User:Travelling salesman/gpx reduce

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This is a python script that removes (almost) unnecessary points from gpx-files. Set your gps receiver to the highest point density available (e.g. 1s) and use this script on your gpx-files before osm-upload. The script yields highly optimized results.

The script must be saved to a file ( and the executable flag should be set. Then it can be executed from the command line. For more documentation, type

./ --help

If the script produces an output that looks non-optimal, this might be because of the third dimension. Otherwise leave me a message.


  • python interpreter
  • scipy scientific tools for python
  • lxml xml library
  • matplotlib

On a debian based linux distribution (such as Ubuntu) you can install these with:

sudo apt-get install python python-scipy python-lxml python-matplotlib


gpx_reduce finds the track that globally optimizes a given cost function under the restriction that no original point may be further away from the resulting track than a certain limit. Points can be deleted but will never be moved or inserted.

The cost function can be easily modified. In the standard setting it consists of the number of remaining points plus (normalized) squared distances to removed points, plus a penalty for large-angle kinks at the remaining trackpoints.

The number of possible tracks is huge and not all of them could be explored individually to find the optimum. Instead the a modified version of the Dijkstra algorithm is utilized. It successively finds the shortest route to each point and then only has to trace back to all possible predecessor points instead of routes. Therefore the best trace is found and the task is performed in a reasonable amount of time. However if the track is very long and offers many possibilities, it may still take several seconds to compute.

Points will never become separated beyond a certain limit (200m by default), since josm would not show lines between further separated points with default settings. There is an additional speed-dependent distance-limit that keeps points more evenly spaced.


#!/usr/bin/env python
# -*- coding: utf8 -*-

gpx_reduce v1.8: removes points from gpx-files to reduce filesize and
tries to keep introduced distortions to the track at a minimum.
Copyright (C) 2011,2012,2013,2015,2016,2017 travelling_salesman on OpenStreetMap

changelog: v1.2: clarity refractoring + speedup for identical points
           v1.3: new track weighting functions, progress display
           v1.4: new track weighting function, restructuring for memory saving
           v1.5: algorithm speedup by roughly a factor of 2 by eliminating some cases.
           v1.6: presets for train etc.
           v1.7: introduced weighting function for elevation errors
           v1.8: speed-dependent distance limit

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <>.

import datetime
import sys
import time
import pylab as pl
import scipy as sc
import numpy.linalg as la
from math import *
from lxml import etree
from optparse import OptionParser

parser = OptionParser('usage: %prog [options] input-file.gpx')
parser.add_option('-v', '--verbose', action='store', type='int',
    dest='verbose', default=1, help='verbose=[0,1]')
parser.add_option('-p', '--plot', action='store_true', dest='plot',
    default=False, help='Show a plot of the result at the end.')
parser.add_option('-d', '--dist', action='store', type='float', dest='max_dist',
    default=0.5, help='Maximum distance of line from original points in meters')
parser.add_option('-o', '--out', action='store', type='string',
    dest='ofname', default=None, help='Output file name')
parser.add_option('-m', '--maxsep', action='store', type='float', dest='max_sep',
    default=200.0, help='Absolute maximum separation of points. No points will be deleted where the resulting distance would become greater than maxsep. Standard JOSM settings will not display points spaced more than 200m. -1 means no limit.')
parser.add_option('-n', '--maxsep0', action='store', type='float', dest='max_sep0',
    default=4.0, help='Maximum separation of points at zero speed.')
parser.add_option('-t', '--maxseptime', action='store', type='float', dest='max_sep_time',
    default=3.0, help='Maximum time separation of points, which will be added to maxsep0. Maximum allowed point separation will be min(m, n + t*v) where v is the speed in m/s.')
parser.add_option('-e', '--ele-weight', action='store', type='float',
    dest='ele_weight', default=0.0,
    help='Weighting of elevation errors vs. horizontal errors. Default is 0.')
parser.add_option('-b', '--bend', action='store', type='float', dest='bend',
    default=1.0, help='Penalty value for large bending angles at each trackpoint. Larger values (1 or more) make the track smoother.')
parser.add_option('-w', '--weighting', action='store', type='string',
    dest='weighting', default='exp',
    help='''Weighting function to be minimized for track reduction:
pnum (number of points),
sqrdistsum (number of points plus sum of squared distances to leftout points),
sqrdistmax (number of points plus sum of squared distances to each maximally separated leftout point per new line segment),
sqrlength (number of points plus sum of squared new line segment lengths normalized by maxsep),
mix (number of points plus sum of squared distances to each maximally separated leftout point per new line segment weighted with corresponding segment length),
exp (number of points plus sum of squared distances to leftout points with exponential weighting of 1/2, 1/4, 1/8... from furthest to closest point). exp=standard''')
(options, args) = parser.parse_args()

if len(args) < 1:

# use the WGS-84 ellipsoid
rE = 6356752.314245 # earth's radius
a = 6378137.0
b = 6356752.314245179

timeformat = '%Y-%m-%dT%H:%M:%SZ'

def distance(p1_, pm_, p2_, ele_weight=1.0):
    # returns distance of pm from line between p1 and p2

    p1, pm, p2 = sc.array(p1_), sc.array(pm_), sc.array(p2_)
    h1, hm, h2 = la.norm(p1), la.norm(pm), la.norm(p2)
    if ele_weight != 1.0 and min(h1, hm, h2) > 0.0:
        hmean = (h1 + hm + h2) / 3.0
        p1 *= (ele_weight + (1.0 - ele_weight) * hmean / h1)
        pm *= (ele_weight + (1.0 - ele_weight) * hmean / hm)
        p2 *= (ele_weight + (1.0 - ele_weight) * hmean / h2)
    line = p2 - p1
    linel = la.norm(line)
    vm = pm - p1
    if linel == 0.0:
        return la.norm(vm)
    linem = line / linel
    position =, linem) / linel
    if position < 0.0:
        return la.norm(vm)
    elif position > 1.0:
        return la.norm(pm - p2)
        return la.norm(vm - line * position)

def rotate(x, y, phi):
    return x*cos(phi) - y*sin(phi), x*sin(phi) + y*cos(phi)

def project_to_meters(lat, lon, latm, lonm):
    # azimuthal map projection centered at average track coordinate
    lon -= lonm
    xyz = latlonele_to_xyz(lat, lon, 0.0)
    zy = rotate(xyz[2], xyz[1], radians(90 - latm))
    lat2 = atan2(zy[0], la.norm([zy[1], xyz[0]]))
    lon2 = atan2(xyz[0], -zy[1])
    x_meters = rE * sin(lon2) * (pi / 2.0 - lat2)
    y_meters = -rE * cos(lon2) * (pi / 2.0 - lat2)
    return x_meters, y_meters

def latlonele_to_xyz(lat, lon, ele):
    s = sin(radians(lat))
    c = cos(radians(lat))
    r = ele + a * b / la.norm([s*a, c*b])
    lon = radians(lon)
    return r * c * sin(lon), r * c * (-cos(lon)), r * s

def xyz_to_latlonele(x, y, z):
    r = la.norm([x, y, z])
    if (r == 0):
        return 0.0, 0.0, 0.0
    lat = degrees(atan2(z, la.norm([x, y])))
    lon = degrees(atan2(x, -y))
    ele = r * (1.0 - a * b / la.norm([a*z, b*x, b*y]))
    return lat, lon, ele

def reduced_track_indices(coordinate_list, timesteps=None):
    # returns a list of indices of trackpoints that constitute the reduced track
    # takes a list of kartesian coordinate tuples
    m = len(coordinate_list)
    if (m == 0): return []
    if timesteps != None and len(timesteps) != len(coordinate_list):
        timesteps = None
    # number of dimensions
    d = len(coordinate_list[0])
    # remove identical entries (can speed up algorithm considerably)
    original_indices = [0]
    points = [{'p': coordinate_list[0], 'weight':1}]
    if timesteps != None: points[0]['t'] = timesteps[0]
    for i in range(1, m):
        if False in [coordinate_list[i-1][j] == coordinate_list[i][j] for j in range(d)]:
            points.append({'p': coordinate_list[i], 'weight':1})
            if timesteps != None: points[-1]['t'] = timesteps[i]
            points[-1]['weight'] += 1
    n = len(points)
    # progress printing initialisations
    progress_printed = False
    progress = None
    tprint = time.time()
    # execute Dijkstra-like algorithm on points
    points[0]['cost'] = 1.0
    points[0]['prev'] = -1
    for i2 in range(1, n):
        penalties = {}
        imin = None
        costmin = float('inf')
        for i1 in reversed(range(i2)):
            p1 = sc.array(points[i1]['p'])
            p2 = sc.array(points[i2]['p'])
            seglength = la.norm(p2 - p1)
            # estimate speed between p1 and p2
            if timesteps != None:
                dt = (points[i2]['t'] - points[i1]['t']).total_seconds()
                v = seglength / max(0.1, dt)
                v = seglength / float(i2 - i1) # assume 1s time spacing
            max_sep = options.max_sep0 + v * options.max_sep_time
            if options.max_dist >= 0:
                max_sep = min(max_sep, options.max_sep)
            if (seglength >= max_sep and i1 != i2 - 1):
                # point separation is too far
                # but always accept direct predecessor i1 = i2 - 1
                if (seglength >= max_sep + options.max_dist):
                    # no chance to find a valid earlier predecessor point
            if points[i1]['cost'] + 1.0 > costmin:
                # the possible predecessor i1 is already too bad.
            i1_i2_segment_valid = True
            lower_i1_possible = True
            distance_squaremax = 0.0
            distance_squaresum = 0.0
            distances_squared = []
            # iterate all medium points between i1 and i2
            for im in range(i1+1, i2):
                pm = sc.array(points[im]['p'])
                d = distance(p1, pm, p2, options.ele_weight)
                if (d <= options.max_dist):
                    d_sq = (d / options.max_dist) ** 2
                    distance_squaremax = max(distance_squaremax, d_sq)
                    distance_squaresum += points[im]['weight'] * d_sq
                    i1_i2_segment_valid = False
                    # check if connection to any further point i1 is impossible
                    d1 = - p2, p1 - p2)
                    d2 = - p2, pm - p2)
                    dd = options.max_dist ** 2
                    d1d2 = - p2, pm - p2)
                    # formula from cosines of point separation angle and cone-opening angles around points
                    if (d1 > dd and d2 > dd and (d1d2 + dd)**2 < (d2 - dd) * (d1 - dd)):
                        lower_i1_possible = False
            if (lower_i1_possible == False):
            if i1_i2_segment_valid:
                if options.weighting == 'sqrdistmax':
                    penalties[i1] = distance_squaremax
                elif options.weighting == 'sqrdistsum':
                    penalties[i1] = distance_squaresum
                elif options.weighting == 'sqrlength':
                    penalties[i1] = (seglength / max_sep) ** 2
                elif options.weighting == 'mix':
                    penalties[i1] = (distance_squaremax * (1.0 + seglength / max_sep))
                elif options.weighting == 'exp':
                    penalties[i1] = 0.5 * sum([0.5**i * d for i, d in
                        enumerate(sorted(distances_squared, reverse=True))])
                    penalties[i1] = 0.0
                # add a penalty for kinks
                if options.bend > 0.:
                    if points[i1]['prev'] != -1:
                        p0 = sc.array(points[points[i1]['prev']]['p'])
                        v0 = p1 - p0
                        v1 = p2 - p1
                        if la.norm(v0) > 0. and la.norm(v1) > 0.:
                            v0 /= la.norm(v0)
                            v1 /= la.norm(v1)
                            kink = (1.0 -, v1)) / 2.0
                            penalties[i1] += options.bend * kink
        # find best predecessor
        imin = None
        costmin = float('inf')
        for prev, penalty in penalties.iteritems():
            # cost function is sum of points used (1.0) plus penalties
            cost = points[prev]['cost'] + 1.0 + penalty
            if cost < costmin:
                imin = prev
                costmin = cost
        points[i2]['cost'] = costmin
        points[i2]['prev'] = imin
        # print progess
        if options.verbose == 1 and (100 * i2) / n > progress and time.time() >= tprint + 1:
            tprint = time.time()
            progress = (100 * i2) / n
            print '\r', progress, '%',
            progress_printed = True
    if progress_printed:
        print '\r',
    # trace route backwards to collect final points
    final_pnums = []
    i = n-1
    while i >= 0:
        final_pnums = [i] + final_pnums
        i = points[i]['prev']
    return [original_indices[i] for i in final_pnums]

############################## main function #################################
for fname in args:
    # initialisations
    tracksegs_old = []
    tracksegs_new = []
    sumx, sumy, sumz = 0.0, 0.0, 0.0
    # import xml data from files
    print 'opening file', fname
    infile = open(fname)
    tree = etree.parse(infile)
    gpx = tree.getroot()
    if gpx.nsmap.has_key(None):
        nsmap = '{' + gpx.nsmap[None] + '}'
        nsmap = ''
    # extract data from xml
    for trkseg in gpx.findall('.//' + nsmap + 'trkseg'):
        trkpts = trkseg.findall(nsmap + 'trkpt')
        n = len(trkpts)
        # extract coordinate values
        lats = [float(trkpt.get('lat')) for trkpt in trkpts]
        lons = [float(trkpt.get('lon')) for trkpt in trkpts]
        eles = [float(trkpt.find(nsmap + 'ele').text) for trkpt in trkpts]
            times = [datetime.datetime.strptime(trkpt.find(nsmap + 'time'
                                       ).text, timeformat) for trkpt in trkpts]
        except Exception as e:
            print e
            times = None
        # save original trackseg for plotting
        if options.plot:
            tracksegs_old.append([[lats[i], lons[i], eles[i]] for i in range(n)])
        # calculate projected points to work on
        coords = []
        for i in range(n):
            x, y, z = latlonele_to_xyz(lats[i], lons[i], eles[i])
            coords.append((x, y, z))
            sumx += x
            sumy += y
            sumz += z
        # execute the reduction algorithm
        final_pnums = reduced_track_indices(coords, times)
        n_new = len(final_pnums)
        print 'number of points:', n, '-', n - n_new, '=', n_new
        # delete certain points from original data
        delete_pnums = [i for i in range(n) if i not in final_pnums]
        for i in reversed(delete_pnums):
            del trkseg[trkseg.index(trkpts[i])]
        # save reduced trackseg for plotting
        if options.plot:
                float(trkpt.get('lon')), float(trkpt.find(nsmap + 'ele').text)]
                for trkpt in trkseg.findall(nsmap + 'trkpt')])
    # export data to file
    if options.ofname != None:
        ofname = options.ofname
    elif fname.endswith('.gpx'):
        ofname = fname[:-4] + '_reduced.gpx'
        ofname = fname + '_reduced.gpx'
    outfile = open(ofname, 'w')
    outfile.write(etree.tostring(tree, xml_declaration=True,
        pretty_print=True, encoding='utf-8'))
    print 'modified copy written to', ofname
    # plot result to screen
    if options.plot:
        latm, lonm, elesum = xyz_to_latlonele(sumx, sumy, sumz)

        for trkseg in tracksegs_old:
            y_old = []
            x_old = []
            for trkpt in trkseg:
                xy = project_to_meters(trkpt[0], trkpt[1], latm, lonm)
            pl.plot(x_old, y_old, 'r.-')
        for trkseg in tracksegs_new:
            y_new = []
            x_new = []
            for trkpt in trkseg:
                xy = project_to_meters(trkpt[0], trkpt[1], latm, lonm)
            pl.plot(x_new, y_new, 'b.-')
        pl.xlabel('x [m]')
        pl.ylabel('y [m]')