s C The intermediate points - can be described in terms of parametric lines through the corner points -.
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1 Irregular ilinear Inerpolaion Irregular bilinear inerpolaion deermine he value a a given poin by aking he weighed average of i four cloe neighbor. Thi algorihm i nonlinear and more compuaionally inenive han andard bilinear inerpolaion. If he daa being inerpolaed i regularly paced ue he andard verion deailed here. The figure below how an irregular poin e. oin repreen he new poin and poin 1-4 are he bounding poin. The diance i he percenage along he verical ide o poin, he diance i he horizonal. oin are inermediae poin ued for calculaion The edge are repreened a parameric line uing he equaion: The inermediae poin - can be decribed in erm of parameric line hrough he corner poin -. oin oin oin oin Where he inermediae variable are defined a: Uing poin - he poin can be olved for by wo differen parameric line. and The value of can be aained by olving eiher pair of hee equaion, bu here are condiion ha only one or he oher e will give a viable anwer. Mehod 1 Uing he fir e of equaion he value of can be formulaed by rearranging he equaion for.
2 The value of can hen be plugged ino he equaion for. Thi equaion i hen rearranged: The value of,,, and are plugged in o place he oluion in erm of. () ( ) ( )() Through a ignifican bi of algebra hi equaion can be condened and facored ino he erm of he econd order polynomial: Where he variable - are: The value of can be olved for uing he quadraic equaion: Since hi equaion i quadraic, i i likely ha wo eparae roo will be found when olving he equaion. When he objec of hi mehod i inerpolaion (no exrapolaion), deermining which facor i real and which i fale i imple. For any inerpolaion he value of mu be beween 0 and 1. Uing he value of he value of can be olved for uing he value of. fer calculaing he and value, he value a poin i calculaed uing he andard bilinear inerpolaion formula. Mehod 2
3 If he verical uprigh of he inerpolaion area are parallel, he oluion for he quadraic equaion will reurn complex value. If hi occur a econd mehod focuing on he horizonal ecion of he viewing area can be ued inead The ame calculaion from he fir mehod are ued bu wih he poin and inead. Following he ame calculaion, he value of he quadraic coefficien become: The value of i hen calculaed uing he quadraic equaion. The value of can be olved for nd and can be ued in he andard bilinear inerpolaion formula lied above. Mehod 3 If he inerpolaion area become a parallelogram, a hird mehod i required o olve for he value a Wih boh pair of edge parallel, he oluion for he poin become linear and only marginally more complicaed han he andard bilinear inerpolaion mehod. Take he original equaion for and :
4 nd he oluion for given and : lugging in he value for and give: u ince edge and are parallel, and. Thi make he cro erm cancel ou and he remaining equaion are: Thi yem can now be olved a a e of linear equaion. [ ] [ ] [ ] Error ondiion menioned earlier, he quadraic oluion for he and variable preen he problem of olving for he incorrec roo. The phenomenology of hi occurrence i hown below The blue dahed line and boxe indicae he reul from he correc roo. The poin i olved for uing value of and ha are beween 0 and 1. The red boxe and dahed line how he reul from he oher oluion o hi quadraic. he value of,,, and he line formed ill inerec a poin, bu he value make le ene. In paricular he horizonal inerecion line ha now become he verical and vie vera. noher problem ha can arie i he condiion of parallel line in he inerpolaion region. Thi condiion i wha
5 drive he need for muliple oluion mehod. Of coure, here i he condiion of a recangular region, or more generally a parallelogram, which would caue boh mehod o fail. Thi condiion mu be checked for explicily. For he verical edge of he earch region, he parallelim condiion can be checked for uing he cro produc of he wo verical edge. If hen Likewie for he horizonal edge he cro produc can be ued o deermine parallelim. If hen The proceing rucure ue hi informaion o deermine which mehod o ue when performing he inerpolaion. The proceing logic i a follow: ropery of ahinon.com La Updaed ecember 10, 2011
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