傅立叶域重建图像
Resampling images in Fourier domain
摘要：
在模拟天空图像时，人们通常会获取由一组像素化样本和插值核定义的星系图像$F(x)$，然后想要产生一个新的采样图像，该图像表示该星系，因为它将以不同的点扩散函数、旋转、剪切或放大和/或不同的像素比例出现。这些操作有时仅在将图像的傅里叶变换$\tilde F(u)$重新采样到与由样本的离散傅里叶变换（DFT）产生的网格不同的$u$空间网格上时才是可能的或最有效地执行。在某些应用中，重要的是，重采样的图像是准确的$10^3$优于1部分，所以在本文中，我们首先使用标准的傅立叶技术，以显示傅立叶域插值与包裹sinc函数产生的确切值的$\tilde F(u)$的输入样本和内核。该操作随图像维度缩放为$N^4$，并且可能非常慢，因此我们接下来研究用紧凑内核近似sinc函数所产生的误差。我们表明，这些近似产生乘法误差加上一对鬼图像（在每个维度）在模拟图像。标准Lanczos或三次插值器，当应用在傅立叶域，产生不可接受的文物。我们发现，$10^3$中的错误$<1$部分可以通过以下方式获得：（1）在执行$x\rightarrow u$ DFT之前对原始图像进行4倍零填充，然后（2）使用我们设计的6点分段五次插值来重新填充到所需的$u$网格，以使重影最小化，然后（3）执行DFT回到$x$域。
英文摘要：
When simulating sky images, one often takes a galaxy image $F(x)$ defined by a set of pixelized samples and an interpolation kernel, and then wants to produce a new sampled image representing this galaxy as it would appear with a different point-spread function, a rotation, shearing, or magnification, and/or a different pixel scale. These operations are sometimes only possible, or most efficiently executed, as resamplings of the Fourier transform $\tilde F(u)$ of the image onto a $u$-space grid that differs from the one produced by a discrete Fourier transform (DFT) of the samples. In some applications it is essential that the resampled image be accurate to better than 1 part in $10^3$, so in this paper we first use standard Fourier techniques to show that Fourier-domain interpolation with a wrapped sinc function yields the exact value of $\tilde F(u)$ in terms of the input samples and kernel. This operation scales with image dimension as $N^4$ and can be prohibitively slow, so we next investigate the errors accrued from approximating the sinc function with a compact kernel. We show that these approximations produce a multiplicative error plus a pair of ghost images (in each dimension) in the simulated image. Standard Lanczos or cubic interpolators, when applied in Fourier domain, produce unacceptable artifacts. We find that errors $<1$ part in $10^3$ can be obtained by (1) 4-fold zero-padding of the original image before executing the $x\rightarrow u$ DFT, followed by (2) resampling to the desired $u$ grid using a 6-point, piecewise-quintic interpolant that we design expressly to minimize the ghosts, then (3) executing the DFT back to $x$ domain.
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