A multiresolution spline with application to image mosaics 《Acm Trans on Graphics》 , 1983 , 2 (4) :217-236

本文主要介紹使用 Multiresolution Spline演算法來消除影象拼接之間的痕跡 A technical problem common to all applications of photomosaics is joining two images so that the edge between them is not visible 如下圖所示：兩個影象拼接線中間有一個明顯痕跡

首先介紹了一下要解決的問題 這裡使用 image spline 來表示消除痕跡的手段 We will use the term image spline to refer to digital techniques for making these adjustments.

這裡我們先描述一個簡化版本的問題，一維訊號拼接 這裡我們介紹 a weighted average splining technique. 在拼接的鄰域乘以一個權重係數，然後疊加兩個影象（sum），這個演算法的關鍵是 T 寬度的選擇，寬度太小 消除痕跡不明顯，仍有痕跡殘留，寬度太大，會將邊界附近的邊緣特徵削弱 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. 寬度過大會造成一個物體重影，類似雙曝光現象 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 所以這個寬度的選擇和影象特徵尺寸大小密切相關。

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.

上面的兩難問題我們可以換一種方式表達：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.

如果影象只分高頻資訊和低頻資訊，那麼在高頻資訊中我們使用較小的 T，在低頻資訊中選擇較大 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. 這裡我們將影象分解為多個 band-pass component images，在每個 band 中進行拼接，最後疊加所有 components

以上就是 multiresolution spline 大致思路。

下面是演算法實現的具體細節 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 [5], so we refer to the sequence Lo, L1 … , LN as the Laplacian pyramid.

multiresolution spline algorithm: