影像多閥值優化 (Image thresholding optimization) 有許多可作為優化評定 標準的方法，其中，本研究所採用的是熵值最大化 (maximum entropy) 作為優化 所提出針對二值化閥值 (bi-level thresholding) 的熵值最大化度量法，而這個度量 法也可延伸為多值化閥值 (multilevel thresholding) 方法。
首先，取得目標圖像的灰階直方圖，最大範圍定義為 L ，因此其範圍表示 為 {0, 1, …, L-1} 、影像的像素個數為 N ，像素機率函數的機率值則可定義為
⁄ , 0 ! 1 (2.1)
其中 h(i) 為灰階函數的分佈值、N 為目標圖像總像素，然而，若閥值數量
Memeplexes, M：群數，在每一次的迭代中會將 P 依序分配至每一群中，詳
圖 2.1 影像多閥值優化之實驗流程圖
在取得最佳閥值後會計算峰值訊噪比 (Peak Signal to Noise Ratio, PSNR) 作 為多閥值影像分割的效能評估，單位為 dB，其計算式如下：
C D 20log$, 255
DICJ (2.4)
其中，RMSE 為均方根誤差 (Root Mean Square Error, RMSE) ，運算如下：
DICJ K∑ ∑P3$ NO3$ L , M L , M %
I (2.5)
在公式 (2.5) 中，I 跟 L 是原始圖像與分割後的圖像，I Q 為圖像大小。
Chapter 3
領域的廣泛應用，這項技術是透過 VC 維 (Vapnik Chervonenkis dimension) 與結 構風險最小化 (Structural risk minimization) 的概念所開發而成的分類工具，後續 的發展中主要可分為兩種：線性 (linear) 與非線性 (nonlinear) 的支援向量機。圖 3.1 SVM 分類概念圖
由 於 上 述 公 式 (3.3) 並 沒 有 封 閉 解 ， 因 此 導 入 拉 格 朗 日 多 項 式 乘 數 (Lagrange’s multipliers) 試圖取得近似最佳解 ` , 1, … , 並使其最大化，如公 式 (3.4)
Iab： !c ` d `
e 3$
12 d ` `OS SOR RO
N
,O3$ (3.4)
` [ 0, 1, … , ∑ ` SN3$ 0
最後，藉由公式 (3.4) 訓練後找出最佳化超平面來進行分類；但是，一般來 說，以線性方式就能將資料完全分離是非常罕見的，大多數的情況都是無法分類 完全，因此為了因應此一問題，便衍生出了非線性支援向量機。
3.3 非 非 非 非線性支援向量機 線性支援向量機 線性支援向量機 線性支援向量機
在現實情況中，要維持原始維度來進行分類其實是很困難的，因此為了解 決這一難點，便延伸出了非線性的分類方法，稱之為 － 非線性支援向量機 (Non-Linear Support Vector Machines)。
與線性支援向量機不同的是，非線性支援向量機在尋找超平面前會先將資料 投射到特徵空間，其概念如圖 3.2：
圖 3.2 特徵空間投射概念圖 其表示式為
Φ：UV ↦ h (3.5)
或表示為
R → Φ R (3.6)
在將訓練資料投射至更高維度後將使得運算式會變得很繁雜並耗費大量資 (support vector, SV)。
' R, `∗, Z∗ d S `∗j R , R
多項式核心函數 ( Polynomial kernel function )：
jnR , ROo R ∙ RO 1 V with d ∈ N (3.12)
高斯徑向基核心函數 ( Gaussian radial basis function )：
jnR , ROo p2qrs52str; (3.13) 正切雙曲線核心函數 ( Tangent hyperbolic kernel function )：
jnR , ROo tanh R ∙ RO Θ (3.14)
Step 1 資料處理
圖 3.4 SVM 參數優化之實驗流程圖
Chapter 4
Microsoft Visual Studio 2008。4.2 影像多閥值優化實驗結果 影像多閥值優化實驗結果 影像多閥值優化實驗結果 影像多閥值優化實驗結果
本研究實驗中所使用的測試圖像如圖 4.1，其中六種是標準測試圖像，分別 為 ” BABOON” 、 ” F16” 、 ” FISHINGBOAT” 、 ” BRANDYROSE” 、 ” SKYLINE_ARCH” 、 ” PILLS” ， 另 外 六 種 為 醫 學 圖 像 則 是 ” 84” 、 ” 12015024—20071105—CT-050016”、 ” 12015024—20071105—CT-050021”、 ” 2008072010045” 、 ” anatomic_imaging_of_the_shoulder_coronal_t1_se” 、 ” brain_mri_transversal_t1_002”。
圖 4.2 – 4.13 分別為各個圖像的多閥值優化結果，圖 4.14 – 4.17 表示為 Baboon 與 F16 圖像於多閥值分割實驗中的收斂情形，表 4.1 是改良式混合蛙跳 演算法對於醫學圖像的實驗結果，表 4.2 則是應用於測試圖像的優化結果，最後 表 4.3 為窮取法、混合蛙跳演算法、改良式混合蛙跳演算法的分別比較結果。
圖 4.1 實驗測試圖像
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.2 84 圖像優化閥值切割結果
Original image
2-level thresholding image
3-level thresholding image
Histogram
level thresholding image The selected thresholds of 2-level threshold
level thresholding image The selected thresholds of 3-level threshold level threshold
evel threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.3 12015024—20071105—CT-050016 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.4 12015024—20071105—CT-050021 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.5 2008072010045 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.6 anatomic_imaging_of_the_shoulder_coronal_t1_se 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.7 brain_mri_transversal_t1_002 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.8 BABOON 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.9 F16 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.10 FISHINGBOAT 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.11 BRANDYROSE 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.12 SKYLINE_ARCH 圖像優化閥值切割結果
Original image Histogram
2-level thresholding image The selected thresholds of 2-level threshold
3-level thresholding image The selected thresholds of 3-level threshold
4-level thresholding image The selected thresholds of 4-level threshold
5-level thresholding image The selected thresholds of 5-level threshold 圖 4.13 PILLS 圖像優化閥值切割結果
圖 4.14 Baboon 收斂實驗曲線圖 (MSFL)
Number of optimal frog replaced in iterations
Number of iterations
Number of optimal frog replaced in iterations
Number of iterations
T=2 T=3 T=4 T=5
圖 4.16 F16 收斂實驗曲線圖 (MSFL)
Number of optimal frog replaced in iterations
Number of iterations
Number of optimal frog replaced in iterations
Number of ierations
T=2 T=3 T=4 T=5
表 4.1 改良式混合蛙跳演算法對於醫學圖像之實驗結果 5 74,109,142,177,210 19.9014 32.9268 0.27
brain_mri_
表 4.2 改良式混合蛙跳演算法對於測試圖像之實驗結果 5 71,107,143,179,220 20.7612 28.4059 0.29
SKYLINE_ARCH (400x594)
2 78,174 12.6668 23.1858 0.26
3 64,131,197 16.0391 25.5107 0.26
4 42,96,147,198 19.1042 26.9944 0.28 5 32,71,114,158,203 21.96716 28.2495 0.29
PILLS
表 4.3 三種演算法之效能比較
Image
MSFLA SFLA Exhaustive
Thresholds Time
(s) Thresholds Time 4 33,73,114,159 0.26 33,73,114,159 0.27 33,73,114,159 15305.578
F16 (512x512)
2 71,173 0.22 71,173 0.23 71,173 1.61
3 69,127,183 0.23 69,127,183 0.23 69,127,183 143.359 4 67,106,145,185 0.25 67,106,145,185 0.25 67,106,145,185 17476.625
FISHINGBOAT (512x512)
2 107,176 0.25 107,176 0.27 107,176 1.75
3 64,119,176 0.26 64,119,176 0.27 64,119,176 155.563 4 48,88,128,181 0.29 48,88,128,181 0.29 48,88,128,181 16623.484
BRANDYROSE (518x744)
2 100,159 0.24 100,159 0.25 100,159 1.719
3 74,121,172 0.25 74,121,172 0.25 74,121,172 152.687 4 71,108,146,184 0.27 71,108,146,184 0.27 71,108,146,184 16464.703
SKYLINE_ARCH (400x594)
2 78,174 0.26 78,174 0.26 78,174 1.766
3 64,131,197 0.26 64,131,197 0.26 64,131,197 155.687 4 42,96,147,198 0.28 42,96,147,198 0.28 42,96,147,198 16654.875
PILLS (800x519)
2 90,159 0.24 90,159 0.24 90,159 1.766
3 56,109,165 0.25 56,109,165 0.26 56,109,165 152.75 4 53,97,144,186 0.26 53,97,144,186 0.27 53,97,144,186 23164.453
4.3 支援向量機之參數優化實驗結果 支援向量機之參數優化實驗結果 支援向量機之參數優化實驗結果 支援向量機之參數優化實驗結果
Data from UCI Machine Learning RepositoryDataset No. of
class
No. of instances
No. of
features Accuracy
1 SPECTF 2 267 44 0.769231
Chapter 5
研究結論與未來展望 研究結論與未來展望 研究結論與未來展望 研究結論與未來展望
本研究以近年來較新的混合蛙跳演算法嘗試應用於影像多閥值與支援向量 機之參數優化上，在影像多閥值方面，從實驗中可以得知以熵值最大化為度量標 準所搜尋出的優化閥值與全域搜尋法結果相同，但效能卻有大幅度的提升；在支 援向量機之參數優化的實驗中，部分資料集的測試結果良好，但其他測試資料則 稍有落差，此部分的詳細過程尚須探討。
在未來，本研究將針對支援向量機參數優化不足之處進行改良，並改為多分 類參數優化。
參考文獻 參考文獻 參考文獻 參考文獻
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