Lecture 31 Change of Basis of Compression Images Transformations Matrix Pixel 州 0 xi grayscale Exit image f 256 8bits 512 E 512 n if R 2 color 2 hi 3 512 overload 512 JPEG Joint Photographic Experts Group age X of basis ⼆ 仇焦 n current basis eueg grayscale o 相近 grayscale 很 Standard basis 7 記錄 的 pixels no use is ill Better basis 1 1 vector can completely give Solid the information on a image Fourier basis coeffi 8x8 f64 Hill 512 wn signal p change the lossless 2懼 basis 7coeffsc lossly compression 512 throw away small weft threshing ˇ 不 记⼼ gbe3，4 Video squenǔduwd Wavelets H lik Wavelets i 州 州 Favier 0 lossless step P GW ⼆ find a it ⼗ GW8 coefficients Cg nwc 吖 111 側 CWP ⼆ lgbasis g ui f些 ⼈ p fast ǐǜu inv LFF NT 20Few is enough orthogonal Orthonormal WL W to good WO Wait find Ǘ a Change Gwnf of basis W the Suppose it o with it Dnewbais basis have with linear new basis vectors comb 以 如 Tiuxn to respect has matrix respect has V1 to V8 8 W8 wi matrix A B What relation B conclusion It similar Bin M t change of basis ⼼比 家 ⼆ w what is A I know using in V8 Tag CMtwvt.MG V8 GTM IX TN t GND t.it CHU anV1 ⼗ Qz Nzt TNDidahtdzz.LV a an A u TND.TN Tcompktly from Bacause every X WVU basis taa V8 ⼗ in ⼗ GSLV8 an a i. k eigenvector basis Tcu what ⼆ is ⼈以 A n. if but find in too expensive ffee 器 6.3 Mdt 6.4 A 6.5 ⼆ Au et and 多 AT Positive definite 6.6 Similar real ⼀ 2enough evectors BMÀM 3 can find orthonormal 赢入 点 NÀM UIÚ A 6，7 a Stpl SVD 器 Au i Example i u general solution 比七 ⼆ GǙX tczeixztcj stp2 MX A is singular Hìi 他可find that A的 ⼆ N 八 ⼆ 0 Ptznw 2入 ⼆ 0 八 Aiantiymmtric 入 pure imaginary o Mr 入⼆ 0 坑 上 snj QU I g ucttcěxitaǚxzt b No at ti Uco 比 Cz XN axlt Geii t GX3 aěytaehtaeiy uu ftp.e 2Th Eht 管 ti Orthogonal when AÁ 2以 E it t ⼆ EN 五元 x erector ÁA Sym Antisym Orthogonal c.ci I if ins A uurěù ětsǒ51 了 y em 5 IQ ⽇了 以 I 八 ⼆ 八七 0 八⼆ 2 州1 啡 ⼼ 啦 a diagonaltable Yes all c.CC b doesn't matter Symmetric all a C Real Positive 咖比 no ⼩ semi Definite Czo e Markov matrix 7 入⼆ 1 其中 ⼀个 others No 1 因為 要 stable 是 projector 7 九 O.rl CO FP ⼆ orz firsts orthogonal vector value so 九 三八 value SVD 3 A every orthogonal Atothngonalkdiag ⼆ UEV ATA_i.lv uai Eui symmuic VEVT ⼆ ⼆ V is the euectov Tiìnf 扯 sgni of ÁA Aiiir NIU 3 2 umpǒyuvzi ⼩ ii 68 a vector not 5 in the I M Null Space NLA SUD 1 i V2 NO 4 A lsymmtric Orthogonal 九 be can sym IN 1 Orthgmd 1 In 入 is real i A posdef I Qx ⼼ No N IQXHMN IMI INNH.IN no 入 ⼆ 重根 有 对 ⾓化 了 Yes 可 Non singular Yes Show 主 Projection isymmw z.pt p 肛 is a Projection 1 I ÌHI i 本 肝 2⽉ 扛 i i A⼆⽉ 埔 例 Ortho f 2⽉ 3 A At Ii ICHI A红 ÍCHI 2⼯ I 肛 肛 I 八 入 i 1 2，0 入 1，0_n.eu