%%%?================================================================%%%
%%%?This?arithmetic?is?programmed?by?Xiang?Li
%%%?Copyright?2005All?Copyright?Reserved?by?Xiang?Li
%%%?You?can?modify?and?distibute?this?programme?under?the?GPL?license
%%%?If?you?have?any?question?about?this?programme
%%%?please?contact?with?XiangLi@uestc.edu.cn
%%%?================================================================%%%


%%%?================================================================%%%
%%%?first?step:
%%%?assume?we?have?the?inverse?alias?functionthen?we?need?to?determine?and
%%%?establish?the?range?of?x?and?y?of?the?original?image?from?this?function
%%%?================================================================%%%

%%%?read?alias?image?into?current?working?memory?space
origImg?=?imread（‘lena_barrel.tif‘‘tif‘）;
figure;
imshow（origImg‘truesize‘）;
title（‘barrel?image‘）;

%%%?get?width?and?height?of?alias?image
origWidth?=?size（origImg1）;
origHeight?=?size（origImg2）;

%%%?calculate?the?middle?coordinate?of?alias?image
origMidOfX?=?round（origWidth/2）;
origMidOfY?=?round（origHeight/2）;

%%%?shift?the?coordinate?to?the?center?of?the?alias?image
[xOrigIndexyOrigIndex]?=?meshgrid（1:origWidth1:origHeight）;
xNewIndex?=?xOrigIndex（:）?-?origMidOfX;
yNewIndex?=?yOrigIndex（:）?-?origMidOfY;

%%%?transform?rectangular?to?polar?coordinatenote?that?theta?is?measured
%%%?in?radiansand?please?note?that?we?can?think?Cartesian?ordinate?is?just
%%%?the?rectangular?coordinatewe?can?of?course?think?it?like?thisanyway
[thetaradius]?=?cart2pol（xNewIndexyNewIndex）;

%%%?a1?and?a2?are?image?alias?coeffientsso?here?we?produce?the?aliasing?image‘s
%%%?radius?in?polar?coordinates.
%%%?noteinverse?formula:newRadius?=?（a1?-?sqrt（a1?^?2?-?4?*?a2?*?radius））/（2?*?a2）
%%%?is?from?formular:
%%%?aliasRadius?=?a1?*?radius?-?a2?*?radius?.^?2
%%%?which?is?the?alias?function?I?guessedand?it?seems?true.
a1?=?2.0;
a2?=?0.005;
newRadius?=?（a1?-?sqrt（a1?^?2?-?4?*?a2?*?radius））/（2?*?a2）;
newTheta?=?theta;

%%%?now?we?can?map?the?aliasing?rectangular?coordinates?to?its?polar
%%%?counterpart
[xAliasIndexyAliasIndex]?=?pol2cart（newThetanewRadius）;

%%%?here?calculate?the?min?and?max?position
xMinAliasIndex?=?double（round（min（xAliasIndex）））;
xMaxAliasIndex?=?double（round（max（xAliasIndex）））;
yMinAliasIndex?=?double（round（min（yAliasIndex）））;
yMaxAliasIndex?=?double（round（max（yAliasIndex）））;

%%%?calculate?the?to-be-fixed?image‘s?width?and?height?according?to?the?min?and?max?position
newWidth?=?xMaxAliasIndex?-?xMinAliasIndex?+?1;
newHeight?=?yMaxAliasIndex?-?yMinAliasIndex?+?1;

%%%?================================================================%%%
%%%?end?first?step
%%%?================================================================%%%


%%%?================================================================%%%
%%%?second?step:
%%%?for?each?point?in?the?to-be-fixed?imagecompute?the?corresponding?position
%%%?in?th