng 0 < α β ( x, y),so that the mappng (x, y) (x + a f, y a g) s one-to-one. The values β (x, y) are chosen at each pont so that the Jacoban, J, of the rescaled mappng (x + β f(x, y), y + β g(x, y)), gven by the equatons J ( β f 1)( βg + 1) - β f g 2 = x + y y x (where the subscrpts denote partal dervatves), s equal to zero. These quadratc equatons can be solved at each pont n the warpng functons for β usng fnte dfference approxmatons for the partal dervatves. We are only nterested n solutons for whch 0 < β 1, because we requre the nterpolated functon to translate the pxels a fracton of the desred shft β > 0 wthout overshootng the target ( β > 1). The quadratc equaton always passes through J=1 for β = 0, whch leaves only three possble cases: (1) No real roots between 0 and 1. In ths case the Jacoban s postve for all α (0,1] so a s not restrcted by ths pont. (2) One real root b between 0 and 1. In ths case J s postve for 0 < α < β. ~4~

5 (3) Two real roots β 0 and β1between 0 and 1. In ths case a must be less than the smaller root or larger than the larger root for postve J. Therefore, we set each β (x, y) equal to the smallest postve real root on (0,1] or 1 f there are no roots on (0,1]. The scalng parameterα s chosen to be less than or equal to the smallest value of b n the warpng functon. Implement steps: (1), Calculate the partal warpng by a chosen method. (2), Calculate the scalng factor for specfed partal warp. (3), Scale the partal warp (4), Add partal warp to the total warp (5), Check for convergence condton, f scalng factor s equal to 1, turn (7), f not, turn (6). (6), Reposton start Ponts for next teraton. (7), End ths procedure, return total warp and apply t to whole mage; How to calculate the scalng factor: For every pont n an mage, calculate a scalng factor by the followng method and fnd the smallest one to return: (1), Estmate partal dervatves of warp functons, whch s f x, f y, g x, g y (2), Solve quadratc functon ( f x g y f y g x,( f x g y ),1 J mn ) and returns the two real roots. (3), Check all the cases to acqure the proper root. RESULTS Inverse dstance weghted nterpolaton methods ~5~

6 No fold-over free control(one Step) Example 1: µ =1 ~6~

7 Example 2: µ =1 Example 3: µ =1 Radal bass functon transform mage warpng ~7~

8 2 2 u / 2 Bass functon: g ( d) = ( d + r ) µ = ± 1, No fold-over free control(one Step) r=10, µ =1 r=25, µ =1 r=80, µ =1 ~8~

9 r=25, µ =-1 r=80, µ =-1 Example 1 Compare the result by dfferent parameters n the case of no overlap free control Fold-over free control a = a = a = ~9~

10 a = a = a = a = a = a = Example 1, One result wth Overlap free control: r=80, µ =1 ~10~

11 a = a = a = Example 2, One result wth Overlap free control: r=25, µ =-1 Specfcaton wthout Overlap free control ~11~

12 a = a = a = a = a = a = Wth Overlap free control ~12~

13 Example 2, Two results wthout and wth Overlap free control: r=40, µ =1 Specfcaton wthout Overlap free control a = a = ~13~

14 a = Wth Overlap free control Example 3, Two results wthout and wth Overlap free control: r=15, µ =1 FUTURE WORK Utlze the trangle dssecton based method to establsh overlap free soluton. ~14~

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