Wavelets

Definition: Wavelets are mathematical functions that let us divide data into different frequency components and then study each component with a resolution appropriate for its overall scale. Wavelets are used in computer imaging, animation, noise reduction and data compression.

In many fields of study, from science and engineering to economics and psychology, we need to analyze data so that we can discover underlying patterns and information. A common way of doing this is to transform the data by applying mathematical functions.

One of the best-known processing techniques is Fourier analysis, in which you can approximate a real-world data stream by adding together a series of sine and cosine curves at different frequencies; the more curves you include in your approximation, the more closely you can replicate the original data. Since we know how to work with these well-defined trigonometric curves, we can often deduce patterns in the data that would otherwise remain hidden.

But Fourier analysis has limitations. It works best when the original data has features that repeat periodically, and it has trouble with transient signals or data that shows abrupt changes, such as the spoken word. Often, we need to be able to change our analytical representation depending on the actual data, so that we can resolve more detail in specific parts of the data stream. In essence, we need a way to change scale at various points, and scale is at the heart of wavelets.

The following explanation is adapted from Dana Mackenzie’s highly recommended article “Wavelets: Seeing the Forest and the Trees” .

Consider how we view a landscape. If you’re looking down from a jet airliner in summer, a forest appears as a solid canopy of green. If you’re in a car driving by, however, you see individual trees. If you stop and move closer, you can make out individual branches and leaves. Up close, you may spot a dewdrop or an insect sitting on a leaf. With a magnifying glass, you can see structural details of the leaf and its veins.

As we get ever closer to an object, our view becomes narrower and we see finer and finer detail. In other words, as our scope becomes smaller, our resolution becomes greater.

Our eyes and mind adapt quickly to these changes in perspective, moving from the macro scale to the micro. Unfortunately, we can’t apply this technique to a photograph or computerized digital image.

If you enlarged a picture of a forest (as if you were trying to get “closer” to a tree), all you’d see is a fuzzier image; you still wouldn’t be able to make out the branch, the leaf or the dewdrop. Regardless of what you might see in the movies, no amount of “sharpening” or processing can help you see detail that hasn’t already been encoded into the image. We can't see anything smaller than a pixel, and the camera can show us only one resolution at a time.

Wavelet algorithms allow us to record or process different areas of a scene at different levels of detail (resolution) and using greater amounts of compression (scale). In essence, they let us take new photos at closer range. If you look at a collection of data (also called a signal) from a broad perspective, you’ll notice large-scale features; using a smaller, closer perspective, you can observe much smaller features.

Unlike the sinusoidal, endlessly repeating waves used in Fourier analysis, wavelets are often irregular and asymmetric, with values that die out to zero as they move farther from a central point. By decomposing a data stream into wavelets, it’s often possible to preserve and even enhance specific local features of the signal and information about its timing.

Wavelets can take almost any shape, and much of the work being done in wavelet applications is based on finding appropriate wavelet functions that work for the type of data being processed.

The first wavelet function was a simple square waveform, developed by mathematician Alfred Harr in the early 1900s. Real advancement in the field, however, began in the mid-1980s, when Jean Morlet, an engineer at a French oil company, developed wavelet-transform analysis to interpret seismic data. He then teamed with physicist Alex Grossmann to formalize the mathematics.

Moving well beyond their geophysical roots, wavelets today are used for a variety of purposes, especially in the areas of digital imaging and compression.

Depending on your needs, for example, you can use different types of compression to reduce the size of a digital image according to how much detail or accuracy you are willing to give up. Wavelet-based compression can be much more efficient than older types. Wavelets also make possible incredibly fine detail and texture mapping, such as the lifelike rendering of hair in the animated film Monsters, Inc., while still keeping file sizes and processing times manageable.

Wavelets are central to a number of image-related compression standards, including the JPEG-2000 standard for color images and WSQ, the wavelet scalar quantization gray-scale fingerprint image compression algorithm that the FBI has used since 1993 for storing its fingerprint database.

The wavelet compression in the MPEG-4 digital video standard offers better-quality Web-based video than JPEG, yet it produces files that are a fraction of the size. MPEG-4 also has several quality layers, allowing servers to adjust their output dynamically according to needed bandwidth.

Wavelets are also being used for noise reduction and image-searching techniques. Scientists are now exploring the use of wavelets for various types of medical diagnostics and for weather forecasting as well.

小波

定義: 小波是數學函數，它讓我們將數據分成不同頻率的分量，然后按與整體尺度相適應的分辨率分析每個分量。小波用于計算機成像、動畫、降噪和數據壓縮。

在很多研究領域，從科學研究與工程技術到經濟學和心理學，我們需要分析數據，從而能發現基本的模式和信息。進行這種分析常用的方法就是利用數學函數做數據變換。

傅里葉分析是其中一個最著名的處理技術，通過將不同頻率上的一系列正弦和余弦曲線迭加起來，你就能逼近真實世界中的數據流。在你的近似計算中曲線越多，就越能更精確地復制原始的數據。由于我們知道如何用它們定義完善的三角函數曲線，所以我們常常能推算出隱藏的數據模式。

但是傅里葉分析也有局限性。它最適合分析周期性重復的原始數據，對瞬態信號或者表現出突然變化的數據(如說的話)，傅里葉分析就有困難。所以我們常常需要隨實際數據改變我們的分析表示法，從而使我們能分辨出數據流中特定部分更多的細節。本質上，我們需要一種能在不同點上改變尺度的方法，而尺度就是小波的核心。

下面的解釋節選于Dana Mackenzie所著、受到高度推崇的“小波: 既見森林又見樹木”一書。

考慮一下我們是如何看風景的。如果你在夏天從飛機上向下看，森林就是鋪天蓋地的綠色。然而，若是你開車從旁邊經過，你見到的是一棵棵的樹木。如果你停下來，走得更靠近一些，你就能看清枝杈和樹葉。再近些，你還要可以看見樹葉上的露珠和昆蟲。而用放大鏡，你就能看清樹葉和其脈絡的構造細節。

當我們更靠近一個物體時，我們的視野就變窄了，看見越來越細微的細節。換言之，當我們的范圍變得更小時，我們的分辨率就更高。我們的眼睛和思維能很快適應視野的變化，從宏觀轉到微觀。可惜我們不能將此技術應用于照片或計算機化的數字圖像。

如果你放大一張森林的照片(好像你在試圖“走近”一棵樹木)，你所見到的是更模糊的圖像;。你不能分辨出枝杈、樹葉或露珠。不管你在電影里看到什么，任何“銳化”或處理都無助你看清細節，這些細節原本就沒有編碼進圖像。我們見不到比像素更小的東西，照相機一次只能給我們提供一種分辨率。

小波算法允許我們以不同等級的細節(分辨率)和利用更大的壓縮(比例尺)，記錄或處理一個場景的不同區域。本質上，它們讓我們在更近的距離上拍攝新的照片。如果你從一個很寬的視野看數據的集合(也稱信號)，你將看到大尺寸的特性，在更小、更靠近的視野上，你能觀察到更細小的特性。

與傅里葉分析中使用的無限重復的正弦波不同，小波