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基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法及其在圖像補(bǔ)全中的應(yīng)用

劉靜 劉涵 黃開(kāi)宇 蘇立玉

劉靜, 劉涵, 黃開(kāi)宇, 蘇立玉. 基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法及其在圖像補(bǔ)全中的應(yīng)用[J]. 電子與信息學(xué)報(bào), 2019, 41(11): 2787-2794. doi: 10.11999/JEIT181076
引用本文: 劉靜, 劉涵, 黃開(kāi)宇, 蘇立玉. 基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法及其在圖像補(bǔ)全中的應(yīng)用[J]. 電子與信息學(xué)報(bào), 2019, 41(11): 2787-2794. doi: 10.11999/JEIT181076
Jing LIU, Han LIU, Kaiyu HUANG, Liyu SU. Automatic Rank Estimation Based Riemannian Optimization Matrix Completion Algorithm and Application to Image Completion[J]. Journal of Electronics & Information Technology, 2019, 41(11): 2787-2794. doi: 10.11999/JEIT181076
Citation: Jing LIU, Han LIU, Kaiyu HUANG, Liyu SU. Automatic Rank Estimation Based Riemannian Optimization Matrix Completion Algorithm and Application to Image Completion[J]. Journal of Electronics & Information Technology, 2019, 41(11): 2787-2794. doi: 10.11999/JEIT181076

基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法及其在圖像補(bǔ)全中的應(yīng)用

doi: 10.11999/JEIT181076
基金項(xiàng)目: 國(guó)家自然科學(xué)基金(61573276)
詳細(xì)信息
    作者簡(jiǎn)介:

    劉靜:女,1975年生,教授,博士生導(dǎo)師,從事壓縮感知、圖像融合、雷達(dá)信號(hào)處理方向的研究

    劉涵:女,1991年生,碩士生,研究方向?yàn)閴嚎s感知、圖像處理、矩陣補(bǔ)全

    黃開(kāi)宇:男,1992年生,博士生,研究方向?yàn)閴嚎s感知、信號(hào)處理、信號(hào)與圖像處理

    蘇立玉:男,1996年生,碩士生,研究方向?yàn)閴嚎s感知、圖像處理、張量補(bǔ)全

    通訊作者:

    劉靜 elelj20080730@gmail.com

  • 中圖分類(lèi)號(hào): TP391.41

Automatic Rank Estimation Based Riemannian Optimization Matrix Completion Algorithm and Application to Image Completion

Funds: The National Natural Science Foundation of China (61573276)
  • 摘要: 矩陣補(bǔ)全(MC)作為壓縮感知(CS)的推廣,已廣泛應(yīng)用于不同領(lǐng)域。近年來(lái),基于黎曼優(yōu)化的MC算法因重構(gòu)精度高、計(jì)算速度快的特點(diǎn),引起了廣泛關(guān)注。針對(duì)基于黎曼優(yōu)化的MC算法需假設(shè)原矩陣秩固定已知,且隨機(jī)選擇迭代起點(diǎn)的特點(diǎn),該文提出一種基于自動(dòng)秩估計(jì)的黎曼優(yōu)化MC算法。該算法通過(guò)優(yōu)化包含秩正則項(xiàng)的目標(biāo)函數(shù),迭代獲取秩估計(jì)值和預(yù)重構(gòu)矩陣。在估計(jì)所得秩對(duì)應(yīng)的矩陣空間上以預(yù)重構(gòu)矩陣為迭代起點(diǎn),利用基于黎曼流形的共軛梯度法進(jìn)行矩陣補(bǔ)全,從而提高重構(gòu)精度。實(shí)驗(yàn)結(jié)果表明,與幾種經(jīng)典的圖像補(bǔ)全方法相比,該文算法圖像重構(gòu)精度顯著提高。
  • 圖  1  基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法的圖像補(bǔ)全

    圖  2  低秩矩陣構(gòu)建示意圖

    圖  3  黎曼流形上的共軛梯度法

    圖  4  30%采樣率下各算法圖像補(bǔ)全結(jié)果

    表  1  自動(dòng)秩估計(jì)算法偽代碼

     算法1 自動(dòng)秩估計(jì)算法
     輸入:${\text{A}} = {{\text{A}}_p} \in {\mathbb {R}^{m \times n}}$,索引矩陣${\text{Ω}}$,正則項(xiàng)系數(shù)$\mu $, $\alpha $,初始秩$\hat k$,最大迭代次數(shù)$K$,容錯(cuò)度${\tau _2}$。
     初始化:執(zhí)行奇異值分解${\text{A}}{\rm{ = }}{\text{U}}{\text{W}}{{\text{V}}^{\rm{T}}}$,將${\text{U}}$的第$r$列單位化記為${{\text{u}}_r}$,將${\text{V}}$的第$r$行單位化記為${{\text{v}}_r}$, ${\text{w}} = \left\{ {{w_r}} \right\}_{r = 1}^{\min \left( {m,n} \right)}$為${\text{W}}$中奇異值組成的     向量。令${\text{Z}} = {\text{0}}$, ${{\text{P}}_{{\Omega ^c}}}\left( {\text{A}} \right) = {\text{0}}$。
     輸出:${\text{Z}} $, $k$。
     (1) for $i = 1,2,·\!·\!·,K$ do:
     (2) ${{\text{A}}_r} = {\text{A}}$;
     (3) 更新${{\text{u}}_r}$, ${{\text{v}}_r}$, ${w_r}$: for $r = 1,2, ·\!·\!· ,\hat k$ do:
              若${w_r} \ne 0$,根據(jù)式(8)、式(9)和式(11)依次更新${{\text{u}}_r}$, ${{\text{v}}_r}$, ${w_r}$,
              ${{\text{A}}_r} = {{\text{A}}_r} - {w_r}{{\text{u}}_r}{\text{v}}_r^{\rm{T}}$,
              end;
     (4) 更新${\text{A}}$:更新${\text{Z}} = {\text{A}} - {{\text{A}}_r}$,令${ {\text{P} }_{ {\varOmega ^c} } }\left( {\text{A} } \right) = { {\text{P} }_{ {\varOmega ^c} } }\left( {\text{Z} } \right)$;
     (5) 更新$k$:for $k = 1,2, ·\!·\!· ,\min\left( {m,n} \right)$ do:
         計(jì)算$f\left( k \right) = {\rm{ } }\mu \left| { { {\text{w} }_r} } \right|_{r = 1}^k + 0.5\parallel {\text{A} } - \sum\limits_{r = 1}^k { {w_r}{ {\text{u} }_r}{ {\text{v} }_r}^{\rm{T} } } \parallel _{\rm F}^2 + \alpha k$,若$f\left( k \right) < f\left( {k + 1} \right)$,則結(jié)束循環(huán),
         end;
     (6) $\hat k = k$;
     (7) 若${{\parallel {{\text{P}}_\Omega }\left( {{\text{A}} - {\text{Z}}} \right){\parallel _{\rm{F}}}} /{\parallel {{\text{P}}_\Omega }\left( {\text{A}} \right){\parallel _{\rm{F}}}}} < {\tau _2}$或${{\parallel {{\text{A}}^{i + 1}} - {{\text{A}}^i}{\parallel _{\rm{F}}}} / {\parallel {{\text{A}}^{i + 1}}{\parallel _{\rm{F}}}}} < {\tau _2}$,則結(jié)束循環(huán);
     (8) end。
    下載: 導(dǎo)出CSV

    表  2  基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法偽代碼

     算法2 基于自動(dòng)秩估計(jì)的黎曼優(yōu)化矩陣補(bǔ)全算法
     輸入:${{\text{X}} _1}{\rm{ = }}Z \in {{\cal M}_k}$(${\text{Z}} $和$k$源于算法1),容錯(cuò)度${\tau _1}$,切向量${{\text{η}} _0}{\rm{ = }}0$。
     輸出:${{\text{X}}^ * }$。
     (1) for $i = 1,2, ·\!·\!· ,K$ do:
     (2) 梯度${\xi _i}: = {\rm{gradf}}\left( {{{\text{X}}_i}} \right)$;             % 計(jì)算黎曼梯度
     (3) 若$\parallel {\xi _i}\parallel \le {\tau _1}$,則停止迭代,令${{\text{X}}^ * }{\rm{ = }}{{\text{X}}_i}$,否則轉(zhuǎn)(4);% 終止條件
     (4) 共軛方向${{\text{η}} _i}: = - {\xi _i} + {\beta _i}{{\cal T}_{{{\text{X}} _{i - 1}} \to {{\text{X}}_i}}}\left( {{{\text{η}}_{i - 1}}} \right)$;      % 計(jì)算共軛方向
     (5) 步長(zhǎng)${t_i} = {{\rm{argmin}}_t}f\left( {{{\text{X}}_i} + t{{\text{η}} _i}} \right)$;          % 計(jì)算步長(zhǎng)
     (6) 執(zhí)行Armijo回溯以找到滿(mǎn)足$f\left( {{{\text{X}}_i}} \right) - f\left( {{R_{{{\text{X}}_i}}}\left( {0.{5^m}{t_i}{{\text{η}} _i}} \right)} \right) \ge - 0.0001 \times 0.{5^m}{t_i}\left\langle {{\xi _i},{{\text{η}} _i}} \right\rangle $且m≥0的最小整數(shù),計(jì)算${X_{i + 1}}: = {R_{{{\text{X}}_i}}}\left( {0.{5^m}{t_i}{{\text{η}} _i}} \right)$;                           % 收縮算子
     (7) end。
    下載: 導(dǎo)出CSV

    表  3  基于各算法的補(bǔ)全后圖像PSNR(dB)/SSIM指標(biāo)評(píng)價(jià)

    采樣率(%)圖像補(bǔ)全算法
    本文算法SVPOptSpaceSVTIALM
    10Barbara25.1371/0.101827.1138/0.074928.4540/0.270326.4330/0.181727.9684/0.2217
    House25.0207/0.073727.0100/0.084528.8611/0.380526.0756/0.212227.3161/0.0319
    20Barbara29.5855/0.618729.4788/0.370529.0277/0.350927.7929/0.363829.0097/0.3175
    House32.1346/0.775030.5881/0.468129.2989/0.409628.0950/0.456929.1008/0.0667
    30Barbara31.8223/0.777530.5821/0.553029.7224/0.413829.0192/0.519329.6337/0.4010
    House34.3279/0.843432.6125/0.685829.8818/0.443730.2986/0.647229.9081/0.4560
    40Barbara33.1805/0.805431.3704/0.624930.4532/0.492230.2063/0.647130.2152/0.4592
    House36.9926/0.917533.4685/0.744930.5393/0.468732.2618/0.771830.6276/0.4546
    50Barbara34.3090/0.854532.3230/0.704531.1457/0.534931.6388/0.760731.0285/0.5060
    House37.9729/0.934234.4193/0.790931.8817/0.585434.2940/0.857531.3316/0.4965
    60Barbara35.5808/0.893233.3609/0.761232.2731/0.595533.4085/0.855231.9375/0.5660
    House39.5723/0.950435.5242/0.829733.5629/0.709936.5579/0.915032.3391/0.4992
    70Barbara37.1206/0.927734.6884/0.812433.4690/0.645335.7766/0.919133.0595/0.6449
    House41.0744/0.962236.8819/0.869034.4479/0.739539.3028/0.952433.4229/0.5724
    80Barbara39.0801/0.952936.4704/0.866535.3219/0.747938.8081/0.956534.7671/0.6462
    House43.1665/0.972838.6710/0.904237.2815/0.828841.8076/0.923435.2485/0.6317
    90Barbara42.3685/0.969939.3773/0.921338.4127/0.865340.6578/0.935738.0598/0.7796
    House46.1068/0.981041.9691/0.944240.3322/0.894342.0364/0.970738.1441/0.7449
    下載: 導(dǎo)出CSV
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  • 收稿日期:  2018-11-23
  • 修回日期:  2019-05-07
  • 網(wǎng)絡(luò)出版日期:  2019-05-20
  • 刊出日期:  2019-11-01

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