Algorithm to locate local maxima

Question

I have data that always looks something like this:

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

I need an algorithm to locate the three peaks.

The x-axis is actually a camera position and the y-axis is a measure of image focus/contrast at that position. There are features at three different distances that can be in focus and I need to determine the x-values for these three points.

The middle hump is always a little harder to pick out, even for a human.

I have a homemade algorithm that mostly works, but I'm wondering if there's a standard way to grab local maxima from a function that can have a little noise in it. The peaks overcome the noise easily though.

Also, being camera data, an algorithm that doesn't require scanning the full range could be useful.

Edit: Posting the Python code that I ended up using. It uses my original code that finds maxima given a search threshold and does a binary search to find a threshold that results in the desired number of maxima.

Edit: Sample data included in code below. New code is O(n) instead of 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

Output:

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

The local maxima would be any x point which has a higher y value than either of its left and right neighbors. To eliminate noise, you could put in some kind of tolerance threshold (ex. x point must have higher y value than n of its neighbors).

To avoid scanning every point, you could use the same approach but go 5 or 10 points at a time to get a rough sense of where the maximum lie. Then come back to those areas for a more detailed scan.

Answer 2

Couldn't you move along the graph, regularly calculating the derivative and if it switches from positive to negative you've found a peak?