正交匹配追踪(Orthogonal Match Pursuit, OMP)算法

介绍

正交匹配追踪(OMP)算法由Y.C. Pati等人于1993年提出，是MP算法的一个变种。OMP算法主要改善求解时的收敛速度。算法与MP不同的是在迭代过程中求解稀疏系数时，选取正交于此前已选取基的基来计算系数。

流程

待分解的信号记作$y$，第$i$个basis记作$b_i$，稀疏系数向量记作$\Phi$，比较MP与OMP如下：

$$
\begin{aligned}
R^{(0)} &= y \\
m(n) &= \underset{i}{\arg\max} |\langle R^{(n)},b_i\rangle| \\
\Phi_{m(n)} &= \langle R^{(n)},b_{m(n)}\rangle/|b_{m(n)}|^2 \\
R^{(n+1)} &= R^{(n)}-b_{m(n)}\Phi_{m(n)}
\end{aligned}
$$

$$
\begin{aligned}
R^{(0)} &= y \\
m(n) &= \underset{i}{\arg\max} |\langle R^{(n)},b_i\rangle| \\
\Phi^{(n)} &= B^{(n)+} y \\
R^{(n+1)} &= y-B^{(n)}\Phi^{(n)}
\end{aligned}
$$

式中¹：

$$
\begin{aligned}
B^+ &= (B^\mathrm{T}B)^{-1}B^\mathrm{T} \\
\Phi^{(n)} &= [\Phi_{m(1)} \cdots \Phi_{m(n)}] \\
B^{(n)} &= [b_{m(1)}\cdots b_{m(n)}]
\end{aligned}
$$

$A^+$代表矩阵$A$的伪逆(摩尔彭若斯广义逆)，可以用于计算向量正交于矩阵空间的投影 ↩︎

MP和OMP的差异只有第3步(这里认为basis已经归一化)：
$$
\begin{split}
\text{MP: } & \Phi_{(n)} &= \langle R^{(n)},b_{m(n)}\rangle \\
\text{OMP: } & \Phi^{(n)} &= B^{(n)+} y
\end{split}
$$
OMP在迭代步骤中会更新已经计算过的系数，但MP不会。

算法终止条件可以选择$n$次迭代或$R^{(n)}<\epsilon$。

实现

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% inputs:
% y: signal
% B: basis matrix [b1 b2 ... bn]
% iter: number of iterations
% output:
% coef: coefficient vector \Phi
function coef = omp(y, B, iter)
    R = y;
    coef = zeros(size(B,2), 1);
    S = [];
    indices = zeros(iter, 1);
    for i = 1:iter
        [~, m] = max(abs(B'*R));
        S(:, i) = normalize(B(:, m), 'norm');
        indices(i) = m;
        coefS = pinv(S) * y;
        R = y - S * coefS;
    end
    for i = 1:iter
        coef(indices(i)) = coefS(i) / norm(B(:, indices(i)));
    end
end

效果

仍然使用高斯包络的回波模型生成时域位置不同的基。执行OMP算法，可以得到实际超声信号在这组基下的稀疏表示。

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clear;

% signal - colum vector
data = [2232,3202,4089,2942,0,0,351,4089,4089,4089,1629,0,0,0,0,4072,4089,4089,4089,4089,0,0,0,0,0,4089,4089,4089,4089,222,0,0,0,4089,4089,4089,4089,4089,0,0,0,0,1397,4089,4089,4089,4089,49,0,0,0,0,3668,4089,4089,4089,4089,3108,0,0,0,0,0,4089,4089,4089,4089,4089,4089,0,0,0,0,0,0,4089,4089,4089,4089,4089,3502,0,0,0,0,1797,848,127,120,1605,4039,4054,4089,4089,4080,1658,814,1330,1920,1915,1214,382,397,975,1751,2744,3466,3746,3662,3218,2738,2335,2013,1733,1471,1236,1067,1009,1077,1249,1467,1680,1858,1990,2099,2212,2350,2513,2689,2850,2950,2955,2821,2572,2258,1974,1822,1859,2060,2326,2544,2612,2523,2327,2118,1952,1859,1816,1780,1728,1664,1609,1596,1637,1730,1864,2015,2167,2298,2400,2476,2523,2546,2544,2519,2472,2408,2330,2248,2166,2091,2025,1979,1942,1915,1905,1912,1932,1966,2009,2054,2092,2118,2133,2151,2171,2203,2243,2276,2285,2261,2209,2145,2095,2068,2072,2103,2143,2171,2174,2147,2103,2059,2035,2035,2056,2090,2125,2163,2186,2188,2177,2156,2137,2126,2123,2127,2137,2141,2151,2149,2147,2136,2124,2110,2093,2092,2136,2216,2262,2198,2017,1802,1711,1870,2257,2696,2955,2854,2367,1666,1073,884,1223,1984,2865,3483,3556,3043,2174,1323,848,918,1442,2147,2731,2986,2892,2564,2174,1873,1732,1747,1858,2011,2149,2248,2296,2296,2259,2200,2131,2074,2041,2039,2064,2106,2148,2177,2187,2172,2150,2128,2115,2108,2104,2107,2117,2135,2148,2154,2147,2134,2125,2121,2124,2126,2131,2138,2142,2141,2130,2118,2113,2110,2121,2139,2164,2173,2161,2135,2105,2078,2068,2086,2123,2165,2196,2206,2180,2141,2098,2075,2073,2088,2115,2145,2170,2177,2165,2143,2124,2110,2107,2111,2119,2125,2130,2134,2140,2139,2138,2137,2134,2127,2125,2124,2123,2125,2131,2134,2139,2145,2147,2143,2133,2113,2099,2094,2103,2126,2154,2179,2182,2168,2135,2101,2074,2074,2091,2122,2151,2174,2183,2178,2155,2127,2098,2081,2081,2097,2118,2145,2166,2175,2170,2154,2126,2102,2095,2101,2114,2134,2146,2150,2146,2136,2125,2121,2123,2127,2126,2123,2120,2126,2138,2151,2157,2158,2149,2128,2103,2083,2082,2095,2122,2158,2184,2194,2184,2147,2101,2063,2056,2077,2128,2176,2197,2189,2157,2117,2085,2075,2090,2118,2152,2176,2180,2159,2131,2101,2085,2090,2109,2132,2147,2143,2147,2176,2228,2244,2161,1984,1791,1739,1930,2326,2728,2915,2740,2214,1539,1043,980,1425,2221,3037,3513,3420,2795,1900,1138,821,1051,1673,2373,2860,2985,2796,2431,2064,1825,1742,1783,1894,2025,2139,2225,2281,2305,2285,2237,2166,2088,2031,2006,2018,2064,2120,2172,2202,2204,2188,2163,2132,2105,2086,2078,2089,2112,2134,2151,2161,2159,2148,2137,2123,2120,2120,2124,2125,2128,2131,2131,2130,2135,2140,2143,2141,2131,2114,2107,2105,2115,2132,2155,2169,2171,2160,2132,2099,2076,2072,2096,2135,2177,2195,2182,2152,2116,2091,2086,2096,2119,2143,2165,2173,2160,2138,2113,2100,2102,2111,2132,2143,2146,2145,2142,2133,2123,2120,2121,2128,2135,2134,2128,2118,2114,2112,2127,2152,2174,2178,2162,2128,2089,2069,2069,2086,2128,2166,2197,2204,2180,2141,2097,2064,2060,2086,2128,2163,2180,2178,2157,2129,2109,2100,2103,2119,2131,2144,2148,2148,2141,2131,2119,2110,2110,2121,2131,2142,2144,2139,2132,2123,2124,2133,2144,2152,2152,2132,2098,2072,2067,2091,2146,2197,2226,2224,2180,2113,2056,2025,2032,2075,2135,2187,2218,2210,2173,2126,2094,2083,2090,2107,2126,2143,2151,2150,2141,2125,2114,2117,2131,2146,2152,2141,2114,2100,2128,2192,2227,2172,2019,1856,1823,2006,2355,2682,2799,2581,2073,1484,1111,1161,1650,2406,3106,3430,3218,2546,1707,1062,883,1215,1869,2533,2938,2980,2719,2323,1960,1745,1693,1772,1919,2080,2213,2299,2339,2329,2279,2194,2096,2015,1975,1985,2040,2119,2189,2227,2228,2199,2155,2105,2074,2066,2083,2110,2145,2163,2167,2154,2138,2120,2113,2112,2121,2131,2139,2140,2133,2126,2124,2124,2131,2140,2148,2145,2131,2106,2086,2090,2114,2147,2178,2192,2179,2154,2113,2076,2062,2072,2107,2149,2185,2195,2178,2146,2112,2089,2088,2103,2123,2144,2155,2154,2137,2123,2117,2117,2125,2136,2144,2146,2140,2128,2117,2115,2116,2120,2130,2143,2152,2154,2141,2118,2092,2084,2101,2140,2181,2208,2198,2154,2096,2046,2029,2051,2107,2176,2225,2236,2208,2146,2083,2046,2047,2081,2129,2171,2185,2174,2152,2124,2111,2112,2118,2129,2133,2133,2131,2129,2124,2123,2127,2135,2138,2142,2144,2136,2131,2121,2111,2108,2116,2136,2164,2185,2180,2148,2099,2049,2026,2051,2116,2196,2258,2269,2225,2143,2057,1995,1986,2035,2118,2199,2248,2245,2200,2131,2080,2059,2064,2100,2137,2159,2161,2147,2128,2117,2116,2127,2140,2150,2151,2140,2120,2099,2099,2135,2204,2229,2166,2012,1870,1855,2041,2362,2649,2716,2485,2005,1486,1201,1315,1815,2509,3097,3304,3010,2339,1583,1071,1018,1414,2050,2636,2936,2894,2597,2208,1880,1710,1706,1820,1982,2140,2258,2330,2355,2324,2242,2135,2032,1972,1970,2020,2100,2176,2222,2226,2196,2155,2114,2092,2089,2096,2114,2129,2140,2143,2142,2138,2132,2128,2126,2130,2136,2138,2137,2132,2120,2112,2111,2124,2147,2168,2166,2145,2114,2082,2076,2089,2124,2167,2197,2202,2176,2128,2079,2050,2058,2091,2139,2183,2204,2193,2157,2114,2084,2074,2091,2124,2152,2164,2162,2147,2124,2110,2106,2115,2134,2144,2155,2148,2132,2117,2106,2110,2121,2134,2147,2153,2153,2140,2119,2092,2084,2093,2127,2172,2213,2221,2185,2118,2047,2006,2014,2072,2153,2227,2266,2250,2188,2107,2043,2013,2043,2102,2161,2198,2201,2174,2132,2102,2088,2096,2120,2144,2154,2150,2139,2126,2114,2113,2121,2132,2145,2151,2146,2137,2121,2107,2102,2110,2131,2159,2184,2189,2164,2115,2057,2018,2023,2080,2170,2257,2297,2275,2191,2079,1981,1943,1978,2074,2187,2273,2290,2246,2164,2075,2024,2021,2064,2127,2176,2197,2180,2147,2116,2099,2103,2122,2143,2155,2153,2136,2112,2092,2095,2142,2207,2233,2166,2019,1880,1876,2061,2368,2618,2650,2405,1953,1503,1294,1458,1950,2577,3055,3155,2819,2178,1517,1125,1168,1597,2199,2696,2905,2792,2474,2106,1830,1714,1753,1885,2041,2182,2280,2332,2334,2288,2203,2098,2014,1972,1989,2053,2131,2196,2226,2218,2180,2138,2104,2082,2080,2100,2121,2141,2152,2152,2139,2133,2128,2125,2126,2126,2132,2133,2133,2132,2126,2116,2121,2134,2150,2165,2164,2137,2102,2073,2064,2087,2133,2187,2218,2216,2179,2118,2054,2023,2035,2088,2160,2218,2233,2202,2146,2090,2061,2067,2097,2133,2166,2174,2162,2137,2112,2100,2103,2123,2143,2157,2155,2137,2117,2106,2110,2125,2135,2146,2152,2150,2145,2129,2106,2087,2084,2100,2141,2187,2215,2211,2163,2096,2034,2007,2028,2097,2182,2255,2274,2238,2157,2065,2004,2002,2053,2130,2197,2229,2212,2161,2107,2069,2070,2097,2136,2169,2180,2167,2133,2103,2087,2093,2120,2149,2165,2157,2143,2125,2110,2101,2099,2113,2140,2170,2194,2193,2156,2098,2039,2007,2038,2109,2204,2278,2296,2245,2144,2032,1953,1940,2009,2129,2248,2310,2294,2210,2100,2024,2000,2040,2106,2169,2201,2200,2171,2128,2095,2085,2095,2125,2149,2165,2159,2137,2111,2090,2100,2141,2201,2223,2161,2035,1913,1912,2085,2354,2563,2569,2328,1921,1542,1398,1593,2058,2605,2984,3009,2647,2053,1492,1215,1326,1761,2303,2707,2831,2673,2360,2034,1817,1748,1806,1936,2081,2210,2291,2323,2315,2259,2169,2073,1998,1979,2015,2082,2155,2210,2226,2208,2164,2113,2081,2073,2093,2121,2147,2157,2147,2133,2120,2114,2122,2138,2149,2150,2141,2127,2119,2110,2111,2116,2129,2151,2167,2172,2157,2125,2088,2063,2062,2099,2151,2206,2228,2209,2157,2089,2035,2024,2054,2119,2186,2227,2222,2179,2120,2071,2055,2074,2113,2153,2176,2175,2158,2134,2110,2099,2099,2120,2143,2158,2156,2139,2123,2113,2118,2123,2137,2140,2149,2150,2139,2123,2099,2084,2088,2114,2158,2205,2221,2203,2150,2072,2016,2001,2038,2117,2204,2268,2272,2219,2126,2041,1998,2015,2078,2155,2216,2233,2199,2143,2093,2063,2067,2098,2144,2175,2182,2162,2128,2097,2087,2103,2130,2158,2170,2160,2140,2119,2100,2093,2100,2120,2153,2186,2204,2187,2134,2072,2021,2009,2055,2143,2238,2297,2294,2220,2100,1989,1929,1951,2050,2179,2283,2318,2274,2178,2071,2000,1997,2048,2122,2185,2216,2206,2166,2120,2085,2076,2092,2126,2158,2171,2161,2140,2113,2092,2097,2135,2188,2208,2153,2045,1948,1952,2103,2333,2503,2492,2275,1925,1612,1509,1705,2121,2582,2880,2864,2522,2000,1528,1323,1462,1869,2346,2678,2747,2572,2279,1998,1828,1796,1868,1987,2109,2211,2276,2302,2289,2236,2154,2066,2008,1992,2025,2096,2165,2218,2225,2196,2145,2100,2076,2079,2103,2134,2157,2165,2154,2131,2113,2109,2115,2131,2144,2151,2145,2132,2117,2111,2108,2117,2129,2152,2171,2176,2158,2120,2076,2055,2062,2105,2163,2212,2230,2205,2151,2086,2034,2022,2055,2123,2193,2230,2219,2170,2111,2071,2062,2085,2119,2153,2170,2170,2155,2130,2109,2098,2108,2125,2147,2157,2155,2138,2120,2106,2106,2119,2132,2146,2157,2158,2146,2124,2097,2077,2081,2113,2161,2208,2224,2199,2138,2065,2014,2013,2058,2138,2218,2263,2256,2192,2104,2026,2000,2029,2104,2181,2227,2229,2189,2130,2078,2058,2072,2105,2146,2174,2177,2159,2130,2106,2094,2106,2128,2152,2160,2156,2141,2119,2103,2092,2094,2116,2151,2192,2211,2195,2133,2058,2008,2006,2062,2160,2252,2299,2281,2199,2081,1981,1935,1972,2075,2196,2293,2312,2256,2155,2051,1990,1999,2058,2137,2203,2229,2207,2157,2098,2068,2064,2095,2138,2171,2183,2164,2133,2103,2091,2099,2136,2184,2197,2146,2046,1962,1980,2126,2334,2464,2431,2222,1914,1659,1606,1801,2172,2556,2791,2740,2418,1965,1578,1432,1587,1956,2361,2622,2654,2489,2226,1993,1855,1836,1906,2016,2128,2216,2273,2289,2271,2213,2131,2053,2006,2010,2051,2116,2179,2213,2213,2183,2142,2105,2083,2081,2099,2128,2157,2168,2158,2138,2116,2106,2111,2126,2142,2149,2148,2139,2126,2108,2101,2108,2129,2160,2182,2185,2153,2108,2064,2048,2068,2116,2174,2221,2226,2195,2135,2069,2030,2034,2080,2140,2199,2224,2212,2163,2104,2060,2052,2078,2126,2167,2189,2182,2154,2120,2096,2087,2095,2123,2150,2165,2161,2144,2124,2111,2106,2116,2129,2144,2156,2157,2140,2115,2088,2076,2087,2125,2172,2215,2223,2190,2122,2050,2004,2014,2072,2155,2233,2270,2255,2181,2092,2020,1998,2033,2108,2182,2226,2224,2182,2125,2085,2065,2079,2106,2141,2167,2174,2163,2136,2111,2097,2099,2122,2145,2158,2156,2139,2117,2102,2096,2102,2128,2161,2190,2200,2175,2122,2059,2016,2021,2080,2171,2253,2292,2263,2175,2060,1969,1945,1995,2103,2222,2300,2300,2233,2129,2033,1993,2012,2078,2154,2209,2224,2201,2148,2100,2070,2071,2098,2137,2164,2170,2157,2132,2110,2095,2102,2135,2178,2196,2153,2062,1985,1998,2122,2300,2416,2385,2202,1932,1715,1679,1861,2191,2522,2708,2645,2350,1949,1623,1520,1682,2016,2364,2573,2575,2416,2193,2003,1895,1882,1940,2028,2122,2201,2255,2274,2258,2206,2131,2055,2012,2013,2058,2123,2186,2216,2208,2172,2122];
data = fft(data);
data(1) = 0;
data = ifft(data);
y = data(156:end)';
assert(iscolumn(y), 'signal should be column vector');

% basis matrix
fsa = 40e6;
tstep = 1 / fsa;
step = tstep;
model = getmodel();
t = 0:tstep:tstep*(length(y) - 1);
steps = 0:step:t(end);
t = t';
U = zeros(length(y), length(steps));
for i = 1:length(steps)
    U(:,i) = model(t - steps(i));
end
assert(size(U,1)==length(y), 'dictionary dim error');

% iterate 20 times
sol = mp(y, U, 20);

% reconstruct signal
y_r = U * sol;
figure;
subplot(2,1,1);
plot(t, y);
hold on;
plot(t, y_r);
subplot(2,1,2);
plot(t, y);
hold on;
plot(steps, sol);

function model = getmodel()
    f = 4e6;
    phi = 0;
    alpha = 5e-7;
    model = @(t) exp(-pi*(t/alpha).^2).*cos(2*pi*f*t + phi);
end

{
    "type": "line",
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    "options": {
        "tooltips": {
            "enabled": false
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        "animation": {
            "duration": 0
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        "elements": {
            "point": {
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            "line": {
                "borderWidth": 1,
                "tension": 0,
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}

同样的信号，同样的basis，MP和OMP的分解系数比较如下：

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    "data": {
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            "order": 1
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            "label": "OMP",
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            "borderColor":"#FFFFFF80",
            "order": 0
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        ]
    },
    "options": {
        "tooltips": {
            "enabled": false
        },
        "animation": {
            "duration": 0
        },
        "elements": {
            "point": {
                "radius": 0
            },
            "line": {
                "borderWidth": 1,
                "tension": 0,
                "backgroundColor":"transparent"
            }
        }
    }
}

正交匹配追踪(Orthogonal Match Pursuit, OMP)算法

介绍

流程

实现

效果

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