Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing. Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of a curvelet transform is approximately 10–20 times that of an FFT, and has the same dependence of for an image of size .
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| - 曲波变换（英语：Curvelet Transform）是一种可以对多尺度信号进行表示的非自适应方法。作为小波变换的推广，曲波变换目前广泛的应用于诸如图像处理和科学计算等领域。 小波通过使用具有时频局域化性质的基对傅里叶变换进行了推广。对于高维信号，通过局域化朝向（Orientation），小波变换可以具有方向信息。曲波变换和包含方向信息的小波变换的区别在于，对于角度的局域化会随着尺度变化。 曲波变换适用于表示图像等除奇异点外光滑的信号，这些曲线具有有界的曲率。卡通、几何和文字等图片都具有这样的性质，这些图片的边缘会随着图片的放大显得越来越直。然而一般的照片不具有类似的特征，它们往往在几乎所有的尺度上都有细节信息。所以在处理一般的照片时，选择具有方向信息的小波变换会在每个尺度上都具有相同的纵横比。 当图像类型适合时，曲波变换可以提供比其他小波变换更稀疏的表示。 通过假设仅使用 个小波作为几何测试图像的最佳逼近，并将近似误差作为 的函数来量化表示的稀疏性。对于傅里叶变换，均方误差的衰减速度约为 。对于包括方向性的和非方向性的一系列小波变换，均方误差的衰减速度约为 。而采用曲波变换则可以使均方误差的衰减速度下降到约为 。 曲波变换的快速算法计算复杂度大约是快速傅里叶变换的10-20倍。对于一张大小为的图片，计算复杂度约为 。 (zh)
- Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing. Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of a curvelet transform is approximately 10–20 times that of an FFT, and has the same dependence of for an image of size . (en)
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| - Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing. Wavelets generalize the Fourier transform by using a basis that represents both location and spatial frequency. For 2D or 3D signals, directional wavelet transforms go further, by using basis functions that are also localized in orientation. A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale j is by so the fine-scale bases are skinny ridges with a precisely determined orientation. Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear increasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be more elongated than the lower resolution curvelets. However, natural images (photographs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale. When the image is of the right type, curvelets provide a representation that is considerably sparser than other wavelet transforms. This can be quantified by considering the best approximation of a geometrical test image that can be represented using only wavelets, and analysing the approximation error as a function of . For a Fourier transform, the squared error decreases only as . For a wide variety of wavelet transforms, including both directional and non-directional variants, the squared error decreases as . The extra assumption underlying the curvelet transform allows it to achieve . Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of a curvelet transform is approximately 10–20 times that of an FFT, and has the same dependence of for an image of size . (en)
- 曲波变换（英语：Curvelet Transform）是一种可以对多尺度信号进行表示的非自适应方法。作为小波变换的推广，曲波变换目前广泛的应用于诸如图像处理和科学计算等领域。 小波通过使用具有时频局域化性质的基对傅里叶变换进行了推广。对于高维信号，通过局域化朝向（Orientation），小波变换可以具有方向信息。曲波变换和包含方向信息的小波变换的区别在于，对于角度的局域化会随着尺度变化。 曲波变换适用于表示图像等除奇异点外光滑的信号，这些曲线具有有界的曲率。卡通、几何和文字等图片都具有这样的性质，这些图片的边缘会随着图片的放大显得越来越直。然而一般的照片不具有类似的特征，它们往往在几乎所有的尺度上都有细节信息。所以在处理一般的照片时，选择具有方向信息的小波变换会在每个尺度上都具有相同的纵横比。 当图像类型适合时，曲波变换可以提供比其他小波变换更稀疏的表示。 通过假设仅使用 个小波作为几何测试图像的最佳逼近，并将近似误差作为 的函数来量化表示的稀疏性。对于傅里叶变换，均方误差的衰减速度约为 。对于包括方向性的和非方向性的一系列小波变换，均方误差的衰减速度约为 。而采用曲波变换则可以使均方误差的衰减速度下降到约为 。 曲波变换的快速算法计算复杂度大约是快速傅里叶变换的10-20倍。对于一张大小为的图片，计算复杂度约为 。 (zh)
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