卫星RPC有理多项式模型原理与正反算实现（python）

摘要

        在航天摄影测量领域，RPC模型是一种通用的卫星几何定位模型。作为基础构件，RPC模型是后续前方交会（三角化）、区域网平差、DSM生成的基础。但是，笔者在实际使用中发现网上的关于RPC模型开源代码良莠不齐，有的介绍详细但没有代码，难以找到一组比较易用的RPC模型库。经过寻找测试，笔者发现s2p作者开源的rpm具有健全的功能与清晰的代码，笔者将其稍微修改与整理，并添加了注释，使其变为依赖少、可应用大部分国产卫星RPC处理的单文件功能库，功能主要包括：1.读取RPC模型；2.RPC模型正投影，即地理坐标计算像素坐标；3.RPC模型逆投影，即像素坐标与高程计算经纬度。后续也将基于这个简单的小库，做一些应用的演示。

        本文首先介绍了RPC几何定位模型的基本知识，然后提供了RPC模型代码的实现并进行了简单的使用示范，最后评估的该代码的精度与性能。

RPC几何定位模型介绍

        卫星遥感影像在成像过程中由于受到诸多复杂因素的影响，使各像点产生了不同程度的几何变形。建立遥感影像几何定位模型可以正确地描述每一个像点坐标与其对应地面点物方坐标间的严格几何关系，以便对原始影像进行高精度的几何纠正及对地目标定位，从而实现由二维影像反演实地表面的平面或空间位置，以满足各种遥感应用的需求。

        现今已建立了各种传感器模型,来描述地面空间坐标与相应像点坐标的数学关系.在传统的摄影测量领域,应用较多的是物理模型.这种模型的处理技术已趋向成熟,定位精度比较高.但由于物理传感器模型涉及传感器物理结构,成像方式及各种成像参数.为了保护技术秘密,一些公司只使用有理函数模型(RFM:Rational Function Model),并向用户提供RFM的参数――有理多项式系数(RPC:Rational Polynomial Coefficients).

        RPC可以将其理解为一个带畸变参数的相机模型，其描述的是从三维的地理坐标到二维的卫星影像坐标之间的转换关系，一般称之为从物方到像方，我们可以理解为3D到2D，这也被称之为正投影，具体为已知lon，lat，h，求得像素坐标s，l。而反向投影的则是从影像坐标系到地理坐标系，即从像方到物方，这里要注意的是，这仍然是一个3D到2D的关系，因为2D是无法升维到3D的，具体为已经s，l，h，求得lon与lat。很多同学会在这个地方犯迷糊，注意已知什么，求什么，就可以清楚把握。

        RPC模型正投影表达式可参考(杨博，王密，皮英冬．仅用虚拟控制点的超大区域无控制区域网平差［Ｊ］．测绘学报）

        一种正投影计算过程参考：

  式中，rn、cn分别为归一化的行列号，使用归一化值是为了减少计算过程中的误差。P、L、H分别为归一化的经度、纬度和高程。计算为：

    P = (Latitude     - LAT_OFF)    / LAT_SCALE
    L = (Longitude    - LONG_OFF)   / LONG_SCALE
    H = (Height       - HEIGHT_OFF) / HEIGHT_SCALE
    rn = (Row         - LINE_OFF)   / LINE_SCALE
    cn = (Column      - SAMP_OFF)   / SAMP_SCALE

        正算式子中的分子、分母分别为三次20项的多项式函数。其中(LINE_NUM_COEF_n, LINE_DEN_COEF_n, SAMP_NUM_COEF_n, SAMP_DEN_COEF_n)分别为包含20个系数，分子分母的函数关系如下：

        一般情况下，GeoTIFF文件中会包含RPC信息，国产卫星也一般有.rpc或.rpb文件来存储RPC信息，该信息中包含了4个多项式的80个系数，以及**_OFF，**_SCALE等10个参数，因此只需要知道高程数据便可求解各个像元的经纬度，从而实现正算。

        逆投影（反投影）则是将地理坐标当做待求解参数，进行平差迭代求解，具体原理各大摄影教科书都有，此处不再赘述。

RPC模型库代码实现

Rpc模型库

 #命名为：geo_rpc.py
 """
 RPC model parsers, localization, and projection
 """
 import numpy as np
 from osgeo import gdal
 ​
 #最大迭代次数超过则报错
 class MaxLocalizationIterationsError(Exception):
     """
     Custom rpcm Exception.
     """
     pass
 ​
 ​
 def apply_poly(poly, x, y, z):
     """
     Evaluates a 3-variables polynom of degree 3 on a triplet of numbers.
     将三次多项式的统一模式构建为一个单独的函数
     Args:
         poly: list of the 20 coefficients of the 3-variate degree 3 polynom,
             ordered following the RPC convention.
         x, y, z: triplet of floats. They may be numpy arrays of same length.
 ​
     Returns:
         the value(s) of the polynom on the input point(s).
     """
     out = 0
     out += poly[0]
     out += poly[1]*y + poly[2]*x + poly[3]*z
     out += poly[4]*y*x + poly[5]*y*z +poly[6]*x*z
     out += poly[7]*y*y + poly[8]*x*x + poly[9]*z*z
     out += poly[10]*x*y*z
     out += poly[11]*y*y*y
     out += poly[12]*y*x*x + poly[13]*y*z*z + poly[14]*y*y*x
     out += poly[15]*x*x*x
     out += poly[16]*x*z*z + poly[17]*y*y*z + poly[18]*x*x*z
     out += poly[19]*z*z*z
     return out
 ​
 ​
 def apply_rfm(num, den, x, y, z):
     """
     Evaluates a Rational Function Model (rfm), on a triplet of numbers.
     执行20个参数的分子和20个参数的除法
     Args:
         num: list of the 20 coefficients of the numerator
         den: list of the 20 coefficients of the denominator
             All these coefficients are ordered following the RPC convention.
         x, y, z: triplet of floats. They may be numpy arrays of same length.
 ​
     Returns:
         the value(s) of the rfm on the input point(s).
     """
     return apply_poly(num, x, y, z) / apply_poly(den, x, y, z)
 ​
 ​
 def rpc_from_geotiff(geotiff_path):
     """
     Read the RPC coefficients from a GeoTIFF file and return an RPCModel object.
     该函数返回影像的Gdal格式的影像和RPCmodel
     Args:
         geotiff_path (str): path or url to a GeoTIFF file
 ​
     Returns:
         instance of the rpc_model.RPCModel class
     """
     # with rasterio.open(geotiff_path, 'r') as src:
     #
     dataset = gdal.Open(geotiff_path, gdal.GA_ReadOnly)
     rpc_dict = dataset.GetMetadata("RPC")
     # 同时返回影像与rpc
     return dataset, RPCModel(rpc_dict,'geotiff')
 ​
 def read_rpc_file(rpc_file):
     """
     Read RPC from a RPC_txt file and return a RPCmodel
     从TXT中直接单独读取RPC模型
     Args:
         rpc_file: RPC sidecar file path
 ​
     Returns:
         dictionary read from the RPC file, or an empty dict if fail
 ​
     """
     with open(rpc_file) as f:
         rpc_content = f.read()
     rpc = read_rpc(rpc_content)
     return RPCModel(rpc)
 ​
 def read_rpc(rpc_content):
     """
     Read RPC file assuming the ikonos format
     解析RPC参数
     Args:
         rpc_content: content of RPC sidecar file path read as a string
 ​
     Returns:
         dictionary read from the RPC file
 ​
     """
     import re
 ​
     lines = rpc_content.split('\n')
 ​
     dict_rpc = {}
     for l in lines:
         ll = l.split()
         if len(ll) > 1:
             k = re.sub(r"[^a-zA-Z0-9_]","",ll[0])
             dict_rpc[k] = ll[1]
 ​
     def parse_coeff(dic, prefix, indices):
         """ helper function"""
         return ' '.join([dic["%s_%s" % (prefix, str(x))] for x in indices])
 ​
     dict_rpc['SAMP_NUM_COEFF']  = parse_coeff(dict_rpc, "SAMP_NUM_COEFF", range(1, 21))
     dict_rpc['SAMP_DEN_COEFF']  = parse_coeff(dict_rpc, "SAMP_DEN_COEFF", range(1, 21))
     dict_rpc['LINE_NUM_COEFF']  = parse_coeff(dict_rpc, "LINE_NUM_COEFF", range(1, 21))
     dict_rpc['LINE_DEN_COEFF']  = parse_coeff(dict_rpc, "LINE_DEN_COEFF", range(1, 21))
 ​
     return dict_rpc
 ​
 class RPCModel:
     def __init__(self, d, dict_format="geotiff"):
         """
         Args:
             d (dict): dictionary read from a geotiff file with
                 rasterio.open('/path/to/file.tiff', 'r').tags(ns='RPC'),
                 or from the .__dict__ of an RPCModel object.
             dict_format (str): format of the dictionary passed in `d`.
                 Either "geotiff" if read from the tags of a geotiff file,
                 or "rpcm" if read from the .__dict__ of an RPCModel object.
         """
         if dict_format == "geotiff":
             self.row_offset = float(d['LINE_OFF'][0:d['LINE_OFF'].rfind(' ')])
             self.col_offset = float(d['SAMP_OFF'][0:d['SAMP_OFF'].rfind(' ')])
             self.lat_offset = float(d['LAT_OFF'][0:d['LAT_OFF'].rfind(' ')])
             self.lon_offset = float(d['LONG_OFF'][0:d['LONG_OFF'].rfind(' ')])
             self.alt_offset = float(d['HEIGHT_OFF'][0:d['HEIGHT_OFF'].rfind(' ')])
 ​
             self.row_scale = float(d['LINE_SCALE'][0:d['LINE_SCALE'].rfind(' ')])
             self.col_scale = float(d['SAMP_SCALE'][0:d['SAMP_SCALE'].rfind(' ')])
             self.lat_scale = float(d['LAT_SCALE'][0:d['LAT_SCALE'].rfind(' ')])
             self.lon_scale = float(d['LONG_SCALE'][0:d['LONG_SCALE'].rfind(' ')])
             self.alt_scale = float(d['HEIGHT_SCALE'][0:d['HEIGHT_SCALE'].rfind(' ')])
 ​
             self.row_num = list(map(float, d['LINE_NUM_COEFF'].split()))
             self.row_den = list(map(float, d['LINE_DEN_COEFF'].split()))
             self.col_num = list(map(float, d['SAMP_NUM_COEFF'].split()))
             self.col_den = list(map(float, d['SAMP_DEN_COEFF'].split()))
 ​
             if 'LON_NUM_COEFF' in d:
                 self.lon_num = list(map(float, d['LON_NUM_COEFF'].split()))
                 self.lon_den = list(map(float, d['LON_DEN_COEFF'].split()))
                 self.lat_num = list(map(float, d['LAT_NUM_COEFF'].split()))
                 self.lat_den = list(map(float, d['LAT_DEN_COEFF'].split()))
 ​
         elif dict_format == "rpcm":
             self.__dict__ = d
 ​
         else:
             raise ValueError(
                 "dict_format '{}' not supported. "
                 "Should be {{'geotiff','rpcm'}}".format(dict_format)
             )
 ​
 ​
     def projection(self, lon, lat, alt):
         """
         Convert geographic coordinates of 3D points into image coordinates.
         正投影：从地理坐标到图像坐标
         Args:
             lon (float or list): longitude(s) of the input 3D point(s)
             lat (float or list): latitude(s) of the input 3D point(s)
             alt (float or list): altitude(s) of the input 3D point(s)
 ​
         Returns:
             float or list: horizontal image coordinate(s) (column index, ie x)
             float or list: vertical image coordinate(s) (row index, ie y)
         """
         nlon = (np.asarray(lon) - self.lon_offset) / self.lon_scale
         nlat = (np.asarray(lat) - self.lat_offset) / self.lat_scale
         nalt = (np.asarray(alt) - self.alt_offset) / self.alt_scale
 ​
         col = apply_rfm(self.col_num, self.col_den, nlat, nlon, nalt)
         row = apply_rfm(self.row_num, self.row_den, nlat, nlon, nalt)
 ​
         col = col * self.col_scale + self.col_offset
         row = row * self.row_scale + self.row_offset
 ​
         return col, row
 ​
 ​
     def localization(self, col, row, alt, return_normalized=False):
         """
         Convert image coordinates plus altitude into geographic coordinates.
         反投影：从图像坐标到地理坐标
         Args:
             col (float or list): x image coordinate(s) of the input point(s)
             row (float or list): y image coordinate(s) of the input point(s)
             alt (float or list): altitude(s) of the input point(s)
 ​
         Returns:
             float or list: longitude(s)
             float or list: latitude(s)
         """
         ncol = (np.asarray(col) - self.col_offset) / self.col_scale
         nrow = (np.asarray(row) - self.row_offset) / self.row_scale
         nalt = (np.asarray(alt) - self.alt_offset) / self.alt_scale
 ​
         if not hasattr(self, 'lat_num'):
             lon, lat = self.localization_iterative(ncol, nrow, nalt)
         else:
             lon = apply_rfm(self.lon_num, self.lon_den, nrow, ncol, nalt)
             lat = apply_rfm(self.lat_num, self.lat_den, nrow, ncol, nalt)
 ​
         if not return_normalized:
             lon = lon * self.lon_scale + self.lon_offset
             lat = lat * self.lat_scale + self.lat_offset
 ​
         return lon, lat
 ​
 ​
     def localization_iterative(self, col, row, alt):
         """
         Iterative estimation of the localization function (image to ground),
         for a list of image points expressed in image coordinates.
         逆投影时的迭代函数
         Args:
             col, row: normalized image coordinates (between -1 and 1)
             alt: normalized altitude (between -1 and 1) of the corresponding 3D
                 point
 ​
         Returns:
             lon, lat: normalized longitude and latitude
 ​
         Raises:
             MaxLocalizationIterationsError: if the while loop exceeds the max
                 number of iterations, which is set to 100.
         """
         # target point: Xf (f for final)
         Xf = np.vstack([col, row]).T
 ​
         # use 3 corners of the lon, lat domain and project them into the image
         # to get the first estimation of (lon, lat)
         # EPS is 2 for the first iteration, then 0.1.
         lon = -col ** 0  # vector of ones
         lat = -col ** 0
         EPS = 2
         x0 = apply_rfm(self.col_num, self.col_den, lat, lon, alt)
         y0 = apply_rfm(self.row_num, self.row_den, lat, lon, alt)
         x1 = apply_rfm(self.col_num, self.col_den, lat, lon + EPS, alt)
         y1 = apply_rfm(self.row_num, self.row_den, lat, lon + EPS, alt)
         x2 = apply_rfm(self.col_num, self.col_den, lat + EPS, lon, alt)
         y2 = apply_rfm(self.row_num, self.row_den, lat + EPS, lon, alt)
 ​
         n = 0
         while not np.all((x0 - col) ** 2 + (y0 - row) ** 2 < 1e-18):
 ​
             if n > 100:
                 raise MaxLocalizationIterationsError("Max localization iterations (100) exceeded")
 ​
             X0 = np.vstack([x0, y0]).T
             X1 = np.vstack([x1, y1]).T
             X2 = np.vstack([x2, y2]).T
             e1 = X1 - X0
             e2 = X2 - X0
             u  = Xf - X0
 ​
             # project u on the base (e1, e2): u = a1*e1 + a2*e2
             # the exact computation is given by:
             #   M = np.vstack((e1, e2)).T
             #   a = np.dot(np.linalg.inv(M), u)
             # but I don't know how to vectorize this.
             # Assuming that e1 and e2 are orthogonal, a1 is given by
             #  / 
             num = np.sum(np.multiply(u, e1), axis=1)
             den = np.sum(np.multiply(e1, e1), axis=1)
             a1 = np.divide(num, den).squeeze()
 ​
             num = np.sum(np.multiply(u, e2), axis=1)
             den = np.sum(np.multiply(e2, e2), axis=1)
             a2 = np.divide(num, den).squeeze()
 ​
             # use the coefficients a1, a2 to compute an approximation of the
             # point on the gound which in turn will give us the new X0
             lon += a1 * EPS
             lat += a2 * EPS
 ​
             # update X0, X1 and X2
             EPS = .1
             x0 = apply_rfm(self.col_num, self.col_den, lat, lon, alt)
             y0 = apply_rfm(self.row_num, self.row_den, lat, lon, alt)
             x1 = apply_rfm(self.col_num, self.col_den, lat, lon + EPS, alt)
             y1 = apply_rfm(self.row_num, self.row_den, lat, lon + EPS, alt)
             x2 = apply_rfm(self.col_num, self.col_den, lat + EPS, lon, alt)
             y2 = apply_rfm(self.row_num, self.row_den, lat + EPS, lon, alt)
 ​
             n += 1
 ​
         return lon, lat
 ​

使用示例

 import geo_rpc
 ​
 rpc = geo_rpc.read_rpc_file('../data/BWDSC_RPC.txt')
 ​
 #取一个在该rpc范围内的经纬度，并假定一个高程值
 lon, lat ,h  = 112.961403969,34.452284651 ,100
 ​
 #测试正投影
 rows ,cols = rpc.projection(lon, lat ,h )
 print (" rows and cols : ", rows ,cols )
 #测试反投影
 rows ,cols = rpc.localization(rows ,cols ,h )
 print ( " lon and lat : ", lon, lat)

输出结果：

  rows and cols :  7442.261718114856 19653.793617557167
  lon and lat :  112.961403969 34.452284651

实验结果

精度评估

经过多组对比，该代码的的计算结果与gdal（C++）的rpc模型坐标转换和组内“传承rpc自用库一致。

性能评估

 import geo_rpc
 from time import *
 ​
 rpc = geo_rpc.read_rpc_file('../data/BWDSC_RPC.txt')
 ​
 #取一个在该rpc范围内的经纬度，并假定一个高程值，可以取dem对应值，详见前文博客
 lon, lat ,h  = 112.961403969,34.452284651 ,100
 ​
 begin_time = time()
 for i in range(10000):
     #测试正投影
     rows ,cols = rpc.projection(lon, lat ,h )
 ​
 end_time = time()
 run_time = end_time-begin_time
 print ('正投影平均耗时：',run_time/10000*1000,'ms')
 ​
 #测试反投影
 begin_time = time()
 for i in range(10000):
     rows ,cols = rpc.localization(rows ,cols ,h )
 ​
 end_time = time()
 run_time = end_time-begin_time
 print ('反投影平均耗时：',run_time/10000*1000,'ms')

测试结果：

 正投影平均耗时： 0.028485536575317383 ms
 反投影平均耗时： 1.0120615482330322 ms

在ThinkPadX1（内存：16G/cpu：Intel-i5）上，进行了一个不太严谨的测试，可以看出，因为迭代的关系，反向投影耗时约为正向的30倍，但数量级也在ms级别，具可用的性能。

附：

经典文章：卫星遥感影像RPC参数求解算法研究_张过

正反算matlab实现：OS-Eval/RPCprocess at main · xym2009/OS-Eval (github.com)

（Python）使用Gdal进行批量的影像RPC正射校正 - 知乎 (zhihu.com)

遥感 RPC， RPB文件相关信息-CSDN博客 介绍RPC和RPB及其读写转换及正算（经纬度->图像坐标）等matlab实现

来源：

（Python）卫星RPC有理多项式模型读取与正反投影坐标计算原理与实现 - 知乎 (zhihu.com)