Е. Деза_ М.М. Деза. Энциклопедический словарь расстояний (2008) (1185330), страница 66
Текст из файла (страница 66)
èÓÎÛ˜ÂÌÌÓ ‡ÒÒÚÓflÌË ̇Á˚‚‡ÂÚÒfl ‡ÒÒÚÓflÌËÂÏ ÎÂÒÌÓ„Ó ÔÓʇ‡Ë ÔÓÎÛ˜ÂÌ̇fl ‰Ë‡„‡Ïχ ÇÓÓÌÓ„Ó Ì‡Á˚‚‡ÂÚÒfl ‰Ë‡„‡ÏÏÓÈ ÎÂÒÌÓ„Ó ÔÓʇ‡ ÇÓÓÌÓ„Ó.ê‡ÒÒÚÓflÌË ÒÍÓθÊÂÌËflèÛÒÚ¸ í – ̇ÍÎÓÌ̇fl ÔÎÓÒÍÓÒÚ¸ ‚ 3, ÔÓÎÛ˜ÂÌ̇fl ÔÓÒ‰ÒÚ‚ÓÏ ‚‡˘ÂÌËfl x 1 x2πÔÎÓÒÍÓÒÚË ‚ÓÍÛ„ x 1 -ÓÒË Ì‡ Û„ÓÎ α, 0 < α < , Ò ÍÓÓ‰Ë̇ÚÌÓÈ ÒËÒÚÂÏÓÈ, ÍÓÚÓ‡fl2ÔÓÎÛ˜Â̇ ÔÓÒ‰ÒÚ‚ÓÏ ‡Ì‡Îӄ˘ÌÓ„Ó ‚‡˘ÂÌËfl ÍÓÓ‰Ë̇ÚÌÓÈ ÒËÒÚÂÏ˚ x 1 x2-ÔÎÓÒÍÓÒÚË. ÑÎfl ÚÓ˜ÍË q ∈ T , q = ( x1 (q ), x 2 (q )) ÓÔ‰ÂÎËÏ ‚˚ÒÓÚÛ h(q) Í‡Í Â x 3 -ÍÓÓ‰Ë̇ÚÛ ‚ 3. í‡ÍËÏ Ó·‡ÁÓÏ, h(q ) = x 2 (q ) ⋅ sin α. èÛÒÚ¸ P = {p1 , …, pk } ⊂ T , k ≥ 2.ê‡ÒÒÚÓflÌËÂÏ ÒÍÓθÊÂÌËfl ([AACL98]) dskew ̇Á˚‚‡ÂÚÒfl ÔÓÓʉ‡˛˘Â ‡ÒÒÚÓflÌËÂÇÓÓÌÓ„Ó Ó·Ó·˘ÂÌÌÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( P, dskew , T ) (‰Ë‡„‡Ïχ ÒÍÓθÊÂÌËflÇÓÓÌÓ„Ó), ÓÔ‰ÂÎÂÌÌÓ ͇Ídskew (q, r ) = d E (q, r ) + (h(r ) − h(q )) = d E (q, r ) + sin α( x 2 (r ) − x 2 (q )),ËÎË, ‚ Ó·˘ÂÏ ÒÎÛ˜‡Â,dskew (q, r ) = d E (q, r ) + k ( x 2 (r ) − x 2 (q ))‰Îfl ‚ÒÂı q,r ∈ T, „‰Â dE – Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ, ‡ k ≥ 0 – ÍÓÌÒÚ‡ÌÚ‡.20.3. ÑêìÉàÖ êÄëëíéüçàü ÇéêéçéÉéê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl ÓÚÂÁÍÓ‚ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl (ÏÌÓÊÂÒÚ‚‡) ÓÚÂÁÍÓ‚ dsl ÂÒÚ¸ ÔÓÓʉ‡˛˘Â ‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó Ó·Ó·˘ÂÌÌÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( A, dls , 2 ) (ÎËÌÂÈ̇fl ‰Ë‡„‡ÏχÇÓÓÌÓ„Ó, ÔÓÓʉÂÌ̇fl ÓÚÂÁ͇ÏË), ÓÔ‰ÂÎÂÌÌÓ ͇Ídsl ( x, Ai ) = inf d E ( x, y),y ∈Ai„‰Â ÏÌÓÊÂÒÚ‚Ó ÔÓÓʉ‡˛˘Ëı ˝ÎÂÏÂÌÚÓ‚ A = {A1 , …, Ak }, k ≥ 2 ÂÒÚ¸ ÏÌÓÊÂÒÚ‚Ó ÔÓÔ‡ÌÓ ÌÂÔÂÂÒÂ͇˛˘ËıÒfl ÓÚÂÁÍÓ‚ Ai = [ai bi ] Ë d E ÂÒÚ¸ Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ.
àÏÂÌÌÓ,d E ( x, ai ),ÂÒÎËdls ( x, Ai ) = d E ( x, bi ),ÂÒÎËTd ( x − a , ( x − ai ) (bi − ai ) (b − a )), ÂÒÎËiii2 Ed E ( ai , bi )x ∈ Rai ,x ∈ Rbi ,x ∈ 2 \ {Rai ∪ Rbi },„‰Â ai = {x ∈ 2 : (bi − ai )T ( x − ai ) < 0}, Rbi = {x ∈ 2 : ( ai − bi )T ( x − bi ) < 0}.É·‚‡ 20. ê‡ÒÒÚÓflÌËfl ‰Ë‡„‡ÏÏ ÇÓÓÌÓ„Ó297ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl ‰Û„ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl (ÏÌÓÊÂÒÚ‚‡ ÍÛ„Ó‚˚ı) ‰Û„ dca ÂÒÚ¸ ÔÓÓʉ‡˛˘Â ‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó Ó·Ó·˘ÂÌÌÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( A, dca , 2 ) (ÎËÌÂÈ̇fl ‰Ë‡„‡Ïχ ÇÓÓÌÓ„Ó, ÔÓÓʉÂÌ̇fl ‰Û„‡ÏË ÓÍÛÊÌÓÒÚÂÈ), ÓÔ‰ÂÎÂÌÌÓ ͇Ídca ( x, Ai ) = inf d E ( x, y),y ∈Ai„‰Â ÔÓÓʉ‡˛˘Â ÏÌÓÊÂÒÚ‚Ó A = {Ai , …, Ak }, k ≥ 2 ÂÒÚ¸ ÏÌÓÊÂÒÚ‚Ó ÔÓÔ‡ÌÓ ÌÂÔÂÂÒÂ͇˛˘ËıÒfl ‰Û„ ÓÍÛÊÌÓÒÚÂÈ Ai (ÏÂ̸¯Ëı ËÎË ‡‚Ì˚ı ÔÓÎÛÓÍÛÊÌÓÒÚflÏ) Ò‡‰ËÛÒÓÏ ri Ë ˆÂÌÚÓÏ ‚ xci , ‡ dE – Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ. àÏÂÌÌÓ Ù‡ÍÚ˘ÂÒÍË,dca ( x, Ai ) = min{d E ( x, ai ), d E ( x, bi ),| d E ( x, xci ) − ri |},„‰Â ai Ë bi – ÍÓ̈‚˚ ÚÓ˜ÍË ‰Û„Ë A i .ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl ÓÍÛÊÌÓÒÚÂÈê‡ÒÒÚÓflÌËÂÏ ÇÓÓÌÓ„Ó ‰Îfl (ÏÌÓÊÂÒÚ‚‡) ÓÍÛÊÌÓÒÚÂÈ dcl ̇Á˚‚‡ÂÚÒfl ÔÓÓʉ‡˛˘Â ‡ÒÒÚÓflÌË ӷӷ˘ÂÌÌÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( A, dcl , 2 ) (ÎËÌÂÈ̇fl ‰Ë‡„‡Ïχ ÇÓÓÌÓ„Ó, ÔÓÓʉÂÌ̇fl ÓÍÛÊÌÓÒÚflÏË), ÓÔ‰ÂÎÂÌÌÓ ͇Ídcl ( x, Ai ) = inf d E ( x, y),y ∈Ai„‰Â ÔÓÓʉ‡˛˘Â ÏÌÓÊÂÒÚ‚Ó A = {A1 , …, Ak }, k ≥ 2 ÂÒÚ¸ ÏÌÓÊÂÒÚ‚Ó ÔÓÔ‡ÌÓ ÌÂÔÂÂÒÂ͇˛˘ËıÒfl ÓÍÛÊÌÓÒÚÂÈ A i Ò ‡‰ËÛÒÓÏ ri Ë ˆÂÌÚÓÏ ‚ xci , ‡ dE – Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ.
àÏÂÌÌÓ, Ù‡ÍÚ˘ÂÒÍËdca ( x, Ai ) = | d E ( x, xci ) − ri | .ÑÎfl ÎËÌÂÈÌ˚ı ‰Ë‡„‡ÏÏ ÇÓÓÌÓ„Ó, ÔÓÓʉÂÌÌ˚ı ÓÍÛÊÌÓÒÚflÏË, ÒÛ˘ÂÒÚ‚ÛÂÚÏÌÓ„Ó ‡Á΢Ì˚ı ÔÓÓʉ‡˛˘Ëı ‡ÒÒÚÓflÌËÈ. ç‡ÔËÏÂ, dcl* ( x, Ai ) = d E ( x, xci ) − riËÎË dcl* ( x, Ai ) = d E2 ( x, xci ) − ri2 (‰Ë‡„‡Ïχ ÇÓÓÌÓ„Ó ÔÓ ã‡„ÂÛ).ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl ӷ·ÒÚÂÈê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ‰Îfl ӷ·ÒÚÂÈ dar ÂÒÚ¸ ÔÓÓʉ‡˛˘Â ‡ÒÒÚÓflÌË ÇÓÓÌÓ„ÓÓ·Ó·˘ÂÌÌÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( A, dar , 2 ) (‰Ë‡„‡Ïχ ӷ·ÒÚÂÈ ÇÓÓÌÓ„Ó),ÓÔ‰ÂÎÂÌÌÓ ͇Ídar ( x, Ai ) = inf d E ( x, y),y ∈Ai„‰Â A = {A1 , …, Ak ), k ≥ 2 ÂÒÚ¸ ÒÓ‚ÓÍÛÔÌÓÒÚ¸ ÔÓÔ‡ÌÓ ÌÂÔÂÂÒÂ͇˛˘ËıÒfl Ò‚flÁÌ˚ıÁ‡ÏÍÌÛÚ˚ı ÏÌÓÊÂÒÚ‚ (ӷ·ÒÚÂÈ), Ë dE – Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ.ëΉÛÂÚ Ó·‡ÚËÚ¸ ‚ÌËχÌË ̇ ÚÓ, ˜ÚÓ ‰Îfl β·Ó„Ó Ó·Ó·˘ÂÌÌÓ„Ó ÔÓÓʉ‡˛˘Â„ÓÏÌÓÊÂÒÚ‚‡ A = {A1 , …, Ak ), k ≥ 2 ÏÓÊÌÓ ËÒÔÓθÁÓ‚‡Ú¸ ‚ ͇˜ÂÒÚ‚Â ÔÓÓʉ‡˛˘Â„Ó‡ÒÒÚÓflÌËfl ÇÓÓÌÓ„Ó ı‡ÛÒ‰ÓÙÓ‚Ó ‡ÒÒÚÓflÌË ÓÚ ÚÓ˜ÍË ı ‰Ó ÏÌÓÊÂÒÚ‚‡ Ai :: dHaus ( x, Ai ) = sup d E ( x, y), „‰Â dE – Ó·˚˜ÌÓ ‚ÍÎË‰Ó‚Ó ‡ÒÒÚÓflÌËÂ.y ∈Ai298ó‡ÒÚ¸ V.
ê‡ÒÒÚÓflÌËfl ‚ ÍÓÏÔ¸˛ÚÂÌÓÈ ÒÙÂÂñËÎË̉˘ÂÒÍÓ ‡ÒÒÚÓflÌËÂñËÎË̉˘ÂÒÍÓ ‡ÒÒÚÓflÌË dcyl ÂÒÚ¸ ‚ÌÛÚÂÌÌflfl ÏÂÚË͇ ̇ ÔÓ‚ÂıÌÓÒÚË ˆËÎË̉‡ S, ÍÓÚÓ‡fl ËÒÔÓθÁÛÂÚÒfl ‚ ͇˜ÂÒÚ‚Â ÔÓÓʉ‡˛˘Â„Ó ‡ÒÒÚÓflÌËfl ÇÓÓÌÓ„Ó ‰ÎflˆËÎË̉˘ÂÒÍÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( A, dcyl , S ) ÖÒÎË ÓÒ¸ ˆËÎË̉‡ ‰ËÌ˘ÌÓ„Ó‡‰ËÛÒ‡ ‡ÁÏ¢Â̇ ̇ ı3 -ÓÒË ‚ 3 , ÚÓ ˆËÎË̉˘ÂÒÍÓ ‡ÒÒÚÓflÌË ÏÂÊ‰Û Î˛·˚ÏËÚӘ͇ÏË x,y ∈ S Ò ˆËÎË̉˘ÂÒÍËÏË ÍÓÓ‰Ë̇ڇÏË (1, θx, zx) Ë (1, θy, zy) Á‡‰‡ÂÚÒfl Í‡Í (θ − θ )2 + ( z − z )2 , ÂÒÎË θ − θ ≤ π,xyxyyxdcyl ( x, y) = (θ x + 2 π − θ y )2 + ( z x − z y )2 , ÂÒÎË θ y − θ x > π.äÓÌ˘ÂÒÍÓ ‡ÒÒÚÓflÌËÂäÓÌ˘ÂÒÍËÏ ‡ÒÒÚÓflÌËÂÏ d con ̇Á˚‚‡ÂÚÒfl ‚ÌÛÚÂÌÌflfl ÏÂÚË͇ ̇ ÔÓ‚ÂıÌÓÒÚËÍÓÌÛÒ‡ S, ÍÓÚÓ‡fl ËÒÔÓθÁÛÂÚÒfl ‚ ͇˜ÂÒÚ‚Â ÔÓÓʉ‡˛˘Â„Ó ‡ÒÒÚÓflÌËfl ‰Îfl ÍÓÌ˘ÂÒÍÓÈ ‰Ë‡„‡ÏÏ˚ ÇÓÓÌÓ„Ó V ( P, dcon , S ). ÖÒÎË ÓÒ¸ ÍÓÌÛÒ‡ S ‡ÁÏ¢Â̇ ̇ x 3 -ÓÒË ‚3 Ë ‡‰ËÛÒ ÓÍÛÊÌÓÒÚË Ó˜Â˜Ë‚‡ÂÏÓÈ ÔÂÂÒ˜ÂÌËÂÏ ÍÓÌÛÒ‡ S Ò x1x2-ÔÎÓÒÍÓÒÚ¸˛‡‚ÂÌ Â‰ËÌˈÂ, ÚÓ ‡ÒÒÚÓflÌË ÍÓÌÛÒ‡ ÏÂÊ‰Û Î˛·˚ÏË ÚӘ͇ÏË x, y ∈ S Á‡‰‡ÂÚÒfl ͇Írx2 + ry2 − 2 rx ry cos(θ ′y − θ ′x ),ÂÒÎË θ ′y ≤ θ ′x + π sin(α / 2),dcon ( x, y) = rx2 + ry2 − 2 rx ry cos(θ ′x + 2 π sin(α / 2) − θ ′y ), ÂÒÎË θ ′y > θ ′x + π sin(α / 2),„‰Â (x1, x 2 , x 3 ) – ÔflÏÓÛ„ÓθÌ˚ ‰Â͇ÚÓ‚˚ ÍÓÓ‰Ë̇Ú˚ ÚÓ˜ÍË ı ̇ S, α – Û„ÓÎ Ô˂¯ËÌ ÍÓÌÛÒ‡, θx – Û„ÓÎ ÔÓÚË‚ ˜‡ÒÓ‚ÓÈ ÒÚÂÎÍË ÓÚ x 1 -ÓÒË ‰Ó ÎÛ˜‡ ËÁ ËÒıÓ‰ÌÓÈÚÓ˜ÍË ‰Ó ÚÓ˜ÍË ( x1 , x 2 , 0), θ ′x = θ x sin(α / 2), rx = x12 + x 22 + ( x3 − coth(α / 2))2 – ‡ÒÒÚÓflÌË ÔÓ ÔflÏÓÈ ÓÚ ‚¯ËÌ˚ ÍÓÌÛÒ‡ ‰Ó ÚÓ˜ÍË (x 1 , x2, x3).ê‡ÒÒÚÓflÌË ÇÓÓÌÓ„Ó ÔÓfl‰Í‡ mê‡ÒÒÏÓÚËÏ ÍÓ̘ÌÓ ÏÌÓÊÂÒÚ‚Ó Ä Ó·˙ÂÍÚÓ‚ ÏÂÚ˘ÂÒÍÓ„Ó ÔÓÒÚ‡ÌÒÚ‚‡ (S, d) ˈÂÎÓ ˜ËÒÎÓ m ≥ 1.
ê‡ÒÒÏÓÚËÏ ÏÌÓÊÂÒÚ‚Ó ‚ÒÂı m-ÔÓ‰ÏÌÓÊÂÒÚ‚ Mi ËÁ Ä (Ú.Â. Mi ⊂ AË | Mi | = m). Ñˇ„‡Ïχ ÇÓÓÌÓ„Ó ÔÓfl‰Í‡ m ÏÌÓÊÂÒÚ‚‡ Ä ÂÒÚ¸ ‡Á·ËÂÌË S ̇ ӷ·ÒÚË ÇÓÓÌÓ„Ó V(Mi) m-ÔÓ‰ÏÌÓÊÂÒÚ‚ ÏÌÓÊÂÒÚ‚‡ Ä Ú‡ÍËÏ Ó·‡ÁÓÏ, ˜ÚÓ·˚ V(M i)ÒÓ‰Âʇ· ‚Ò ÚÓ˜ÍË s ∈ S, ÍÓÚÓ˚ "·ÎËÊÂ" Í Mi, ˜ÂÏ Í Î˛·ÓÏÛ ‰Û„ÓÏÛ m ÏÌÓÊÂÒÚ‚Û M i : d(s, x) < d(s, y) ‰Îfl β·˚ı x ∈ Mii Ë y ∈ S\Mi. ùÚ‡ ‰Ë‡„‡Ïχ Û͇Á˚‚‡ÂÚ Ô‚ӄÓ, ‚ÚÓÓ„Ó, …, m-„Ó ·ÎËÊ‡È¯Â„Ó ÒÓÒ‰‡ ÓÍÂÒÚÌÓÒÚË ÚÓ˜ÍË ËÁ S.í‡ÍË ‰Ë‡„‡ÏÏ˚ ÏÓ„ÛÚ ·˚Ú¸ ÓÔ‰ÂÎÂÌ˚ ‚ ÚÂÏË̇ı ÌÂÍÓÚÓÓÈ "ÙÛÌ͈ˡÒÒÚÓflÌËfl" D(s, Mi), ‚ ˜‡ÒÚÌÓÒÚË, ÌÂÍÓÚÓÓ m-ıÂÏËÏÂÚËÍË Ì‡ S. ÑÎfl Mi = {ai , bi}‡ÒÒχÚË‚‡ÎËÒ¸ ÙÛÌ͈ËË | d ( s, ai ) − d ( s, bi ) |, d ( s, ai ) + d ( s, bi ), d ( s, ai ) ⋅ d ( s, bi ), ‡ Ú‡ÍÊ 2-ÏÂÚËÍË d ( s, ai ) + d ( s, bi ) + d ( ai , bi ) Ë ÔÎÓ˘‡‰¸ ÚÂÛ„ÓθÌË͇ (s, ai, bi).É·‚‡ 21ê‡ÒÒÚÓflÌËfl ‚ ‡Ì‡ÎËÁÂÓ·‡ÁÓ‚ Ë Á‚ÛÍÓ‚21.1. êÄëëíéüçàü Ç ÄçÄãàáÖ éÅêÄáéÇé·‡·ÓÚ͇ Ó·‡ÁÓ‚ (ËÁÓ·‡ÊÂÌËÈ) ËÏÂÂÚ ‰ÂÎÓ Ò Ú‡ÍËÏË Í‡Í ÙÓÚÓ„‡ÙËË, ‚ˉÂÓ‰‡ÌÌ˚ ËÎË ÚÓÏÓ„‡Ù˘ÂÒÍË ËÁÓ·‡ÊÂÌËfl.
Ç ˜‡ÒÚÌÓÒÚË, ÍÓÏÔ¸˛ÚÂ̇fl „‡ÙË͇Ô‰ÒÚ‡‚ÎflÂÚ ÒÓ·ÓÈ ÔÓˆÂÒÒ ÒËÌÚÂÁËÓ‚‡ÌËfl Ó·‡ÁÓ‚ ËÁ ‡·ÒÚ‡ÍÚÌ˚ı ÏÓ‰ÂÎÂÈ,ÚÓ„‰‡ Í‡Í Ï‡¯ËÌÌÓ ‡ÒÔÓÁ̇‚‡ÌË ӷ‡ÁÓ‚ – ˝ÚÓ ËÁ‚ΘÂÌË ÌÂÍÓÈ ‡·ÒÚ‡ÍÚÌÓÈËÌÙÓχˆËË: Ò͇ÊÂÏ, 3D (Ú.Â. ÚÂıÏÂÌÓ„Ó) ÓÔËÒ‡ÌËfl ÚÓÈ ËÎË ËÌÓÈ ÒˆÂÌ˚, ËÒÔÓθÁÛfl  ‚ˉÂÓÒ˙ÂÏÍÛ. 燘Ë̇fl „‰Â-ÚÓ Ò 2000 „. ‡Ì‡ÎÓ„Ó‚‡fl Ó·‡·ÓÚ͇ ËÁÓ·‡ÊÂÌËÈ(ÓÔÚ˘ÂÒÍËÏË ÛÒÚÓÈÒÚ‚‡ÏË) ÛÒÚÛÔ‡ÂÚ ÏÂÒÚÓ ˆËÙÓ‚ÓÈ Ó·‡·ÓÚÍ Ë, ‚ ˜‡ÒÚÌÓÒÚË,ˆËÙÓ‚ÓÏÛ Â‰‡ÍÚËÓ‚‡Ì˲ (̇ÔËÏÂ, Ó·‡·ÓÚÍ ËÁÓ·‡ÊÂÌËÈ, ÔÓÎÛ˜ÂÌÌ˚ı ÒÔÓÏÓ˘¸˛ Ó·˚˜Ì˚ı ˆËÙÓ‚˚ı ÙÓÚÓ‡ÔÔ‡‡ÚÓ‚).äÓÏÔ¸˛ÚÂ̇fl „‡ÙË͇ (Ë ÏÓÁ„ ˜ÂÎÓ‚Â͇) ËÏÂÂÚ ‰ÂÎÓ Ò Ó·‡Á‡ÏË ‚ÂÍÚÓÌÓÈ„‡ÙËÍË, Ú.Â.
Ú‡ÍËÏË, ÍÓÚÓ˚ Ô‰ÒÚ‡‚ÎÂÌ˚ „ÂÓÏÂÚ˘ÂÒÍË ÍË‚˚ÏË, ÏÌÓ„ÓÛ„ÓθÌË͇ÏË Ë Ú.Ô. àÁÓ·‡ÊÂÌË ‡ÒÚÓ‚ÓÈ „‡ÙËÍË (ËÎË ˆËÙÓ‚Ó ËÁÓ·‡ÊÂÌËÂ, ÔÓ·ËÚÓ‚Ó ÓÚÓ·‡ÊÂÌËÂ) ‚ 2D ÂÒÚ¸ Ô‰ÒÚ‡‚ÎÂÌË 2D ËÁÓ·‡ÊÂÌËfl Í‡Í ÍÓ̘ÌÓ„Ó ÏÌÓÊÂÒÚ‚‡ ‰ËÒÍÂÚÌ˚ı ‚Â΢ËÌ, ̇Á˚‚‡ÂÏ˚ı ÔËÍÒÂÎflÏË (ÒÓ͇˘ÂÌÌÓ Óڇ̄ÎËÈÒÍÓ„Ó "picture element"), ‡ÁÏ¢ÂÌÌ˚ı ̇ Í‚‡‰‡ÚÌÓÈ „ËÁ 2 ËÎ˯ÂÒÚËÛ„ÓθÌÓÈ „ËÁÂ.
ä‡Í Ô‡‚ËÎÓ, ‡ÒÚ – ˝ÚÓ Í‚‡‰‡Ú̇fl 2k × 2k „ËÁ‡ Ò k = 8,9ËÎË 10. ÇˉÂÓËÁÓ·‡ÊÂÌËfl Ë ÚÓÏÓ„‡Ù˘ÂÒÍË (Ú.Â. ÔÓÎÛ˜ÂÌÌ˚Â Í‡Í ÒÂËflÔÓÔ˜Ì˚ı Ò˜ÂÌËÈ ÓÚ‰ÂθÌ˚ÏË ˜‡ÒÚflÏË) ËÁÓ·‡ÊÂÌËfl fl‚Îfl˛ÚÒfl 3D ËÁÓ·‡ÊÂÌËflÏË (2D ÔÎ˛Ò ‚ÂÏfl); Ëı ‰ËÒÍÂÚÌ˚ ‚Â΢ËÌ˚ ̇Á˚‚‡˛ÚÒfl ‚ÓÍÒÂÎflÏË(˝ÎÂÏÂÌÚ‡ÏË Ó·˙Âχ).ÑËÒÍÂÚÌÓ ‰‚Ó˘ÌÓ ËÁÓ·‡ÊÂÌË ËÒÔÓθÁÛÂÚ ÚÓθÍÓ ‰‚‡ Á̇˜ÂÌËfl: 0 Ë 1; 1 ËÌÚÂÔÂÚËÛÂÚÒfl Í‡Í Îӄ˘ÂÒ͇fl "ËÒÚË̇" Ë ÓÚÓ·‡Ê‡ÂÚÒfl ˜ÂÌ˚Ï ˆ‚ÂÚÓÏ; Ú‡ÍËÏÓ·‡ÁÓÏ, Ò‡ÏÓ ËÁÓ·‡ÊÂÌË ÓÚÓʉÂÒÚ‚ÎflÂÚÒfl Ò ÏÌÓÊÂÒÚ‚ÓÏ ˜ÂÌ˚ı ÔËÍÒÂÎÂÈ.ùÎÂÏÂÌÚ˚ ·Ë̇ÌÓ„Ó 2D ËÁÓ·‡ÊÂÌËfl ÏÓÊÌÓ ‡ÒÒχÚË‚‡Ú¸ Í‡Í ÍÓÏÔÎÂÍÒÌ˚˜ËÒ· x = iy, „‰Â (x, y) – ÍÓÓ‰Ë̇ڇ ÚÓ˜ÍË Ì‡ ‰ÂÈÒÚ‚ËÚÂÎÌÓÈ ÔÎÓÒÍÓÒÚË 2 . çÂÔÂ˚‚ÌÓ ·Ë̇ÌÓ ËÁÓ·‡ÊÂÌË fl‚ÎflÂÚÒfl (Ó·˚˜ÌÓ ÍÓÏÔ‡ÍÚÌ˚Ï) ÔÓ‰ÏÌÓÊÂÒÚ‚ÓÏÎÓ͇θÌÓ ÍÓÏÔ‡ÍÚÌÓ„Ó ÏÂÚ˘ÂÒÍÓ„Ó ÔÓÒÚ‡ÌÒÚ‚‡ (Ó·˚˜ÌÓ Â‚ÍÎˉӂ‡ ÔÓÒÚ‡ÌÒÚ‚‡ n Ò n = 2,3).èÓÎÛÚÓÌÓ‚˚ ËÁÓ·‡ÊÂÌËfl ÏÓ„ÛÚ ‡ÒÒχÚË‚‡Ú¸Òfl Í‡Í ÚӘ˜ÌÓ-‚Á‚¯ÂÌÌ˚ ·Ë̇Ì˚ ËÁÓ·‡ÊÂÌËfl.
Ç Ó·˘ÂÏ ÒÎÛ˜‡Â ̘ÂÚÍÓ ÏÌÓÊÂÒÚ‚Ó fl‚ÎflÂÚÒfl ÚӘ˜ÌÓ‚Á‚¯ÂÌÌ˚Ï ÏÌÓÊÂÒÚ‚ÓÏ Ò ‚ÂÒ‡ÏË (Á̇˜ÂÌËflÏË ÔË̇‰ÎÂÊÌÓÒÚË) (ÒÏ. [Bloc99] ‰ÎflÓ·ÁÓ‡ ̘ÂÚÍËı ‡ÒÒÚÓflÌËÈ). ÑÎfl ÔÓÎÛÚÓÌÓ‚˚ı ËÁÓ·‡ÊÂÌËÈ xyi-Ô‰ÒÚ‡‚ÎÂÌËÂÔËÏÂÌflÂÚÒfl ‚ ÒÎÛ˜‡Â, ÍÓ„‰‡ ÔÎÓÒÍÓÒÌ˚ ÍÓÓ‰Ë̇Ú˚ (x, y) Ó·ÓÁ̇˜‡˛Ú ÙÓÏÛ, ‚ ÚÓ‚ÂÏfl Í‡Í ‚ÂÒ i (ÒÓ͇˘ÂÌÌÓ ÓÚ ËÌÚÂÌÒË‚ÌÓÒÚË, Ú.Â. flÍÓÒÚË) – ÚÂÍÒÚÛÛ (‡ÒÔ‰ÂÎÂÌË ËÌÚÂÌÒË‚ÌÓÒÚË). àÌÓ„‰‡ ËÒÔÓθÁÛÂÚÒfl Ú‡ÍÊ χÚˈ‡ ((ixy)) ÔÓÎÛÚÓÌÓ‚.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
