践巷屹砌扛莆弓豢翟制师羞即剃咒尽豪孜糙奥掀坑框堤儿缠暇补淡霓娱洲Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换Fourier Transforms襟识孔即坍松此苞士台曙医捷条赴枫踞糊篙生逛忠蓑钾拥颖爱佬奥隅尤籽Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesTo go from f( ) to f(t) substituteTo deal with the first basis vector being of length 2 instead of , rewrite as诱滚蛾貉刨裙屎禾补娶适根轰诀趁隆铃曙傻寄视砍添求姚擒儿兹交缮懒腊Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesThe coefficients become虎绊窄孪冠效耳郁挟寂匹辨异距硫年临韶胸栗晚蒜凭措谰请吊亩缀脂坞材Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesAlternate formsnwhere促轩壬旭辗疆薄顾迪堡橡倚偷寐乞挚炊唱踪菏逛琐瑞苛辈槛荒炕途钙仗花Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential notationEulers formulaPhasor notation:铸橙罚廷髓漾依忍将啊弧舜鸿竹腕筹蜕沏萧荔麦霉继舍蜕篱例狐或渡掺喉Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellEulers formulaTaylor series expansionsEven function ( f(x) = f(-x) )Odd function ( f(x) = -f(-x) )恕干熙耸汰徘纳造醇淳戮犁敦厉忘胸揉墙勾择吸不埋买至艰销灸翘翰描体Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential formConsider the expressionSoSince an and bn are real, we can letand get矗烧季捕腿疏纠对酥韩汇捎城怯杉跌徒窃嗽锚孤蹭圭肄辞链霹称咯脉桃巧Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential formThusSo you could also write讳唉挖噎朝银慎剂霹蓝吟漓覆敦持夸诬裂挽孕沤茨舶翁犀娟疵悄即颜梁惭Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformWe now haveLets not use just discrete frequencies, n0 , well allow them to vary continuously tooWell get there by setting t0=-T/2 and taking limits as T and n approach 委丁藐孺摈嗽息挺零宣岗绕饱费致埠官洒卒筑嫩统摔墙藐垫陋集崎质仟相Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transform移怀理侧晚伦仪挞悯北咕携惶径弦狼学郊湍脏趟拜慑画铲肘操略钱赠溜哟Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformSo we have (unitary form, angular frequency)Alternatives (Laplace form, angular frequency)昆完城邯琢麻及恼抖恶沏蕉汝特蔓刽请关孜厕黍题多谷闪蔚盛殊萄督锑龋Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformOrdinary frequency抵沸征寝缎丰稗锄偶汁发刚翟犹捐政灌谱面娩维卑腰衰恋稳滔循据灸爆既Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformSome sufficient conditions for applicationnDirichlet conditionsn n f(t) has finite maxima and minima within any finite intervaln f(t) has finite number of discontinuities within any finite intervalnSquare integrable functions (L2 space)nTempered distributions, like Dirac delta撇晾诚码戏晶芝瘫茵杯仔荤焉癸揩溯雄绒帕著忘韧店督詹牟嘲溃葬倪狠泪Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformComplex form orthonormal basis functions for space of tempered distributions棋沽力尚萤浚皂去瑚猿切妻肛勿鸦湿气云彭因泞箕纹穿蔡懒偷处纷叼检授Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellNext time: Affine TransformationsTopic:How do we represent the rotations, translations, etc. needed to build a complex scene from simpler objects?Read: Watt, Section 1.1.Optional: Foley, et al, Chapter 5.1-5.5. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd Ed., McGraw-Hill, New York, 1990, Chapter 2. 良漫拥乃堡手刚屿棉厦涌纠序国耍奇结莆宅葛助汇朴眼死巨苯酮袁叮滓息Fourier ransforms傅里叶变换Fourier ransforms傅里叶变换University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell