陈人杰
配准的时候控制点的数⽬该如何确定？
配准的时候控制点的数⽬该如何确定？
⼀次多项式，控制点⾄少选择3个(6个未知数)；
⼆次多项式，控制点⾄少选择6个(12个未知数)；
三次多项式，控制点⾄少选择10个(20个未知数)；
四次多项式，控制点⾄少选择15个(30个未知数)；
n次多项式，控制点⾄少选择(n+1)*(n+2)/2；
证明：
设⼀次多项式：系数个数为a1；⼆次多项式：系数个数为a2；n次多项式，系数个数为an；
a2-a1=6即(2*3); (1)
a3-a2=8即(2*4); (2)
a4-a3=10即(2*5) (3)
。。。。。。干煸豆角的做法
an-an-1=2n+2即[2*(n+1)] ....n
以上n个式⼦相加以后不难得出：an-a1=(n+4)*(n-1) 其中a1=6
所以an=(n+1)*(n+2)即控制点⾄少选择an/2=(n+1)*(n+2)/2个
Georeferencing⼯具的Linktable中残差是如何计算得出？
根据如下公式进⾏计算残差
其中Xorig和Yorig是影像中GCP的原始⾏列坐标，X'和Y'是由原始影像中计算或者估计出的坐标。⽅差的平⽅根代表这个GCP的精度测度。
栅格配准的转换⽅式有如下⼏种：
⼀次多项式（仿射）1st Order Polynomial(Affine)
⼆次多项式2nd Order Polynomial
三次多项式3rd Order Polynomial
校正Adjust
样条Spline
补充维生素a
应⽤不同的转换⽅式会得到不同的配准结果，但是各种转换⽅式有什么区别？
Transforming the raster
管理信息系统论文When you've created enough links, you can transform—or warp—the raster datat to permanently match the map coordinates of the target data. You have the choice of using a polynomial, spline, or adjust transformation to determine the correct map coordinate location for each cell in the raster.
The polynomial transformation us a polynomial built on control points and a least squares fitting (LSF) algorithm. It is optimized for global accuracy but does not guarantee local accuracy. The polynomial transformation yields two formulas: one for computing the output x-coordinate for an input (x,y) location and one for computing the y-coordinate for an input (x,y) location. The goal of the least squares fitting algorithm is to derive a general formula that can be applied to all points, usually at the expen of slight movement of the to positions of the control points. The number of the noncor
related control points
required for this method must be 3 for a first order, 6 for a cond order, and 10 for a third order. The first-order polynom ial transformation is commonly ud to georeference an image.
Below is the equation to transform a raster datat using the affine (first order) polynomial transformation. You can e how six parameters define how a raster's rows and columns transform onto map coordinates.
U a first-order or affine transformation to shift, scale, and rotate a raster datat. This generally results in straight lines on the raster datat mapped as straight lines in the warped raster datat. Thus, squares and rectangles on the raster datat are commonly changed into parallelograms of arbitrary scaling and angle orientation.
With a minimum of three links, the mathematical equation ud with a first-order transformation can exactly map each raster point to the target location. Any more than three links introduces errors, or residuals, that are distributed throughout all the links. However, you should add more than three links
becau if one link is positionally wrong, it has a much greater impact on the transformation. Thus, e
ven though the mathe matical transformation error may increa as you create more links, the overall accuracy of the transformation will increa as well.
The higher the transformation order, the more complex the distortion that can be corrected. However, transformations higher than third order are rarely needed. Higher-order transformations require more links and, thus, will involve progressively more processing time. In general, if your raster datat needs to be stretched, scaled, and rotated, u a first-order transformation. If, however, the raster datat must be bent or curved, u a cond- or third-order transformation.
The spline transformation is a true rubber sheeting method and optimizes for local accuracy but not global accuracy. It is bad on a spline function—a piecewi polynomial that maintains continuity and smoothness between adjacent polynomials. Spline transforms the source control points exactly to target control points; the pixels that are a distance from the control points are not guaranteed to be accurate. This transformation is uful when the control points are important and it is required that they be registered precily. Adding more control points can increa overall accuracy of the spline transformation. Spline requires a minimum of ten control points.
The adjust transformation optimizes for both global LSF and local accuracy. It is built on an algorithm
that combines a polynomial transformation and triangulated irregular network (TIN) interpolation techniques. The adjust transformation performs a polynomial transformation using two ts of control points and adjusts the control points locally to better match the target control points using a TIN interpolation technique. Adjust requires a minimum of three control points.
栅格配准关于春天的图画
当你创建了⾜够多的链节，你能让栅格影像与有地理坐标系的⽬标图层相匹配。配准的⽅法有多项式、样条或adjust配准
多项式配准
多项式配准使⽤多项式，通过最⼩⼆乘法拟合对控制点进⾏操作。它能优化全局精度，但是不能保证局部精度。多项式配准产⽣两种公式：⼀个是计算X坐标，⼀个是计算Y坐标。最⼩⼆乘法产⽣⼀个能适⽤于所有点的通⽤公式，但是会导致控制点位置的微⼩变动。多项式的次数若为1，最少要求3个控制点；若为2，6个控制点；若为3，10个控制点。经常⽤⼀次多项式来配准影像。
下图是使⽤⼀次仿射多项式配准⼀幅栅格图的公式，可以看出影像的⾏和列通过六个参数的设定被配准的过程。
梦使⽤⼀次多项式或仿射变换移动、放⼤缩⼩、旋转⼀幅影像，通常是待校正影像和参考影像在直线段上的匹配。因此，影像上的矩形经常会被改变成任意⾓度和尺度的平⾏四边形。
通过⼀次多项式转换的数学公式，选三个控制点能使它们准确地匹配到⽬标影像上。选更多的点就会形成残差或误差。应该多选择⼀些点。若点较少的话，则如果有⼀个点有误，就会产⽣很⼤的影响。⽽多选⼀些控制点的话，即使误差会增⼤但是整体的精度还是会提⾼。
多次项的次数越⾼，变形越厉害。次数⼤于三的多项式⼏乎不必的。次数⾼的多项式要求的控制点更多，校正所需的时间也会较长。总的来说，若你要对影像进⾏拉伸，缩放和旋转，使⽤⼀次多项式。若你想使影像弯曲变形，选择⼆次或三次多项式。
样条函数
样条配准就是⼀个橡⽪条配准⽅法，也是⼀个局部优化⽽⾮整体的优化的⽅法。它是基于⼀个样条函数——⼀个保持连续性和平滑性的分段多项式。样条法使源控制点和⽬标点完全匹配，但是⾮控制点的精度不能保证。当控制点是⾮常重要和它们要求完全匹配时，这种⽅法是适⽤的。增加控制点能提⾼整体精度。这种⽅法最少需要⼗个控制点。
铬鞣Adjust配准
adjust配准法优化了全局最⼩⼆乘法配准和局部精度。它融合了多项式校正和三⾓⽹插值技术。通过三⾓⽹插值技样，它利⽤两套控制点来调整局部的控制点更好的匹配⽬标控制点。这种⽅法要求最少三个控制点。
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