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摘要: 当前景或背景的灰度分布呈现为非正态分布特征时，比如极值、瑞利、贝塔或均匀分布，将所选阈值与最优阈值之差控制在10个灰度级内并非易事.为了在统一框架内处理不同灰度分布情形下的阈值选择问题，提出了一种归一化互信息量最大化导向的自动阈值选择方法.该方法先采用多尺度梯度乘变换规范化输入图像，获得具有单峰长拖尾灰度分布的规范图像；然后对不同阈值对应的二值图像进行轮廓提取，获得不同的轮廓图像；最后计算规范图像和不同轮廓图像之间的归一化互信息量，并以最大值对应的阈值作为最终阈值.在具有不同灰度分布模式的9幅合成图像和59幅真实世界图像上，将提出的方法和1种人工阈值方法及4种自动阈值方法进行了比较.实验结果表明，提出的方法虽然在计算效率方面不优于4个自动方法，但在分割的适应性和精确度方面优势明显：对前述不同灰度分布情形，其所选阈值与最优阈值之差都在9个灰度级内.Abstract: When the gray level distribution of foreground or background appears a non-normal distribution, such as extreme value, Rayleigh, beta or uniform distribution, it is challenging to keep the difference between the selected threshold and the optimal threshold under control within 10 gray levels. To deal with the issue of threshold selection in different gray level distributions within a unified framework, we propose an automatic method of threshold selection that is guided by maximizing normalized mutual information. Firstly, this method applies a multiscale gradient multiplication to an input image, which produces a normalized image with a single-peaked trailing distribution. Then, the binary images corresponding to different thresholds are subjected to contour extraction to produce different contour images. Finally, normalized mutual information between the normalized image and different contour images is calculated and the threshold corresponding to maximum is taken as the final threshold. The proposed method is compared with a manual thresholding method and 4 automatic ones on 9 synthetic images and 59 real-world ones with different distributions. The results show that the proposed method is not superior to these 4 automatic methods in computational efficiency, but it has a significant advantage in the adaptability and accuracy of segmentation:in the case of the aforementioned different gray level distributions, the differences between its selected thresholds and the optimal ones are all within 9 gray levels.1) 本文责任编委 胡清华
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图 1 MNMI方法所涉及的关键概念和步骤的图示((a)原始图像$X$的灰度直方图; (b)原始图像$X$; (c)使用$t$阈值化原始图像$X$所得的二值图像${{Y}_{t}}$; (d)对原始图像$X$进行多尺度梯度乘变换后所得图像${{M}_{X}}$的灰度直方图; (e)图像${{M}_{X}}$; (f)从二值图像${{Y}_{t}}$中提取轮廓后所得轮廓图像${{C}_{{{Y}_{t}}}}$.注意, 为了能更清楚地显示灰度区间[0, 255]内灰度值出现的频率, (d)中灰度直方图在频率为80处进行了截断.)
Fig. 1 Graphic illustration of crucial concepts and steps in MNMI method ((a) Gray level histogram of original image $X$; (b) original image $X$; (c) binary image ${{Y}_{t}}$ obtained by thresholding original image $X$ with a gray level $t$; (d) gray level histogram of image ${{M}_{X}}$ produced by applying a multiscale gradient multiplication transformation to original image $X$; (e) image ${{M}_{X}}$; (f) contour image ${{C}_{{{Y}_{t}}}}$ extracted from binary image ${{Y}_{t}}$. The gray level histogram in (d) is truncated at the frequency 80 for more clearly showing the frequency of gray level in the range [0, 255].)
图 2 灰度直方图呈现出双峰特征的4幅合成图像.每幅子图的左边显示了合成图像, 右边显示了对应的灰度直方图, 黑色曲线显示了提出的MNMI方法计算阈值的目标函数曲线, 虚线及旁边的数字标示了MNMI方法计算出的阈值(下同) ((a)正态分布情形; (b)极值分布情形; (c)瑞利分布情形; (d)贝塔分布情形)
Fig. 2 4 synthetic images with bimodal gray level histogram. In each sub-figure, a synthetic image is shown on the left; on the right, the gray level histogram is shown, the objective function of MNMI method is illustrated with a black curve, and the threshold obtained by MNMI method is indicated with a dashed line and a number (the same below) ((a) Normal distribution, (b) Extreme value distribution, (c) Rayleigh distribution, (d) Beta distribution)
图 3 灰度直方图呈现出单峰特征的3幅合成图像((a)正态分布情形; (b)极值分布情形; (c)瑞利分布情形)
Fig. 3 3 synthetic images with unimodal gray level histogram ((a) Normal distribution, (b) extreme value distribution, (c) Rayleigh distribution)
图 4 灰度直方图呈现出无峰特征的合成图像
Fig. 4 A synthetic image with a uniform distribution of gray level
图 5 灰度直方图呈现出多峰特征的合成图像 (灰度直方图的灰度区间[0, 50]由瑞利分布和均匀分布组合而成, 区间[51, 100]为均匀分布,区间[101, 150]由极值分布和均匀分布组合而成, 区间[151, 200]由贝塔分布和正态分布组合而成,区间[201, 255]为正态分布)
Fig. 5 A synthetic image with multimodal gray level histogram (The gray level histogram in the range [0, 50] is combined by a Rayleigh distribution and a uniform distribution, [51, 100] by a uniform distribution, [101, 150] by an extreme value distribution and a uniform distribution, [151, 200] by a beta distribution and a normal distribution, and [201, 255] by a normal distribution.)
图 6 6个阈值方法在59幅真实世界图像上分割精度的量化比较(在每幅子图中, 各条水平虚线标示了对应情形下ME值的平均值)
Fig. 6 Quantification comparisons of segmentation accuracy for 6 thresholding methods on 59 real-world images (In each sub-figure, each horizontal dashed lines indicate the corresponding average ME, respectively)
图 7 对59幅真实世界图像, MNMI, GET, ITT, TRT, CKMI方法和IT方法获得的阈值之差(在每幅子图中, 每个黑点旁的数字标示了相应的阈值差)
Fig. 7 The differences between MNMI, GET, ITT, TRT, CKMI and IT methods in segmentation thresholds for 59 real-world images (In each sub-figure, the number beside each black point labels the specific difference of segmentation threshold)
表 1 6个阈值分割方法在灰度直方图呈现出双峰特征的4幅合成图像上的阈值$ t$和ME值
Table 1 Threshold values $t$ and ME values of 6 thresholding methods on 4 synthetic images with bimodal gray level histogram
阈值方法 正态分布$t$, ME (%) 极值分布$t$, ME (%) 瑞利分布$t$, ME (%) 贝塔分布$t$, ME (%) IT 155, 0.18 162, 0.17 201, 1.23 249, 0.20 GET 186, 2.25 144, 16.99 175, 4.17 175, 27.50 ITT 149, 0.33 133, 44.34 142, 16.94 128, 39.16 TRT 150, 0.28 152, 4.33 164, 6.94 174, 27.81 CKMI 139, 1.42 135, 39.36 133, 22.80 124, 40.05 MNMI 155, 0.18 162, 0.17 200, 1.26 249, 0.20 
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表 2 6个阈值分割方法在灰度直方图呈现出单峰特征的3幅合成图像上的阈值$t$和ME值
Table 2 Threshold values $t$ and ME values of 6 thresholding methods on 3 synthetic images with unimodal gray level histogram
阈值方法 正态分布$t$, ME (%) 极值分布$t$, ME (%) 瑞利分布$t$, ME (%) IT 168, 0.03 142, 0.04 120, 0.05 GET 179, 0.14 134, 14.98 93, 14.79 ITT 102, 44.29 122, 63.90 80, 38.00 TRT 146, 0.59 134, 14.98 103, 5.40 CKMI 100, 48.62 126, 48.67 75, 49.70 MNMI 168, 0.03 142, 0.04 120, 0.05
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表 3 6个阈值分割方法在灰度直方图分别呈现出无峰特征和多峰特征的合成图像上的阈值$t$和ME值
Table 3 Threshold values $t$ and ME values of 6 thresholding methods on synthetic images with uniform gray level histogram and multimodal one
阈值方法 均匀分布(无峰)$t$, ME (%) 混合分布(多峰)$t$, ME (%) IT 230, 0.21 204, 0.25 GET 180, 19.71 180, 17.75 ITT 126, 40.61 110, 55.60 TRT 174, 22.24 154, 27.89 CKMI 128, 39.82 108, 55.90 MNMI 230, 0.21 204, 0.25
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表 4 5个自动阈值分割方法的计算效率比较
Table 4 Quantitative comparisons of 5 automatic thresholding methods in computational efficiency
阈值方法 合成图像上耗时(秒) 真实世界图像上耗时(秒) 均值 标准偏差 均值 标准偏差 ITT 0.007 0.002 0.008 0.005 GET 0.039 0.031 0.045 0.028 CKMI 0.081 0.008 0.127 0.073 TRT 0.116 0.061 0.109 0.093 MNMI 0.215 0.068 0.327 0.205
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表 5 不同步长下MNMI方法的计算效率和误分类率
Table 5 Computational efficiency and ME of MNMI method with different steps
步长$\rho $ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 合成图像 平均耗时(秒) 0.215 0.138 0.113 0.103 0.096 0.093 0.090 0.088 0.088 0.087 0.087 0.086 0.088 0.087 0.088 平均ME (%) 0.267 0.267 0.267 0.267 0.267 0.267 0.267 0.319 0.319 0.267 0.267 10.653 0.267 0.267 0.267 真实世界图像 平均耗时(秒) 0.327 0.190 0.147 0.125 0.115 0.108 0.105 0.102 0.104 0.104 0.101 0.101 0.103 0.106 0.108 平均ME (%) 0.371 0.371 0.373 0.371 0.373 0.377 1.038 2.705 0.372 2.001 1.994 3.235 2.113 3.277 3.776
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