韓樂(lè) 江怡華

魯棒截?cái)?-2全變分稀疏恢復(fù)模型

韓樂(lè) 江怡華

（華南理工大學(xué) 數(shù)學(xué)學(xué)院，廣東 廣州 510640）

魯棒壓縮感知；截?cái)嗳兎?；非凸非光滑?yōu)化；稀疏噪聲；結(jié)構(gòu)稀疏

隨著計(jì)算機(jī)科學(xué)與技術(shù)的高速發(fā)展，越來(lái)越多的大規(guī)模數(shù)據(jù)庫(kù)的出現(xiàn)，推動(dòng)著信號(hào)處理領(lǐng)域的變革。然而，仍然存在著小規(guī)模數(shù)據(jù)領(lǐng)域，如醫(yī)學(xué)圖像領(lǐng)域，由于受輻射劑量限制，只能采集到少量的CT圖像和MRI圖像等［1］。

受硬件條件的限制，信號(hào)在獲取的過(guò)程中不可避免地會(huì)被噪聲污染。除了常見的高斯噪聲外，還有非高斯噪聲，如脈沖噪聲、柯西噪聲、泊松噪聲等，其中脈沖噪聲和柯西噪聲均具有稀疏屬性。魯棒稀疏信號(hào)恢復(fù)［11］就是處理稀疏噪聲影響下的稀疏信號(hào)恢復(fù)問(wèn)題。相關(guān)的優(yōu)化模型通常是用向量零范數(shù)的各種近似來(lái)表示原始信號(hào)和噪聲的稀疏性［12］。

1　預(yù)備知識(shí)

2　全變分稀疏恢復(fù)模型

注意到，可以利用指示函數(shù)將問(wèn)題（1）等價(jià)轉(zhuǎn)化為無(wú)約束問(wèn)題

式中，指示函數(shù)定義為

雖然優(yōu)化問(wèn)題（2）是非凸非光滑的，但可以采用鄰近交替線性化算法求解。文中采用有加速的鄰近交替線性化算法GiPALM［13］求解問(wèn)題（2）。

Do

定義3 稱滿足

證畢。

3　子問(wèn)題求解

GiPALM算法中的子問(wèn)題（3）可寫為

Do

證畢。

最優(yōu)解可以由軟閾值算子［22］給出，即

4　數(shù)值實(shí)驗(yàn)及結(jié)果分析

文中使用峰值信噪比（PSNR，PSN）和結(jié)構(gòu)相似度（SSIM，SIM）作為噪聲圖像復(fù)原評(píng)價(jià)標(biāo)準(zhǔn)，

4.1　高斯噪聲灰度圖像恢復(fù)

實(shí)驗(yàn)采用的原始無(wú)噪聲灰度圖像見圖1，其中灰度圖像1-4的大小為256×256，灰度圖像5和6的大小為512×512。在原始灰度圖像上疊加高斯噪聲，生成含噪聲的觀測(cè)數(shù)據(jù)，其中高斯噪聲由Matlab中Imnoise函數(shù)按照均值為0、標(biāo)準(zhǔn)差為0.01生成。

圖1　原始灰度圖像

表1　恢復(fù)含高斯噪聲灰度圖像的PSNR、SIMM與運(yùn)行時(shí)間

Table 1　PSNR， SSIM and running time of recovering grey images with Gaussian noise

圖像序號(hào)PSNR/dBSSIM/dB運(yùn)行時(shí)間/s TVT0TVT1TVT0TVT1TVT0TVT1 126.8225.6227.160.760 00.506 90.802 80.020.0145.25 228.1026.7230.190.760 00.438 40.818 00.020.0143.30 326.8426.9629.080.800 00.508 90.880 70.020.0144.01 428.9027.6831.210.820 00.692 10.896 10.020.0146.86 526.9226.5129.380.860 00.807 30.910 00.100.05282.85 623.9824.9024.900.860 00.806 70.834 00.100.05733.15

4.2　混合噪聲彩色圖像恢復(fù)

圖2　原始彩色圖像

文中分別測(cè)試了在原始彩色圖像中添加稀疏噪聲和混合噪聲兩種情形。稀疏噪聲的添加是在每幅圖像中隨機(jī)選取30%的像素，將其設(shè)置為0～255中的隨機(jī)數(shù)（將圖像每個(gè)通道對(duì)應(yīng)位置的值都進(jìn)行破壞），形成噪聲圖像?；旌显肼暤奶砑觿t是在稀疏噪聲的基礎(chǔ)上，用Imnoise函數(shù)疊加均值為0、標(biāo)準(zhǔn)差為0.02的高斯噪聲。

表2　恢復(fù)含噪彩色圖像的PSNR

Table 2　PSNR of recovering color images with noises

圖像序號(hào)含稀疏噪聲的PSNR/dB含混合噪聲的PSNR/dB TRPCATVT0TVT1TRPCATVT0 TVT1 123.2824.3824.7119.1821.4821.74 224.4924.6724.9819.2121.4021.40 327.7028.0027.5020.6823.2222.71 424.4324.6224.4019.5221.3621.08 529.7330.1129.8721.7921.7923.73 631.5231.8929.8320.6321.1220.75

表3　恢復(fù)含噪彩色圖像的平均運(yùn)行時(shí)間與迭代次數(shù)

Table 3　Running time and iteration number of recovering color images with noises

圖像序號(hào)噪聲每次迭代的平均運(yùn)行時(shí)間/s總迭代次數(shù) TRPCATVT0TVT1TRPCATVT0TVT1 1稀疏噪聲0.0020.9934.95643150150 20.0010.9603.81542150150 30.0011.0253.9624384113 40.0021.0433.90943127150 50.0010.9704.0294489143 60.0011.0203.72043116101 1混合噪聲0.0011.1493.474401501 20.0021.1273.087401501 30.0011.0993.071401501 40.0021.0563.140401501 50.0011.1193.092411501 60.0011.1083.25740150150

表4　含混合噪聲彩色圖像的恢復(fù)效果

Table 4　Recovering results of color images with mixed noise

圖像序號(hào)混合噪聲圖像恢復(fù)結(jié)果 TRPCATVT0TVT1 1 2 4

4.3　混合噪聲視頻恢復(fù)

表5　含高斯噪聲視頻恢復(fù)結(jié)果的PSNR

Table 5　PSNR of recovering video with Gaussian noise

幀數(shù)PSNR/dB TRPCA中值濾波TVT0TVT1 123.3824.8525.6725.64 721.0322.5223.4423.48 1320.7721.9322.9222.92 1820.6521.7422.5322.45 2420.4721.5122.3422.26 3020.6021.4422.1022.10 3620.6121.3222.0922.12 4219.3120.5520.8620.84 4821.2922.8823.9323.86

圖4　混合噪聲下視頻恢復(fù)的幀數(shù)-PSNR曲線

5　結(jié)語(yǔ)

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（School of Mathematics，South China University of Technology，Guangzhou 510640，Guangdong，China）

In addition to Gaussian noise, there is sparse noise with impulsive properties in the signal acquisition process. The common robust sparse signal recovery models can recover the original sparse signal under sparse noise environment. However, in many practical applications, the structural sparsity of the original signal, for example, gradient sparsity needs to be considered. In order to recover the sparse structure of the original high-dimensional signal from the coexistence of sparse noise and Gaussian noise, this paper proposed two nonconvex and nonsmooth optimization models based on truncated1-2total variation (TV) and 3D truncated1-2TV, respectively. These optimization models were solved by the proximal alternating linearized minimization algorithm with extrapolation, and the sub-problems involved were solved by the proximal convex difference algorithm with extrapolation. Under the assumption that the potential function has Kurdyka-Lojasiewicz (KL) property, the convergence analysis of these algorithms was given. The numerical experiments test grey images with Gaussian noise, color images with mixed noise, grey video with mixed noise and so on. The peak signal-to-noise ratio (PSNR) was used as the evaluation criterion for recovered quality. The experimental results show that the new models can correctly recover the original structured sparse signal, and have better PSNR values in the same noisy environment.

robust compressed sensing；truncated total variation；nonconvex and nonsmooth optimization；sparse noise；structural sparse

Supported by the General Program of the National Natural Science Foundation of China （11971177），the Basic and Applied Basic Research Foundation of Guangdong Province （2021A1515010210） and the Degree and Graduate Education Reform Research Foundation of Guangdong Province （2022JGXM011）

10.12141/j.issn.1000-565X.220485

2022?08?01

國(guó)家自然科學(xué)基金面上項(xiàng)目（11971177）；廣東省基礎(chǔ)與應(yīng)用基礎(chǔ)研究基金資助項(xiàng)目（2021A1515010210）；廣東省學(xué)位與研究生教育改革研究項(xiàng)目（2022JGXM011）

韓樂(lè)（1977-），女，副教授，主要從事矩陣優(yōu)化、圖像處理研究。E-mail：hanle@scut.edu.cn

TP391.41

1000-565X（2023）05-0045-09