定位局部最大值的算法

Question

我的数据总是看起来像这样:

alt text http://michaelfogleman.com/static/images/chart.png

我需要一个算法来定位这三个峰。

x 轴实际上是相机位置，y 轴是该位置图像焦点/对比度的度量。三个不同距离的特征可以对焦，我需要确定这三个点的 x 值。

中间的驼峰总是更难辨认出来，即使对于人类也是如此。

我有一个大部分有效的自制算法，但我想知道是否有一种标准方法可以从一个可能有一点噪音的函数中获取局部最大值。不过，峰值很容易克服噪音。

此外，作为相机数据，不需要扫描整个范围的算法可能很有用。

编辑:发布我最终使用的 Python 代码。它使用我的原始代码，在给定搜索阈值的情况下找到最大值，并进行二分搜索以找到一个阈值，从而产生所需数量的最大值。

编辑:以下代码中包含示例数据。新代码是 O(n) 而不是 O(n^2)。

def find_n_maxima(data, count):
    low = 0
    high = max(data) - min(data)
    for iteration in xrange(100): # max iterations
        mid = low + (high - low) / 2.0
        maxima = find_maxima(data, mid)
        if len(maxima) == count:
            return maxima
        elif len(maxima) < count: # threshold too high
            high = mid
        else: # threshold too low
            low = mid
    return None # failed

def find_maxima(data, threshold):
    def search(data, threshold, index, forward):
        max_index = index
        max_value = data[index]
        if forward:
            path = xrange(index + 1, len(data))
        else:
            path = xrange(index - 1, -1, -1)
        for i in path:
            if data[i] > max_value:
                max_index = i
                max_value = data[i]
            elif max_value - data[i] > threshold:
                break
        return max_index, i
    # forward pass
    forward = set()
    index = 0
    while index < len(data) - 1:
        maximum, index = search(data, threshold, index, True)
        forward.add(maximum)
        index += 1
    # reverse pass
    reverse = set()
    index = len(data) - 1
    while index > 0:
        maximum, index = search(data, threshold, index, False)
        reverse.add(maximum)
        index -= 1
    return sorted(forward & reverse)

data = [
    1263.900, 1271.968, 1276.151, 1282.254, 1287.156, 1296.513,
    1298.799, 1304.725, 1309.996, 1314.484, 1321.759, 1323.988,
    1331.923, 1336.100, 1340.007, 1340.548, 1343.124, 1353.717,
    1359.175, 1364.638, 1364.548, 1357.525, 1362.012, 1367.190,
    1367.852, 1376.275, 1374.726, 1374.260, 1392.284, 1382.035,
    1399.418, 1401.785, 1400.353, 1418.418, 1420.401, 1423.711,
    1425.214, 1436.231, 1431.356, 1435.665, 1445.239, 1438.701,
    1441.988, 1448.930, 1455.066, 1455.047, 1456.652, 1456.771,
    1459.191, 1473.207, 1465.788, 1488.785, 1491.422, 1492.827,
    1498.112, 1498.855, 1505.426, 1514.587, 1512.174, 1525.244,
    1532.235, 1543.360, 1543.985, 1548.323, 1552.478, 1576.477,
    1589.333, 1610.769, 1623.852, 1634.618, 1662.585, 1704.127,
    1758.718, 1807.490, 1852.097, 1969.540, 2243.820, 2354.224,
    2881.420, 2818.216, 2552.177, 2355.270, 2033.465, 1965.328,
    1824.853, 1831.997, 1779.384, 1764.789, 1704.507, 1683.615,
    1652.712, 1646.422, 1620.593, 1620.235, 1613.024, 1607.675,
    1604.015, 1574.567, 1587.718, 1584.822, 1588.432, 1593.377,
    1590.533, 1601.445, 1667.327, 1739.034, 1915.442, 2128.835,
    2147.193, 1970.836, 1755.509, 1653.258, 1613.284, 1558.576,
    1552.720, 1541.606, 1516.091, 1503.747, 1488.797, 1492.021,
    1466.720, 1457.120, 1462.485, 1451.347, 1453.224, 1440.477,
    1438.634, 1444.571, 1428.962, 1431.486, 1421.721, 1421.367,
    1403.461, 1415.482, 1405.318, 1399.041, 1399.306, 1390.486,
    1396.746, 1386.178, 1376.941, 1369.880, 1359.294, 1358.123,
    1353.398, 1345.121, 1338.808, 1330.982, 1324.264, 1322.147,
    1321.098, 1313.729, 1310.168, 1304.218, 1293.445, 1285.296,
    1281.882, 1280.444, 1274.795, 1271.765, 1266.857, 1260.161,
    1254.380, 1247.886, 1250.585, 1246.901, 1245.061, 1238.658,
    1235.497, 1231.393, 1226.241, 1223.136, 1218.232, 1219.658,
    1222.149, 1216.385, 1214.313, 1211.167, 1208.203, 1206.178,
    1206.139, 1202.020, 1205.854, 1206.720, 1204.005, 1205.308,
    1199.405, 1198.023, 1196.419, 1194.532, 1194.543, 1193.482,
    1197.279, 1196.998, 1194.489, 1189.537, 1188.338, 1184.860,
    1184.633, 1184.930, 1182.631, 1187.617, 1179.873, 1171.960,
    1170.831, 1167.442, 1177.138, 1166.485, 1164.465, 1161.374,
    1167.185, 1174.334, 1186.339, 1202.136, 1234.999, 1283.328,
    1347.111, 1679.050, 1927.083, 1860.902, 1602.791, 1350.454,
    1274.236, 1207.727, 1169.078, 1138.025, 1117.319, 1109.169,
    1080.018, 1073.837, 1059.876, 1050.209, 1050.859, 1035.003,
    1029.214, 1024.602, 1017.932, 1006.911, 1010.722, 1005.582,
    1000.332, 998.0721, 992.7311, 992.6507, 981.0430, 969.9936,
    972.8696, 967.9463, 970.1519, 957.1309, 959.6917, 958.0536,
    954.6357, 954.9951, 947.8299, 953.3991, 949.2725, 948.9012,
    939.8549, 940.1641, 942.9881, 938.4526, 937.9550, 929.6279,
    935.5402, 921.5773, 933.6365, 918.7065, 922.5849, 939.6088,
    911.3251, 923.7205, 924.8227, 911.3192, 936.7066, 915.2046,
    919.0274, 915.0533, 910.9783, 913.6773, 916.6287, 907.9267,
    908.0421, 908.7398, 911.8401, 914.5696, 912.0115, 919.4418,
    917.0436, 920.5495, 917.6138, 907.5037, 908.5145, 919.5846,
    917.6047, 926.8447, 910.6347, 912.8305, 907.7085, 911.6889,
]

for n in xrange(1, 6):
    print 'Looking for %d maxima:' % n
    indexes = find_n_maxima(data, n)
    print indexes
    print ', '.join(str(data[i]) for i in indexes)
    print

输出:

Looking for 1 maxima:
[78]
2881.42

Looking for 2 maxima:
[78, 218]
2881.42, 1927.083

Looking for 3 maxima:
[78, 108, 218]
2881.42, 2147.193, 1927.083

Looking for 4 maxima:
[78, 108, 218, 274]
2881.42, 2147.193, 1927.083, 936.7066

Looking for 5 maxima:
[78, 108, 218, 269, 274]
2881.42, 2147.193, 1927.083, 939.6088, 936.7066

Answer 1

局部最大值将是任何 x 点，它的 y 值高于其左右邻居中的任何一个。为了消除噪音，您可以设置某种容差阈值(例如 x 点的 y 值必须高于其邻居的 n)。

为避免扫描每个点，您可以使用相同的方法，但一次扫描 5 或 10 个点，以大致了解最大值所在的位置。然后返回这些区域进行更详细的扫描。