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{{Short description|Lossy image compression algorithm}}
'''Embedded zerotrees of wavelet transforms''' ('''EZW''') is a lossy [[image compression]] [[algorithm]]. At low bit rates, i.e. high compression ratios, most of the coefficients produced by a [[Sub-band coding|subband transform]] (such as the [[wavelet transform]]) will be zero, or very close to zero. This occurs because "real world" images tend to contain mostly low frequency information (highly correlated). However where high frequency information does occur (such as edges in the image) this is particularly important in terms of human perception of the image quality, and thus must be represented accurately in any high quality coding scheme.
By considering the transformed coefficients as a [[Tree (graph theory)|tree]] (or trees) with the lowest frequency coefficients at the root node and with the children of each tree node being the spatially related coefficients in the next higher frequency subband, there is a high probability that one or more subtrees will consist entirely of coefficients which are zero or nearly zero, such subtrees are called '''zerotrees'''. Due to this, we use the terms node and coefficient interchangeably, and when we refer to the children of a coefficient, we mean the child coefficients of the node in the tree where that coefficient is located. We use '''children''' to refer to directly connected nodes lower in the tree and '''descendants''' to refer to all nodes which are below a particular node in the tree, even if not directly connected.
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== Introduction ==
'''Embedded zerotree wavelet algorithm''' (EZW) as developed by J. Shapiro in 1993, enables scalable image transmission and decoding. It is based on four key concepts: first, it should be a discrete wavelet transform or hierarchical subband decomposition; second, it should predict the absence of significant information when exploring the [[self-similarity]] inherent in images; third, it has entropy-coded successive-approximation quantization, and fourth, it is enabled to achieve universal lossless data compression via adaptive arithmetic coding.
Besides, the EZW algorithm also contains the following features:
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(5) Adaptive multilevel arithmetic coding which is a fast and efficient method for entropy coding strings of symbols.
== Embedded
=== A. Encoding a coefficient of the significance map ===
In a significance map, the coefficients can be
==== 1. Zerotree root ====
If the magnitude of a coefficient is less than a threshold T, and all its descendants are less than T, then this coefficient is called zerotree root. And if a coefficient has been labeled as zerotree root, it means that all of its descendants are
==== 2. Isolated zero ====
If the magnitude of a coefficient
==== 3. Positive significant coefficient ====
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=== B. Defining threshold ===
The threshold
==== 1. Initial threshold T<sub>0</sub>
==== 2. Threshold T<sub>i</sub> is iteratively reduced to half of the value of the previous threshold
▲==== 1. Initial threshold T<sub>0</sub>: (Assume C<sub>max</sub> is the largest coefficient.) ====
▲<span>[[File:Threshold-0119.png|126x126px]]</span>
▲==== 2. Threshold T<sub>i</sub> is reduced to half of the value of the previous threshold. ====
▲[[File:Threshold-01192.png|frameless|133x133px]]
=== C. Scanning order for coefficients ===
'''[[Raster
=== D. Two-pass bitplane coding ===
==== (1) Refinement pass (or subordinate pass) ====
This determine that if the coefficient is in the
In this method, it will visit the significant coefficients according to the magnitude and raster order within subbands.
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==== (2) Significant pass (or dominant pass) ====
This method will code a bit for each coefficient that is not yet be seen as significant. Once a determination of significance has been made, the significant coefficient is included in a list for further refinement in the refinement pass. And if any coefficient already known to be zero, it will not be coded again.
== Example ==
<pre>
DCT data ZeroTree scan order (EZW)
63 -34 49 10 7 13 -12 7 A B BE BF E1 E2 F1 F2
-31 23 14 -13 3 4 6 -1 C D BG BH E3 E4 F3 F4
15 14 3 -12 5 -7 3 9 CI CJ DM DN G1 G2 H1 H2
-9 -7 -14 8 4 -2 3 2 CK CL DO DP G3 G4 H3 H4
-5 9 -1 47 4 6 -2 2 I1 I2 J1 J2 M1 M2 N1 N2
3 0 -3 2 3 -2 0 4 I3 I4 J3 J4 M3 M4 N3 N4
2 -3 6 -4 3 6 3 6 K1 K2 L1 L2 O1 O2 P1 P2
5 11 5 6 0 3 -4 4 K3 K4 L3 L4 O3 O4 P3 P4
D1: pnzt p ttt tztt tttttptt (20 codes)
PNZT P(t) TTT TZTT TPTT (D1 by M-EZW, 16 codes)
PNZT P(t) Z(t) TZ(p) TPZ(p) (D1 by NM-EZW, 11 codes)
P N (t), P or N above zerotree scan
P N Z(t p), p=pair T, t=triple T, P/N + TT/TTT in D1 code
S1: 1010
D2: ztnp tttttttt
S2: 1001 10 (Shapiro PDF end here)
D3: zzzz zppnppnttnnp tpttnttttttttptttptttttttttptttttttttttt
S3: 1001 11 01111011011000
D4: zzzzzzztztznzzzzpttptpptpnptntttttptpnpppptttttptptttpnp
S4: 1101 11 11011001000001 110110100010010101100
D5: zzzzztzzzzztpzzzttpttttnptppttptttnppnttttpnnpttpttppttt
S5: 1011 11 00110100010111 110101101100100000000 110110110011000111
D6: zzzttztttztttttnnttt
( http://www.polyvalens.com/wavelets/ezw/ )
Detailed: (new S is first, other computed by before cycles)
s-step 1 21 321
val D1 S1 R1 D2 S2 R2 D3 S3. ... R3 ... D4,S4...
A 63 P 1 >=48 56 Z .1 >=56 60 Z ..1 >=60 62
B -34 N 0 <48 -40 T .0 <40 -36 Z ..0 <36 -36
C -31 IZ <32 0 N 1. >=24 -28 Z .1. >=28 -30
D 23 T <32 0 P 0. <24 20 Z .1. >=20 22
BE 49 P 1 >=48 56 .0 <56 52 Z ..0 <52 50
BF 10 T <32 0 P 0 <12 10
BG 14 T <32 0 P 1 >=12 14
BH -13 T <32 0 N 1 >=12 -14
CI 15 T <32 0 T <16 0 P 1 >=12 14
CJ 14 IZ <32 0 T <16 0 P 1 >=12 14
CK -9 T <32 0 T <16 0 N 0 <12 -10
CL -7 T <32 0 T <16 0 T <8 0
DM 3 T <16 0 T <8 0
DN -12 T <16 0 N 1 >=12 -14
DO -14 T <16 0 N 1 >=12 -14
DP 8 T <16 0 P <12 10
E1 7 T <32 0 .E,F,G,H(1,2,3,4)
E2 13 T <32 0 .I,J,K(1,2,3,4)
E3 3 T <32 0 .N,O,P(1,2,3,4)
E4 4 T <32 0 .
J1 -1 T <32 0 .
J2 47 P 0 >48 40 1 >=40 44 .
J3 -3 T <32 0
J4 2 T <32 0
D = dominant pass (P=positive, N=negative, T=ZeroTree, IZ=Izolated zero)
S = subordinate pass;
(R = back reconstructed value)
</pre>
==See also==
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==References==
*{{cite Q|Q56883112}}
==External links==
{{commons category|EZW}}
*{{cite web |url=http://www.polyvalens.com/blog/wavelets/ezw/ |title=EZW encoding |author=Clemens Valens |date=2003-08-24 |archive-url=https://web.archive.org/web/20090203181451/http://pagesperso-orange.fr/polyvalens/clemens/ezw/ezw.html |archive-date=2009-02-03 |url-status=live }}
{{Compression methods}}
[[Category:Image compression]]
[[Category:Lossless compression algorithms]]▼
[[Category:Trees (data structures)]]
[[Category:
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