A multiresolution spline with application to image mosaics 《Acm Trans on Graphics》 , 1983 , 2 (4) :217-236
本文主要介绍使用 Multiresolution Spline算法来消除图像拼接之间的痕迹 A technical problem common to all applications of photomosaics is joining two images so that the edge between them is not visible 如下图所示：两个图像拼接线中间有一个明显痕迹
这里使用 image spline 来表示消除痕迹的手段 We will use the term image spline to refer to digital techniques for making these adjustments.
这里我们先描述一个简化版本的问题，一维信号拼接 这里我们介绍 a weighted average splining technique.
在拼接的邻域乘以一个权重系数，然后叠加两个图像（sum），这个算法的关键是 T 宽度的选择，宽度太小 消除痕迹不明显，仍有痕迹残留，宽度太大，会将边界附近的边缘特征削弱 If T is small compared to image features, then the boundary may still appear as a step in image gray level, albeit a somewhat blurred step. 宽度过大会造成一个物体重影，类似双曝光现象 If, on the other hand, T is large compared to image features, features from both images may appear superimposed within the transition zone, as in a photographic double exposure.
Clearly, the size of the transition zone, relative to the size of image features, plays a critical role in image splining 所以这个宽度的选择和图像特征尺寸大小密切相关。
To eliminate a visible edge the transition width should be at least comparable in size to the largest prominent features in the image. On the other hand, to avoid a double exposure effect, the zone should not be much larger than the smallest prominent image features. There is no choice of T which satisfies both requirements in the star images of Figure 3 because these contain both a diffuse background and small bright stars.
上面的两难问题我们可以换一种方式表达：image spatial frequency content. In particular, a suitable T can only be selected if the images to be splined occupy a relatively narrow spatial frequency band.
如果图像只分高频信息和低频信息，那么在高频信息中我们使用较小的 T，在低频信息中选择较大 T
The approach proposed here is that such images should first be decomposed into a set of band-pass component images. A separate spline with an appropriately selected T can then be performed in each band. Finally, the splined band-pass components are recombined into the desired mosaic image. We call this approach the multiresolution spline. 这里我们将图像分解为多个 band-pass component images，在每个 band 中进行拼接，最后叠加所有 components
以上就是 multiresolution spline 大致思路。
下面是算法实现的具体细节 2. Basic Pyramid Operations A sequence of low-pass filtered images Go, G1 … , GN can be obtained by repeatedly convolving a small weighting function with an image
Convolution with a Gaussian has the effect of low-pass filtering the image. Pyramid construction is equivalent to convolving the image with a set of Gaussian-like functions to produce a corresponding set of filtered images. Because of the importance of the multiple filter interpretation, we shall refer to this sequence of images Go,G1 … GN as the Gaussian pyramid.
The Gaussian pyramid is a set of low-pass filtered images. In order to obtain the band-pass images required for the multiresolution spline we subtract each level of the pyramid from the next lowest level.
This difference of Gaussian-like functions resembles the Laplacian operators commonly used in the image processing , so we refer to the sequence Lo, L1 … , LN as the Laplacian pyramid.
multiresolution spline algorithm:
The idea behind multi-band blending is to blend low frequencies over a large spatial range, and high frequencies over a short range.