何為抽稀
在處理向量化資料時，記錄中往往會有很多重複資料，對進一步資料處理帶來諸多不便。多餘的資料一方面浪費了較多的儲存空間，另一方面造成所要表達的圖形不光滑或不符合標準。因此要通過某種規則，在保證向量曲線形狀不變的情況下， 最大限度地減少資料點個數，這個過程稱為抽稀。
通俗的講就是對曲線進行取樣簡化，即在曲線上取有限個點，將其變為折線，並且能夠在一定程度保持原有形狀。比較常用的兩種抽稀演算法是：道格拉斯-普克(Douglas-Peuker)演算法和垂距限值法。
道格拉斯-普克(Douglas-Peuker)演算法
Douglas-Peuker演算法(DP演算法)過程如下:
- 1、連線曲線首尾兩點A、B；
- 2、依次計算曲線上所有點到A、B兩點所在曲線的距離；
- 3、計算最大距離D，如果D小於閾值threshold,則去掉曲線上出A、B外的所有點；如果D大於閾值threshold,則把曲線以最大距離分割成兩段；
- 4、對所有曲線分段重複1-3步驟，知道所有D均小於閾值。即完成抽稀。
這種演算法的抽稀精度與閾值有很大關係，閾值越大，簡化程度越大，點減少的越多；反之簡化程度越低，點保留的越多，形狀也越趨於原曲線。
下面是Python程式碼實現:
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# -*- coding: utf-8 -*- """ ------------------------------------------------- File Name： DouglasPeuker Description : 道格拉斯-普克抽稀演算法 Author : J_hao date： 2017/8/16 ------------------------------------------------- Change Activity: 2017/8/16: 道格拉斯-普克抽稀演算法 ------------------------------------------------- """ from __future__ import division from math import sqrt, pow __author__ = 'J_hao' THRESHOLD = 0.0001 # 閾值 def point2LineDistance(point_a, point_b, point_c): """ 計算點a到點b c所在直線的距離 :param point_a: :param point_b: :param point_c: :return: """ # 首先計算b c 所在直線的斜率和截距 if point_b[0] == point_c[0]: return 9999999 slope = (point_b[1] - point_c[1]) / (point_b[0] - point_c[0]) intercept = point_b[1] - slope * point_b[0] # 計算點a到b c所在直線的距離 distance = abs(slope * point_a[0] - point_a[1] + intercept) / sqrt(1 + pow(slope, 2)) return distance class DouglasPeuker(object): def __init__(self): self.threshold = THRESHOLD self.qualify_list = list() self.disqualify_list = list() def diluting(self, point_list): """ 抽稀 :param point_list:二維點列表 :return: """ if len(point_list) < 3: self.qualify_list.extend(point_list[::-1]) else: # 找到與收尾兩點連線距離最大的點 max_distance_index, max_distance = 0, 0 for index, point in enumerate(point_list): if index in [0, len(point_list) - 1]: continue distance = point2LineDistance(point, point_list[0], point_list[-1]) if distance > max_distance: max_distance_index = index max_distance = distance # 若最大距離小於閾值，則去掉所有中間點。 反之，則將曲線按最大距離點分割 if max_distance < self.threshold: self.qualify_list.append(point_list[-1]) self.qualify_list.append(point_list[0]) else: # 將曲線按最大距離的點分割成兩段 sequence_a = point_list[:max_distance_index] sequence_b = point_list[max_distance_index:] for sequence in [sequence_a, sequence_b]: if len(sequence) < 3 and sequence == sequence_b: self.qualify_list.extend(sequence[::-1]) else: self.disqualify_list.append(sequence) def main(self, point_list): self.diluting(point_list) while len(self.disqualify_list) > 0: self.diluting(self.disqualify_list.pop()) print self.qualify_list print len(self.qualify_list) if __name__ == '__main__': d = DouglasPeuker() d.main([[104.066228, 30.644527], [104.066279, 30.643528], [104.066296, 30.642528], [104.066314, 30.641529], [104.066332, 30.640529], [104.066383, 30.639530], [104.066400, 30.638530], [104.066451, 30.637531], [104.066468, 30.636532], [104.066518, 30.635533], [104.066535, 30.634533], [104.066586, 30.633534], [104.066636, 30.632536], [104.066686, 30.631537], [104.066735, 30.630538], [104.066785, 30.629539], [104.066802, 30.628539], [104.066820, 30.627540], [104.066871, 30.626541], [104.066888, 30.625541], [104.066906, 30.624541], [104.066924, 30.623541], [104.066942, 30.622542], [104.066960, 30.621542], [104.067011, 30.620543], [104.066122, 30.620086], [104.065124, 30.620021], [104.064124, 30.620022], [104.063124, 30.619990], [104.062125, 30.619958], [104.061125, 30.619926], [104.060126, 30.619894], [104.059126, 30.619895], [104.058127, 30.619928], [104.057518, 30.620722], [104.057625, 30.621716], [104.057735, 30.622710], [104.057878, 30.623700], [104.057984, 30.624694], [104.058094, 30.625688], [104.058204, 30.626682], [104.058315, 30.627676], [104.058425, 30.628670], [104.058502, 30.629667], [104.058518, 30.630667], [104.058503, 30.631667], [104.058521, 30.632666], [104.057664, 30.633182], [104.056664, 30.633174], [104.055664, 30.633166], [104.054672, 30.633289], [104.053758, 30.633694], [104.052852, 30.634118], [104.052623, 30.635091], [104.053145, 30.635945], [104.053675, 30.636793], [104.054200, 30.637643], [104.054756, 30.638475], [104.055295, 30.639317], [104.055843, 30.640153], [104.056387, 30.640993], [104.056933, 30.641830], [104.057478, 30.642669], [104.058023, 30.643507], [104.058595, 30.644327], [104.059152, 30.645158], [104.059663, 30.646018], [104.060171, 30.646879], [104.061170, 30.646855], [104.062168, 30.646781], [104.063167, 30.646823], [104.064167, 30.646814], [104.065163, 30.646725], [104.066157, 30.646618], [104.066231, 30.645620], [104.066247, 30.644621], ]) |
垂距限值法
垂距限值法其實和DP演算法原理一樣，但是垂距限值不是從整體角度考慮，而是依次掃描每一個點，檢查是否符合要求。
演算法過程如下:
- 1、以第二個點開始，計算第二個點到前一個點和後一個點所在直線的距離d；
- 2、如果d大於閾值，則保留第二個點，計算第三個點到第二個點和第四個點所在直線的距離d;若d小於閾值則捨棄第二個點，計算第三個點到第一個點和第四個點所在直線的距離d;
- 3、依次類推，直線曲線上倒數第二個點。
下面是Python程式碼實現：
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# -*- coding: utf-8 -*- """ ------------------------------------------------- File Name： LimitVerticalDistance Description : 垂距限值抽稀演算法 Author : J_hao date： 2017/8/17 ------------------------------------------------- Change Activity: 2017/8/17: ------------------------------------------------- """ from __future__ import division from math import sqrt, pow __author__ = 'J_hao' THRESHOLD = 0.0001 # 閾值 def point2LineDistance(point_a, point_b, point_c): """ 計算點a到點b c所在直線的距離 :param point_a: :param point_b: :param point_c: :return: """ # 首先計算b c 所在直線的斜率和截距 if point_b[0] == point_c[0]: return 9999999 slope = (point_b[1] - point_c[1]) / (point_b[0] - point_c[0]) intercept = point_b[1] - slope * point_b[0] # 計算點a到b c所在直線的距離 distance = abs(slope * point_a[0] - point_a[1] + intercept) / sqrt(1 + pow(slope, 2)) return distance class LimitVerticalDistance(object): def __init__(self): self.threshold = THRESHOLD self.qualify_list = list() def diluting(self, point_list): """ 抽稀 :param point_list:二維點列表 :return: """ self.qualify_list.append(point_list[0]) check_index = 1 while check_index < len(point_list) - 1: distance = point2LineDistance(point_list[check_index], self.qualify_list[-1], point_list[check_index + 1]) if distance < self.threshold: check_index += 1 else: self.qualify_list.append(point_list[check_index]) check_index += 1 return self.qualify_list if __name__ == '__main__': l = LimitVerticalDistance() diluting = l.diluting([[104.066228, 30.644527], [104.066279, 30.643528], [104.066296, 30.642528], [104.066314, 30.641529], [104.066332, 30.640529], [104.066383, 30.639530], [104.066400, 30.638530], [104.066451, 30.637531], [104.066468, 30.636532], [104.066518, 30.635533], [104.066535, 30.634533], [104.066586, 30.633534], [104.066636, 30.632536], [104.066686, 30.631537], [104.066735, 30.630538], [104.066785, 30.629539], [104.066802, 30.628539], [104.066820, 30.627540], [104.066871, 30.626541], [104.066888, 30.625541], [104.066906, 30.624541], [104.066924, 30.623541], [104.066942, 30.622542], [104.066960, 30.621542], [104.067011, 30.620543], [104.066122, 30.620086], [104.065124, 30.620021], [104.064124, 30.620022], [104.063124, 30.619990], [104.062125, 30.619958], [104.061125, 30.619926], [104.060126, 30.619894], [104.059126, 30.619895], [104.058127, 30.619928], [104.057518, 30.620722], [104.057625, 30.621716], [104.057735, 30.622710], [104.057878, 30.623700], [104.057984, 30.624694], [104.058094, 30.625688], [104.058204, 30.626682], [104.058315, 30.627676], [104.058425, 30.628670], [104.058502, 30.629667], [104.058518, 30.630667], [104.058503, 30.631667], [104.058521, 30.632666], [104.057664, 30.633182], [104.056664, 30.633174], [104.055664, 30.633166], [104.054672, 30.633289], [104.053758, 30.633694], [104.052852, 30.634118], [104.052623, 30.635091], [104.053145, 30.635945], [104.053675, 30.636793], [104.054200, 30.637643], [104.054756, 30.638475], [104.055295, 30.639317], [104.055843, 30.640153], [104.056387, 30.640993], [104.056933, 30.641830], [104.057478, 30.642669], [104.058023, 30.643507], [104.058595, 30.644327], [104.059152, 30.645158], [104.059663, 30.646018], [104.060171, 30.646879], [104.061170, 30.646855], [104.062168, 30.646781], [104.063167, 30.646823], [104.064167, 30.646814], [104.065163, 30.646725], [104.066157, 30.646618], [104.066231, 30.645620], [104.066247, 30.644621], ]) print len(diluting) print(diluting) |
最後
其實DP演算法和垂距限值法原理一樣，DP演算法是從整體上考慮一條完整的曲線，實現時較垂距限值法複雜，但垂距限值法可能會在某些情況下導致區域性最優。另外在實際使用中發現採用點到另外兩點所在直線距離的方法來判斷偏離，在曲線弧度比較大的情況下比較準確。如果在曲線弧度比較小，彎曲程度不明顯時，這種方法抽稀效果不是很理想，建議使用三點所圍成的三角形面積作為判斷標準。下面是抽稀效果:
