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- 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 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 (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. In zerotree based image compression scheme such as EZW and SPIHT, the intent is to use the statistical properties of the trees in order to efficiently code the locations of the significant coefficients. Since most of the coefficients will be zero or close to zero, the spatial locations of the significant coefficients make up a large portion of the total size of a typical compressed image. A coefficient (likewise a tree) is considered significant if its magnitude (or magnitudes of a node and all its descendants in the case of a tree) is above a particular threshold. By starting with a threshold which is close to the maximum coefficient magnitudes and iteratively decreasing the threshold, it is possible to create a compressed representation of an image which progressively adds finer detail. Due to the structure of the trees, it is very likely that if a coefficient in a particular frequency band is insignificant, then all its descendants (the spatially related higher frequency band coefficients) will also be insignificant. EZW uses four symbols to represent (a) a zerotree root, (b) an isolated zero (a coefficient which is insignificant, but which has significant descendants), (c) a significant positive coefficient and (d) a significant negative coefficient. The symbols may be thus represented by two binary bits. The compression algorithm consistsof a number of iterations through a dominant pass and a subordinate pass, the threshold is updated (reduced by a factor of two) after each iteration. The dominant pass encodes the significance of the coefficients which have not yet been found significant in earlier iterations, by scanning the trees and emitting one of the four symbols. The children of a coefficient are only scanned if the coefficient was found to be significant, or if the coefficient was an isolated zero. The subordinate pass emits one bit (the most significant bit of each coefficient not so far emitted) for each coefficient which has been found significant in the previous significance passes. The subordinate pass is therefore similar to bit-plane coding. There are several important features to note. Firstly, it is possible to stop the compression algorithm at any time and obtain an approximation of the original image, the greater the number of bits received, the better the image. Secondly, due to the way in which the compression algorithm is structured as a series of decisions, the same algorithm can be run at the decoder to reconstruct the coefficients, but with the decisions being taken according to the incoming bit stream. In practical implementations, it would be usual to use an entropy code such as arithmetic code to further improve the performance of the dominant pass. Bits from the subordinate pass are usually random enough that entropy coding provides no further coding gain. The coding performance of EZW has since been exceeded by SPIHT and its many derivatives. (en)
- EZW (embedded zerotree wavelet) is een compressie-algoritme met verlies van kwaliteit bij het coderen van afbeeldingen. Het EZW-algoritme probeert een zo goed mogelijke kwaliteit van de afbeelding te behouden volgens een bepaalde bitrate en doet dit op een embedded manier.Embedded betekent dat het een bitstream genereert die op elk mogelijk punt afbreekbaar is en zo dus gebruikt kan worden voor progressieve codering. Bij een embedded code zijn de bits volgens belang geordend in de bitstream. Zo kan de encoder het coderen op ieder moment stopzetten terwijl slechts de minder belangrijkere bits verloren gaan.Het principe van het EZW-algoritme wordt ook gebruikt bij het algoritme Set Partitioning in Hierarchical Trees (SPIHT), dat een uitbreiding is op het EZW-algoritme. Daarnaast is ook het algoritme Embedded Block Coding with Optimized Truncation (EBCOT) op EZW gebaseerd. Het EBCOT-algoritme wordt gebruikt bij de JPEG 2000 standaard.EZW bestaat uit twee centrale componenten met name een zerotree datastructuur en de methode van successive approximation quantization. (nl)
- 嵌入式零樹小波 Embedded Zerotrees of Wavelet transforms (EZW)是一種有损的影像壓縮演算法。在高壓縮率的情況下，大部分次頻帶轉換(如小波轉換)所產生的的係數都會是0，或是非常接近0。這背後的原因是在於真實世界的影像大部分集中在低頻的成分。然而，也確實會有一些高頻成分，例如影像中邊緣的部分。人眼的感知對於這些高頻成分特別的敏感，因此在高畫質的影像壓縮編碼中，這些資訊需要被妥當地重建並呈現在解壓縮後的影像中。 演算法一開始先把最低頻的係數當作樹狀架構的根節點，每一個節點的子代為其更高層相對應次帶位置的係數。由於每個節點都會有其各自延伸的子代，因此又可以稱這些節點為子樹。由於大部分的資訊被集中在低頻，因此越往高頻成分走係數的值越容易會是0。如果一個子樹的係數的值全部都是0(或者是低於某個門檻值)，那麼我們就會稱此子樹為零樹。基於這個原因，在後文中節點和係數會被交錯使用，且當提及一個係數的子代時，意指的是該子代所相對應位置的係數值。一個節點的子代意指的是該節點下一層相對應的係數，一個節點的後代指的則是一個節點往後的所有衍生的節點，意即包括其子代的子代，即便與該節點並無直接連結。 基於零樹演算法的壓縮演算法(如EZW和)的目的是在於說希望能夠利用每個次帶間的統計特性來更有效的對重要的係數進行編碼。由於大部分的係數會是近於0的值，我們會花上大部分的位元在表示那些重要的係數值。一個係數的重要性(significance)是由其值的大小是否超過一個門檻值做決定。在編碼程序的一開始我們可以先將門檻值定為相當接近整張圖片最大的係數值，往後在每個編碼的循環中逐步降低該門檻值，如此產生一個漸進式的表示方式。基於次帶轉換的特性，如果一個係數在其本身的次帶為不重要的，那麼其後代皆為0的可能性也會相當的大。 EZW有一些重要的特性：第一，這個演算法可以停在編碼程序中的任何一點，並產生一個合適的重建影像。因此可以想成是，我們是透過更多的位元數去逐步精煉輸出的重建影像；第二，其演算法基本上是透過相當多的漸進式決策手續，因此可以使用算術編碼來更進一步的提高它的壓縮效率。然而，即便沒有使用算術編碼，它的編碼程序所產生的符號值其實已經相當接近隨機分佈了，因此通常加上熵編碼(如算術編碼)的幫助也會有限。 EZW目前已經有很多衍生的演算法，諸如等等，都可以達到比它更好的壓縮效能。 (zh)
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- 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 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. (en)
- EZW (embedded zerotree wavelet) is een compressie-algoritme met verlies van kwaliteit bij het coderen van afbeeldingen. Het EZW-algoritme probeert een zo goed mogelijke kwaliteit van de afbeelding te behouden volgens een bepaalde bitrate en doet dit op een embedded manier.Embedded betekent dat het een bitstream genereert die op elk mogelijk punt afbreekbaar is en zo dus gebruikt kan worden voor progressieve codering. (nl)
- 嵌入式零樹小波 Embedded Zerotrees of Wavelet transforms (EZW)是一種有损的影像壓縮演算法。在高壓縮率的情況下，大部分次頻帶轉換(如小波轉換)所產生的的係數都會是0，或是非常接近0。這背後的原因是在於真實世界的影像大部分集中在低頻的成分。然而，也確實會有一些高頻成分，例如影像中邊緣的部分。人眼的感知對於這些高頻成分特別的敏感，因此在高畫質的影像壓縮編碼中，這些資訊需要被妥當地重建並呈現在解壓縮後的影像中。 演算法一開始先把最低頻的係數當作樹狀架構的根節點，每一個節點的子代為其更高層相對應次帶位置的係數。由於每個節點都會有其各自延伸的子代，因此又可以稱這些節點為子樹。由於大部分的資訊被集中在低頻，因此越往高頻成分走係數的值越容易會是0。如果一個子樹的係數的值全部都是0(或者是低於某個門檻值)，那麼我們就會稱此子樹為零樹。基於這個原因，在後文中節點和係數會被交錯使用，且當提及一個係數的子代時，意指的是該子代所相對應位置的係數值。一個節點的子代意指的是該節點下一層相對應的係數，一個節點的後代指的則是一個節點往後的所有衍生的節點，意即包括其子代的子代，即便與該節點並無直接連結。 EZW目前已經有很多衍生的演算法，諸如等等，都可以達到比它更好的壓縮效能。 (zh)
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- Embedded Zerotrees of Wavelet transforms (en)
- Ezw (nl)
- 嵌入式零樹小波 (zh)
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