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Progressive compression of digital elevation data using meshes

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(8)  Kivanc Kosea , Erdal Yilmazb, A. Enis Cetina 6982;A (;6C2?@6AFa !61192 .@A '205;60.9 (;6C2?@6AFb    ; A56@ =.=2? . ;2D 646A.9 92C.A6<; !.= ! 6:.42 0<: =?2@@6<; .94<?6A5: 6@ =?<=<@21 ! 6:.42 0.; /2 A5?2.A21 .@ . 4?.F@0.92 6:.42 D5<@2 =6E29 C.9B2@ .?2 A52 292C.A6<; C.9B2@ <3 A52 :.= =<6;A@ '52 4?.F@0.92 ! 6:.42 6@ 0<: =?2@@21 B@6;4 .; .1.=A6C2 D.C292A /.@21 6:.42 0<:=?2@@6<; .94<?6A5: '52 :2A5<1 D5605 6@ .; 2EA2;@6<; <3 A52 =?<4?2@ @6C2 :2@5 0<:=?2@@6<; A.82@ .1C.;A.42 <3 A52 :B9A6?2@<9BA6<; =?<=2?AF <3 A52 D.C292A@ D5692 0<16;4 A52 :.= 6:.42@ '56@ :.82@ 6A =<@@6/92 A< 120<12 16332?2;A ?2@<9BA6<;@ <3 A52 :.= 3?<: A52 2;0<121 /6A @A?2.: =?<C616;4 . :B9A6?2@<9BA6<; 16@ =9.F <3 . 46C2; :.= E=2?6:2;A.9 ?2@B9A@ .?2 =?2@2;A21     &2C2?.9 16332?2;A & .A. <?:.A@ 2E6@A , - '52@2 1.A. 3<? :.A@ 0.; @A<?2 6;3<?:.A6<;@ 6;09B16;4 • 42<4?.=560 6;3<?:.A6<; D5605 =?<C612@ A52 =<@6A6<; .;1 @5.=2@ <3 @=2060 42<4?.=560 32.AB?2@ • .AA?6/BA2 6;3<?:.A6<; D5605 =?<C612@ .116A6<;.9 ;<; 4?.=560 6;3<?:.A6<; ./<BA 2.05 32.AB?2 .;1 • 16@=9.F 6;3<?:.A6<; D5605 12@0?6/2@ 5<D A52 32.AB?2@ D699 .==2.? <; A52 @0?22; @ @22; <; 64B?2  . AF=60.9 1646A.9 292C.A6<; ?2=?2@2; A.A6<; 6; . 1646A.9 :.= 0<;@6@A@ <3

(9)  =<6;A 09<B1@  . .;1 A526? 0<;;20A6C6AF 6;3<?:.A6<;  / '52?23<?2 42<@=.A6.9 1.A. 6@ @6:69.? A<

(10)  :2@5 ?2=?2@2;A.A6<; 6; :.;F D.F@ '52 6;3<? :.A6<; 6; . :.= 0<??2@=<;1 A< A52 42<:2A?F .;1 0<;;20A6C6AF 6;3<?:.A6<; <3

(11)  :2@5 :<129@ ?2@=20A6C29F , - <<?16;.A2 X, Y 9<0.A6<;@ <3 . Digital Elevation Model (DEM) 0.; /2 @A<?21 .@ A52 X, Y 0<<?16;.A2 <3 . :2@5 C2?A2E .;1 :<?2 <C2? A52 292C.A6<; <3 A52 =.?A60B9.? =<6;A 0.; /2 @A<?21 .@ A52 Z 0<<?16;.A2 <3 A52 C2?A2E '52?23<?2 1646A.9 292C.A6<; 1.A. 0.; /2 A?2.A21 .@ :2@5 :<129@  :2@5 9682 ?2=?2@2;A.A6<; <3 . 1646A.9 292C.A6<; 1.A. 5.@ @2C2?.9 .1C.;A.42@ <? 2E.:=92 0.90B9.A6<; <3 A52 @9<=2 .;1 A52 .@=20A D5605 .?2 0?B06.9 3<? A52 .==960.A6<;@ 9682 ?2 =?<=<4.A6<; 0.90B9.A6<; ,

(12) - 0.; /2 2.@69F 1<;2 3?<: .

(13)  :2@5 9682 ?2=?2@2;A.A6<; <3 . :.= @ A52 .?2. ?2=?2@2;A21 /F A52 :.= 92 42A@ 9.?42? A52 .:<B;A <3 1.A. @A<?21 6; A52 :.= 92 .9@< 6;0?2.@2@ '52?2 3<?2 D.F@ <3 23062;A9F @A<?6;4 .;1 A?.;@:6AA6;4 A5<@2 1.A.. 978-1-4244-3395-7/09/$25.00 ©2009 IEEE. /20<:2 . 0?B06.9 6@@B2 &6;02 A52 1646A.9 292C.A6<; 1.A. 0.; /2 ?2=?2@2;A21 .@ .

(14)  :2@5 @A.A60 :2@5 0<:=?2@@6<; A205 ;6>B2@ 46C2; 6; , - D5605 0<:=?2@@2@

(15)  :2@5 :<129@ B@6;4 6:.42 0<:=?2@@6<; A<<9@ 6@ C2?F @B6A./92 3<? A52 0<:=?2@@6<; <3 A56@ 1.A. ; , - 6A 6@ =?<=<@21 A5.A

(16)  :2@5 :<129@ 0.; /2 0<;C2?A21 A< 6:.42@ B@6;4 <?A5<4?.=560 =?<720A6<; A5.A 0.; A52; /2 0<:=?2@@21 B@6;4 @A.;1.?1 6:.42 1.A. 0<:=?2@@6<; .94<?6A5:@ 646A.9 292C.A6<; 1.A. 6@ C2?F @B6A./92 3<? <?A5< 4?.=560 =?<720A6<; 6; 6A@ ;.AB?2 #;2 0.; 16?20A9F B@2 A52 292 C.A6<; 1.A. .@ A52 =6E29 C.9B2@ <3 A52 =?<720A21 6:.42 .;1 A52 X, Y C.9B2 <3 2.05 ;<12 .@ A52 =6E29 9<0.A6<; @ . ?2@B9A 6A 6@ @A?.645A3<?D.?1 A< 0?2.A2  6:.42@ 3?<: :.=@ 0AB .99F 4?619682 1646A.9 292C.A6<; 1.A. 6@ 6; . ?.@A2? 3<?:.A 6A@293 '52 0<;;20A6C6AF 6;3<?:.A6<; <3 . :.= 0.; .9@< /2 12;21 16?20A9F 3<? A52 =6E29 ;264/<?5<<1 A52?23<?2 A52?2 6@ ;< 2EA?. ;221 3<? A52 A?.@:6@@6<; <3 A52 0<;;20A6C6AF 6;3<?:.A6<; 9682 6;

(17)  :2@52@ 2?2 .; .1.=A.A6<; <3 A52 0<:=?2@@6<; .94<?6A5: 46C2; 6; , - D699 /2 B@21 '52 ! 6:.42 6@ A?.;@3<?:21 B@6;4 .1.= A6C2 D.C292A A?.;@3<?: 46C2; 6; ,-,-,- '52; A52 A?.;@ 3<?:21 6:.42 6@ 321 6;A< &$' 2;0<12? .;1 =?<4?2@@6C29F 2;0<121 '52 2;0<121 /6A@A?2.: 6@ =?<4?2@@6C2 .@ 16332?2;A ?2@<9BA6<;@ <3 A52 ! 6:.42 0.; /2 ?20<;@A?B0A21 3?<: 163 32?2;A 92;4A5 <3 A52 /6A@A?2.: '56@ =.=2? 6@ <?4.;6G21 .@ 3<99<D@ ; &20A6<;  A52 ?29. A6<; /2AD22; A52 @A?B0AB?2 <3

(18)  :2@5 :<129@ .;1 A52 1646A.9 :.=@ 6@ 2E=9.6;21 ; &20A6<;

(19)  :.= 0<:=?2@@6<; .94<?6A5: 6@ =?2@2;A21 <:=?2@@6<; ?2@B9A@ .?2 46C2; 6; &20A6<;   

(20) 

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(25)  :2@5 :<129 6@ 0<:=<@21 <3 AD< /.@6@ =.?A@ 42<:2 A?F .;1 0<;;20A6C6AF 6;3<?:.A6<; 2<:2A?F 6;3<?:.A6<; 12 @0?6/2@ A52 =9.02@ <3 C2?A2E =<6;A@ D5605 0.; /2 0<B;A21 .@ A52 @64;.9 @.:=92 =<6;A@ <;;20A6C6AF 1.A. 12;2@ 5<D A52@2 @.:=92 =<6;A@ .?2 0<;;20A21 A< 2.05 <A52? 6; <A52? D<?1@ 5<D A5<@2 =<6;A@ 0?2.A2 @B?3.02@ @ @22; 6; 64B?2@ .;1  C2? A2E =<6;A@ <; /<A5 .

(26)  :2@5 :<129 .;1 . A2??.6; :.= 1< ;<A ;221 A< /2 ?24B9.?9F =<@6A6<;21 .?42 =9.;.? @B?3.02@ 6; /<A5

(27)  :<129@ .;1 :.=@ 0.; /2 ?2=?2@2;A21 /F . 32D 9.?42 A?6.;492@ .;1 A52 .?2.@ D6A5 ;2 12A.69@ 0.; /2 ?2=?2@2;A21 /F . 9.?42 ;B:/2? <3 @:.99 A?6.;492@ '52?23<?2 A52 @.:=96;4 @AB0AB?2 :.F ;<A /2 ?24B9.? '56@ 6??24B9.? @A?B0AB?2 6@ <;2 <3 A52 ?2.@<; A5.A @A.;1.?1 ;.AB?.9 6:.42 0<:=?2@@6<; .94<. IV - 502. IGARSS 2009.

(28)   ??24B9.?

(29)  :2@5 :<129 <3 . 0<D. .. /.   ??24B9.? :2@5 9682 ?2=?2@2;A.A6<; <3 1646A.9 292C.A6<; 1.A. . &.:=92 =<6;A@ .@ =<6;A 09<B1@ / 0<;;20A6C6AF 6; 3<?:.A6<;. 1.B/20562@ 2A0 0.; /2 B@21 1B?6;4 A56@ :B9A6?2@<9BA6<; .;.9 F@6@ <3 A52 6;=BA 6:.42 <D2C2? .@ A52 ?29.A6<; /2AD22; A52 =6E29@ <3 A52 =?<720A21 :.= 6:.42 6@ 16332?2;A 3?<: ;.AB?.9 6:.42@ A52 16?20A .==960.A6<; <3 ;.AB?.9 6:.42 0<:=?2@@6<; :2A5<1@ D<B91 ;<A 23062;A9F D<?8 ; A56@ =.=2? D2 6;A?<1B02 .; .1.=A6C2 =?2160A6<; @A?B0AB?2 @6:69.? A< A52 <;2 6;A?<1B021 6; , - A 2.05 =?2160A6<; 92C29 A52 =6E29@ .?2 =?2160A21 3?<: A526? 0<;;20A21 ;2645/<?@ .A A52 9<D =.@@ 05.;;29 62 A52 6:.42 <3 A52 *' '52 =?2160A6<; @A?B0AB?2 6; . 963A6;4 *' 5.@ A< /2 .1.=A6C2 /20.B@2 . 4?61 =<6;A . =6E29 :.F ;<A 5.C2 . 0<;@A.;A ;B:/2? <3 ;2645/<?@ .A A52 6:.42 @ . ?2@B9A A52 =?2160A6<; @A?B0AB?2 .9D.F@ 05.;42@ , . ?6A5:@ ?2>B6?6;4 ?24B9.? @.:=921 1.A. 0.; ;<A /2 16?20A9F B@21 3<? 0<16;4 :.= 1.A. <D2C2? 6A 6@ =<@@6/92 A< B@2 @A.;1.?1 6:.42 0<16;4 :2A5<1@ D6A5 6??24B9.? :.= 1.A. /F =?<02@@6;4 A52 1.A. 6; . =?<=2? :.;;2? @@B:2 A5.A .

(30)  C2?A2E =<6;A V 6;

(31)  @=.02 6@ ?2=?2@2;A21 .@ 3<99<D@. $ G  . '2??.6;!.= &64;.9. .  G GG . V = [x, y, z].. , , . . e. $  $ .  e  e.  . F <?A5<4<;.99F =?<720A6;4 A56@ =<6;A A< . @2920A21 =9.;2 D2 0.; 0?2.A2 . ?24B9.?9F =9.021 1.A. <? 2E.:=92 .@B:2 A5.A <B? =?<720A6<; =9.;2 6@ A52 5<?6G<;A.9 EF =9.;2 ; A56@ 0.@2 A52 =<@6A6<; <3 A52 6:.42 =6E29 D699 /2 12A2?:6;21 /F x, y 0<:=<;2;A@ <3 A52 C2?A2E V .;1 A52 =6E29 C.9B2 D699 0<? ?2@=<;1 A< A52 z 0<:=<;2;A .@ 3<99<D@ I(x, y) = z.. . , .  . #/C6<B@9F A< </A.6; 16@0?2A2 =6E29 9<0.A6<;@ <; . ?24B9.? 4?61 x, y C.9B2@ 6; >B.A6<; @5<B91 /2 >B.;A6G21  :<?2 12A.6921 2E=9.;.A6<; <3 A52 6:.42 3<?:.A6<; =?<021B?2 0.; /2 3<B;1 6; , -.   $<9F=5.@2 @A?B0AB?2 .;1 A52 =?2160A6<; @A.42 A5.A 6@ B@21 6; A52 D.C292A A?.;@3<?:.A6<; *' '52 =<9F=5.@2 @A?B0AB?2 6; A52 963A6;4 D.C292A A?.;@3<?:. A6<; A5.A D2 B@21 6; A52 0<:=?2@@6<; <3 A52 A2??.6; :.=@ 6@ 46C2; 6; 64B?2

(32)  '52 6:.42 6@ B@21 .@ A52 I9<D=.@@ 05.; ;29J .;1 .99 =?2160A6<;@ .?2 1<;2 B@6;4 A52 C.9B2@ 6; A56@ 05.; ;29 3A2? <;2 92C29 <3 D.C292A A?.;3<?:.A6<; 6:.42 0<; A.6;@ A52 =6E29@ I(2x, 2y), where x ∈ [0, N/2 − 1] and y ∈ [0, M/2 − 1] N, M .?2 A52 @6G2 <3 A52 6:.42 6; 5<?6G<;A.9 .;1 C2?A60.9 16?20A6<;@ ?2@=20A6C29F '52 =?2160A6<; .;1 =?2 160A6<; 2??<? 3<? . =6E29 6; 5<?6G<;A.9 .;1 C2?A60.9 16?20A6<;@ 6@ 0.90B9.A21 .@ 3<99<D@.    

(33) . LHp (x, y) = 0.5(LL(x, y) + LL(x + 1, y)),. 

(34) . '52 :.6; 0<:=?2@@6<; .94<?6A5: A5.A 6@ B@21 52?2 6@ &2A $.? A6A6<;6;4 6; 62?.?056.9 '?22@ &$' ,- &$' 6@ D.C292A A?.;@3<?:.A6<; *' /.@21 0<:=?2@@6<; .94<?6A5: A5.A B@2@ . :B9A6?2@<9BA6<; 120<:=<@6A6<; <3 A52 6;=BA 6:.42 6332? 2;A 9A2? /.;8 @A?B0AB?2@ @B05 .@ <?A5<4<;.9 /6<?A5<4<;.9. LHe (x, y) = LH(x, y) − LHp (x, y), HLp (x, y) = 0.5(LL(x, y) + LL(x, y + 1)),.   . LHe (x, y) = LH(x, y) − LHp (x, y).. . IV - 503. #; A52 <A52? 5.;1 @6;02 A52 =6E29 6; A52  @B//.;1 5.C2.

(35) ;2645/<B?@ 6; A52 05.;;29 6;@A2.1 <3 AD< .@ A52 0.@2 3<?  .;  05.;;29@ 52?2 A52 9A2? @A?B0AB?2 05.;42@ ;@A2.1 <3 46C6;4 2>B.9 D2645A A< 2.05 ;2645/<B? .; .1.=A6C2 @052:2 ,-,- D5<@2 D2645A@ .?2 1206121 .00<?6;4 A< A52 12?6C.A2@ 6; A52 16.4<;.9 16?20A6<; 6@ 12;21 <? A52 =6E29@ 6; A52  05.;;29@ 12?6C.A6C2@ <;  124?22 .;1

(36)  124?22 16?20A6<;@ .?2 0.90B9.A21 .@ 3<99<D@. 800. JPEG SPIHT Original. 700. Compressed File Size (KB). 600 500 400 300 200 100. ΔHH45 (x, y) = |(LL(x, y) − LL(x + 1, y + 1))|, ΔHH135 (x, y) = |(LL(x + 1, y) − LL(x, y + 1))|..  . '52; A52 =?2160A6<; .;1 A52 =?2160A6<; 2??<? 3<? . =6E29 6;  05.;;29 6@. 0 25. ΔHH135 (x, y)(LL(x, y) + LL(x + 1, y + 1)) ΔHH45 (x, y) + ΔHH135 (x, y) ΔHH45 (x, y)(LL(x + 1, y) + LL(x, y + 1)) , (9) + ΔHH45 (x, y) + ΔHH135 (x, y) HH(x, y)e = HH(x, y) − HH(x, y)p . (10).  

(37)  '52 .94<?6A5: :2;A6<;21 6; &20A6<;

(38) 6@ A2@A21 <; AD< 3?22 A2??.6; :.= :<129@ 0.9921 bay .;1 SAfrica <? 0<:=.?6@<; D2 .9@< =2?3<?:21 A2@A@ B@6;4 A52 $ @A.;1.?1 <::2? 06.9 $ 2;0<12?@ 0.; 2;0<12  /6A 4?.F@0.92 6:.42@ <D 2C2? A52 =6E29 C.9B2@ <3 A52 :.= 6:.42 .?2 2E02216;4 /6A@ '52?23<?2 A52 :.= 6:.42 6@ 16C612 6;A< AD< 6:.42@ <;2 <3 D5605 @A<?2@ A52 :<@A @64;60.;A  /6A@ <3 A52 :.= .;1 A52 <A52? <;2 @A<?2@ A52 92.@A @64;60.;A  /6A@ '52@2 AD< 6:.42@ .?2 0<:=?2@@21 @2=2?.A29F $&"% <3 A52 6:.42 @A<?6;4 A52 :<@A @64;60.;A  /6A@ <3 A52 :.= .3320A@ A52 A<A.9 $&"% :<?2 A5.; A52 $&"% <A52? 6:.42 '52?23<?2 A52 ?@A 6:.42 6@ 0<:=?2@@21 6; . 9<@@92@@ :.;;2? '52 0<:=?2@@6<; ?2@B9A@ 3<? bay .;1 SAfrica :.= :<129@ B@6;4 A52 =?<=<@21 .94<?6A5: .;1 $ .?2 46C2; 6; 64B?2@ .;1  ?<: A52 4B?2@ 6A 6@ @22; A5.A A52 =?<=<@21 .94<?6A5: 6@ @B=2?6<? A< A52 $ 0<12?. 35. 40 PSNR. 45. 50. 55.   .F A2??.6; :<129 0<:=?2@@6<; ?2@B9A@ '52 G6= 0<: =?2@@21 A2??.6; :<129 5.@ . 92 @6G2 <3  . HH(x, y)p =. 69A2? 46C2; 6; >B.A6<;  .1.=A@ 6A@293 6;C2?@29F =?<=<?A6<;.9 A< A52 12?6C.A6C2@ D5605 :2.;@ 6A 3.C<?@ A52 @:<<A5 A?.;@6 A6<;@ <; A52 :.= &6;02 A52?2 .?2 ;< <009B@6<;@ <; A52 :.= 6:.42 6A 92@@ =?</./92 A< 2;0<B;A2? @B112; 7B:=@ .:<;4 A52 =6E29 C.9B2@ '52?23<?2 3.C<?6;4 A52 @:<<A5 A?.;@6A6<;@ D<B91 46C2 /2AA2? ?2@B9A@ 3A2? 0?2.A6;4 A52 D.C292A A?.;@3<?:21 :.= 6:.42 6A 6@ 321 A< A52 &$' 0<12? .;1 A52 1.A. 6@ /6A=9.;2 2;0<121 ,- ?<: 16332?2;A 92;4A5@ <3 A52 &$' /6A@A?2.: 16332?2;A ?2@ <9BA6<;@ <3 A52 1646A.9 :.= 0.; /2 ?20<;@A?B0A21 ,- '52?2 3<?2 &$' /.@21 0<16;4 <3 A52 :.= 6:.42@ 6@ . =?<4?2@@6C2 0<:=?2@@6<; @052:2 '52 ?2@B9A@ <3 ?20<;@A?B0A6<; <3 A52 :.= 6:.42 3?<: 16332?2;A 92;4A5 <3 A52 &$' /6A@A?2.: 6@ 46C2; 6; &20A6<;  99 A52 <=2?.A6<;@ 12;21 ./<C2 .?2 ?2C2?@6/92 '52?23<?2 =2?320A ?20<;@A?B0A6<; 6@ =<@@6/92. 30. 1200. JPEG SPIHT Original. 1000. Compressed File Size (KB). 800. 600. 400. 200. 0 10. 15. 20. 25. 30. 35 PSNR. 40. 45. 50. 55. 60.   &3?60. A2???.6; :<129 0<:=?2@@6<; ?2@B9A@ '52 G6= 0<:=?2@@21 A2??.6; :<129 5.@ . 92 @6G2 <3   @ :2;A6<;21 6; &20A6<;

(39)  A52 =?<=<@21 &$' /.@21 0<:=?2@@6<; .94<?6A5: 6@ . =?<4?2@@6C2 0<12? @ A52 120<12? ?2026C2@ :<?2 /6A@ 3?<: A52 2;0<12? 6A 0.; =?<4?2@6C299F B= 1.A2 A52 :.= 6:.42 .;1 0?2.A2 . ;2? ?2@<9BA6<; ?20<;@A?B0 A6<; ; 64B?2  :.= 6:.42@ ?20<;@A?B0A21 3<?: 16332?2;A @6G2@ <3 &$' /6A@A?2.: 0.; /2 @22; 64B?2  .9@< @5<D@ A5.A A52 ?20<;@A?B021 6:.42 D6A5 $&"% <3

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

(41) .. /. 0. 1.   !.= 6:.42 ?20<;@A?B0A21 3?<: .

(42)   /

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