外文翻譯--圖像分割.doc

中文4440字畢業(yè)設(shè)計(jì)/論文外 文 文 獻(xiàn) 翻 譯系 別 計(jì)算機(jī)與電子系 專 業(yè) 班 級(jí) 計(jì)算機(jī)科學(xué)與技術(shù) 姓 名 原 文 出 處 Digital Image Processing 2/E評(píng) 分 指 導(dǎo) 教 師 2012 年 3 月圖像分割 前一章的資料使我們所研究的圖像處理方法開(kāi)始發(fā)生了轉(zhuǎn)變。從輸人輸出均為圖像的處理方法轉(zhuǎn)變?yōu)檩斎藶閳D像而輸出為從這些圖像中提取出來(lái)的屬性的處理方法這方面在1.1節(jié)中定義過(guò))。圖像分割是這一方向的另一主要步驟。分割將圖像細(xì)分為構(gòu)成它的子區(qū)域或?qū)ο?。分割的程度取決于要解決的問(wèn)題。就是說(shuō)當(dāng)感興趣的對(duì)象已經(jīng)被分離出來(lái)時(shí)就停止分割。例如，在電子元件的自動(dòng)檢測(cè)方面，我們關(guān)注的是分析產(chǎn)品的圖像，檢測(cè)是否存在特定的異常狀態(tài)，比如，缺失的元件或斷裂的連接線路。超過(guò)識(shí)別這此元件所需的分割是沒(méi)有意義的。 異常圖像的分割是圖像處理中最困難的任務(wù)之一。精確的分割決定著計(jì)算分析過(guò)程的成敗。因此，應(yīng)該特別的關(guān)注分割的穩(wěn)定性。在某些情況下，比如工業(yè)檢測(cè)應(yīng)用，至少有可能對(duì)環(huán)境進(jìn)行適度控制的檢測(cè)。有經(jīng)驗(yàn)的圖像處理系統(tǒng)設(shè)計(jì)師總是將相當(dāng)大的注意力放在這類可能性上。在其他應(yīng)用方面，比如自動(dòng)目標(biāo)采集，系統(tǒng)設(shè)計(jì)者無(wú)法對(duì)環(huán)境進(jìn)行控制。所以，通常的方法是將注意力集中于傳感器類型的選擇上，這樣可以增強(qiáng)獲取所關(guān)注對(duì)象的能力，從而減少圖像無(wú)關(guān)細(xì)節(jié)的影響。一個(gè)很好的例子就是，軍方利用紅外線圖像發(fā)現(xiàn)有很強(qiáng)熱信號(hào)的目標(biāo)，比如移動(dòng)中的裝備和部隊(duì)。 圖像分割算法一般是基于亮度值的不連續(xù)性和相似性兩個(gè)基本特性之一。第一類性質(zhì)的應(yīng)用途徑是基于亮度的不連續(xù)變化分割圖像，比如圖像的邊緣。第二類的主要應(yīng)用途徑是依據(jù)事先制定的準(zhǔn)則將圖像分割為相似的區(qū)域，門(mén)限處理、區(qū)域生長(zhǎng)、區(qū)域分離和聚合都是這類方法的實(shí)例。 本章中，我們將對(duì)剛剛提到的兩類特性各討論一些方法。我們先從適合于檢測(cè)灰度級(jí)的不連續(xù)性的方法展開(kāi)，如點(diǎn)、線和邊緣。特別是邊緣檢測(cè)近年來(lái)已經(jīng)成為分割算法的主題。除了邊緣檢測(cè)本身，我們還會(huì)討論一些連接邊緣線段和把邊緣“組裝”為邊界的方法。關(guān)于邊緣檢測(cè)的討論將在介紹了各種門(mén)限處理技術(shù)之后進(jìn)行。門(mén)限處理也是一種人們普遍關(guān)注的用于分割處理的基礎(chǔ)性方法，特別是在速度因素占重要地位的應(yīng)用中。關(guān)于門(mén)限處理的討論將在幾種面向區(qū)域的分割方法展開(kāi)的討論之后進(jìn)行。之后，我們將討論一種稱為分水嶺分割法的形態(tài)學(xué)圖像分割方法。這種方法特別具有吸引力，因?yàn)樗鼘⒈菊碌谝徊糠痔岬降膸追N分割屬性技術(shù)結(jié)合起來(lái)了。我們將以圖像分割的應(yīng)用方面進(jìn)行討論來(lái)結(jié)束本章。10.1間斷檢測(cè)在本節(jié)中，我們介紹幾種用于檢測(cè)數(shù)字圖像中三種基本的灰度級(jí)間斷技術(shù):點(diǎn)、線和邊緣。尋找間斷最一般的方法是以3.5節(jié)中描述的方式對(duì)整幅圖像使用一個(gè)模板進(jìn)行檢測(cè)。圖10-1所示的3x3模板，這一過(guò)程包括計(jì)算模板所包圍區(qū)域內(nèi)灰度級(jí)與模板系數(shù)的乘積之和。就是說(shuō)，關(guān)于式(3.5.3)，在圖像中任意點(diǎn)的模板響應(yīng)由下列公式給出：（10.1.1）圖10-1 一個(gè)一般的3*3模板 這里Zi是與模板系數(shù)Wi相聯(lián)系的像素的灰度級(jí)。照例，模板響應(yīng)是它的中心位置。有關(guān)執(zhí)行模板操作的細(xì)節(jié)在3.5節(jié)中討論。10.1.1點(diǎn)檢測(cè)在一幅圖像中，孤立點(diǎn)的檢測(cè)在理論上是簡(jiǎn)單的。使用如圖10-2(a)所示的模板，如果|R| T (10.1.2)我們說(shuō)在模板中心的位置上已經(jīng)檢測(cè)到一個(gè)點(diǎn)。這里T是一個(gè)非負(fù)門(mén)限，R由式(10.1.1)給出?；旧?，這個(gè)公式是測(cè)量中心點(diǎn)和它的相鄰點(diǎn)之間加權(quán)的差值?；舅枷刖褪?如果一個(gè)孤立的點(diǎn)(此點(diǎn)的灰度級(jí)與其背景的差異相當(dāng)大并且它所在的位置是一個(gè)均勻的或近似均勻的區(qū)域)與它周圍的點(diǎn)很不相同，則很容易被這類模板檢測(cè)到。注意，圖10-2(a)中的模板同圖3.39(d)中給出的模板在拉普拉斯操作方而是相同的。嚴(yán)格地講，這里強(qiáng)調(diào)的是點(diǎn)的檢測(cè)。即我們著重考慮的差別是那些足以識(shí)別為孤立點(diǎn)的差異(由T決定)。注意，模板系數(shù)之和為零表示在灰度級(jí)為常數(shù)的區(qū)域，模板響應(yīng)為零。-1-1-1-18-1-1-1-1（a）（b） （c） （d）圖10-2 （a）點(diǎn)檢測(cè)模板，（b）帶有通孔的渦輪葉片的X射線，（c）點(diǎn)檢測(cè)的結(jié)果，（d）使用式（10.1.2）得到的結(jié)果（原圖由X-TEK系統(tǒng)公司提供）例10.1圖像中孤立點(diǎn)的檢瀏 我們以圖10-2(b)功為輔助說(shuō)明如何從一幅圖中將孤立點(diǎn)分割出來(lái).這幅X射線圖顯示了一個(gè)帶有通孔的噴氣發(fā)動(dòng)抓渦槍葉片，通孔位于圈像的右上象限。在孔中只嵌有一個(gè)黑色像素。圖10-2(c)是將點(diǎn)檢測(cè)模板應(yīng)用于X射線圖像后得到的結(jié)果.圖10-2(d)顯示了當(dāng)T取圖10-2(c)中像素最高絕襯值的90%時(shí)，應(yīng)用式(10.1.2)所得的結(jié)果(門(mén)限選擇將在10.3節(jié)中詳細(xì)討論)。圖中的這個(gè)單一的像素清晰可見(jiàn)(這個(gè)像素被人為放大以便印刷后可以看到)。由于這類檢測(cè)是基于單像素間斷，并且檢測(cè)器模板的區(qū)域有一個(gè)均勻的背景，所以這個(gè)檢測(cè)過(guò)程是相當(dāng)有專用性的當(dāng)這一條件不能滿足時(shí)，本章中計(jì)論的其他方法會(huì)更適合檢測(cè)灰度級(jí)間斷10.1.2線檢測(cè)復(fù)雜程度更高一級(jí)的檢測(cè)是線檢測(cè)，考慮圖10-3中顯示的模板。如果第l個(gè)模板在圖像中移動(dòng)，這個(gè)模板將對(duì)水平方向的線條(一個(gè)像素寬度)有更強(qiáng)的響應(yīng)。在一個(gè)不變的背景上，當(dāng)線條經(jīng)過(guò)模板的中間一行時(shí)會(huì)產(chǎn)生響應(yīng)的最大值。畫(huà)一個(gè)元素為1的簡(jiǎn)單陣列，并且使具有不同灰度級(jí)(如5)的一行水平穿過(guò)陣列，可以很容易驗(yàn)證這一點(diǎn)。同樣的實(shí)驗(yàn)可以顯示出圖10-3中的第2個(gè)模板對(duì)于45方向線有最佳響應(yīng);第3個(gè)模板對(duì)于垂直線有最佳響應(yīng);第4個(gè)模板對(duì)于-45方向線有最佳響應(yīng);這些方向也可以通過(guò)注釋每個(gè)模板的優(yōu)選方向來(lái)設(shè)置，即在這些方向上用比別的方向更大的系數(shù)(為2)設(shè)置權(quán)值。注意每個(gè)模板系數(shù)相加的總和為零，表示在灰度級(jí)恒定的區(qū)域來(lái)自模板的響應(yīng)為零。-1214-12-1-1-12-12-122-1-1-1-1-12-12-12-12-1-12-1-1-1-12-1-1-1 Horizontal +45 Vertical -45圖10-3 線模板 令R1，R2，R3和R4。從左到右代表圖10-3中模板的響應(yīng)，這里R的值由式(10.1.1)給出。假設(shè)4個(gè)模板分別應(yīng)用于一幅圖像，在圖像中心的點(diǎn)，如果|Ri|Rj| ,ji，則此點(diǎn)被認(rèn)為與在模板i方向上的線更相關(guān)。例如，如果在圖中的一點(diǎn)有|Ri|Rj| ,j=2,3,4，我們說(shuō)此特定點(diǎn)與水平線有更大的聯(lián)系。 換句話說(shuō)，我們可能對(duì)檢測(cè)特定方向上的線感興趣。在這種情況下，我們應(yīng)使用與這一方向有關(guān)的模板，并設(shè)置該模板的輸出門(mén)限，如式(10.1.2)所示。換句話說(shuō)，如果我們對(duì)檢測(cè)圖像中由給定模板定義的方向上的所有線感興趣.只需要簡(jiǎn)單地通過(guò)整幅圖像運(yùn)行模板，并對(duì)得到的結(jié)果的絕對(duì)值設(shè)置門(mén)限即可。留下的點(diǎn)是有最強(qiáng)響應(yīng)的點(diǎn)。對(duì)于一個(gè)像素寬度的線，這些響應(yīng)最靠近模板定義的對(duì)應(yīng)方向。下列例子說(shuō)明了這一過(guò)程。例 10.2特定方向上的線檢測(cè) 圖10-4(a)顯示了一幅電路接線模板的數(shù)字化(二值的)圖像。假設(shè)我們要找到一個(gè)像素寬度的并且方向?yàn)?45的線條。基于這個(gè)假設(shè)，使用圖10-3中最后一個(gè)模板。圖10-4(b)顯示了得到的結(jié)果的絕對(duì)值。注意，圖像中所有水平和垂直的部分都被除去了。并且在圖10-4(b)中所有原圖中接近-45方向的部分產(chǎn)生了最強(qiáng)響應(yīng)。（a） （b） （c）圖10-4 線檢測(cè)的說(shuō)明。（a）二進(jìn)制電路接線模板，（b）使用-45線檢測(cè)器處理后得到的絕對(duì)值，（c）對(duì)圖像（b）設(shè)置門(mén)限得到的結(jié)果 為了決定哪一條線擬合模板最好，只需要簡(jiǎn)單地對(duì)圖像設(shè)置門(mén)限。圖10-4(c)顯示了使門(mén)限等于圖像中最大值后得到的結(jié)果。對(duì)于與這個(gè)例子類似的應(yīng)用，讓門(mén)限等于最大值是一個(gè)好的選擇，因?yàn)檩斎雸D像是二值的，并且我們要尋找的是最強(qiáng)響應(yīng)。圖10-4(c)顯示了在白色區(qū)所有通過(guò)門(mén)限檢測(cè)的點(diǎn)。此時(shí)，這一過(guò)程只提取了一個(gè)像素寬且方向?yàn)?45的線段(圖像中在左上象限中也有此方向上的圖像部分，但寬度不是一個(gè)像素)。圖10-4(c)中顯示的孤立點(diǎn)是對(duì)于模板也有相同強(qiáng)度響應(yīng)的點(diǎn)。在原圖中，這些點(diǎn)和與它們緊接著的相鄰點(diǎn)，是用模板在這些孤立位置上生成最大響應(yīng)的方法來(lái)定向的。這些孤立點(diǎn)也可以使用圖10-2(a)中的模板進(jìn)行檢測(cè)，然后刪除，或者使用下一章中討論的形態(tài)學(xué)腐蝕法刪除。10.1.3邊緣檢側(cè) 盡管在任何關(guān)于分割的討論中，點(diǎn)和線檢測(cè)都是很重要的，但是邊緣檢測(cè)對(duì)于灰度級(jí)間斷的檢測(cè)是最為普遍的檢測(cè)方法。本節(jié)中，我們討論實(shí)現(xiàn)一階和二階數(shù)字導(dǎo)數(shù)檢測(cè)一幅圖像中邊緣的方法。在3.7節(jié)介紹圖像增強(qiáng)的內(nèi)容中介紹過(guò)這些導(dǎo)數(shù)。本節(jié)的重點(diǎn)將放在邊緣檢測(cè)的特性上。某些前面介紹的概念在這里為了敘述的連續(xù)性將進(jìn)行簡(jiǎn)要的重述?；菊f(shuō)明 在3.7.1節(jié)中我們非正式地介紹過(guò)邊緣。本節(jié)中我們更進(jìn)一步地了解數(shù)字化邊緣的概念。直觀上，一條邊緣是一組相連的像素集合。這些像素位于兩個(gè)區(qū)域的邊界上。然而，我們已經(jīng)在2.5.2節(jié)中用一定的篇幅解釋了一條邊緣和一條邊界的區(qū)別。從根本上講，如我們將要看到的，一條邊緣是一個(gè)“局部”概念，而由于其定義的方式，一個(gè)區(qū)域的邊界是一個(gè)更具有整體性的概念。給邊緣下一個(gè)更合理的定義需要具有以某種有意義的方式測(cè)量灰度級(jí)躍變的能力。 我們先從直觀上對(duì)邊緣建模開(kāi)始。這樣做可以將我們引領(lǐng)至一個(gè)能測(cè)量灰度級(jí)有意義的躍變的形式體系中。從感覺(jué)上說(shuō)，一條理想的邊緣具有如圖10-5(a)所示模型的特性。依據(jù)這個(gè)模型生成的完美邊緣是一組相連的像素的集合(此處為在垂直方向上)，每個(gè)像素都處在灰度級(jí)躍變的一個(gè)垂直的臺(tái)階上(如圖形中所示的水平剖面圖)。實(shí)際上，光學(xué)系統(tǒng)、取樣和其他圖像采集的不完善性使得到的邊緣是模糊的，模糊的程度取決于諸如圖像采集系統(tǒng)的性能、取樣率和獲得圖像的照明條件等因素。結(jié)果，邊緣被更精確地模擬成具有“類斜面”的剖面，如圖10-5(b)所示。斜坡部分與邊緣的模糊程度成比例。在這個(gè)模型中，不再有細(xì)線(一個(gè)像素寬的線條)。相反，現(xiàn)在邊緣的點(diǎn)是包含于斜坡中的任意點(diǎn)，并且邊緣成為一組彼此相連接的點(diǎn)集。邊緣的“寬度”取決于從初始灰度級(jí)躍變到最終灰度級(jí)的斜坡的長(zhǎng)度。這個(gè)長(zhǎng)度又取決于斜度，斜度又取決于模糊程度。這使我們明白:模糊的邊緣使其變粗而清晰的邊緣使其變得較細(xì)。 圖10-6(a)顯示的圖像是從圖10-5(b)的放大特寫(xiě)中提取出來(lái)的。圖10-6(b)顯示了兩個(gè)區(qū)域之間邊緣的一條水平的灰度級(jí)剖面線。這個(gè)圖形也顯示出灰度級(jí)剖面線的一階和二階導(dǎo)數(shù)。當(dāng)我們沿著剖面線從左到右經(jīng)過(guò)時(shí)，在進(jìn)人和離開(kāi)斜面的變化點(diǎn)，一階導(dǎo)數(shù)為正。在灰度級(jí)不變的區(qū)域一階導(dǎo)數(shù)為零。在邊緣與黑色一邊相關(guān)的躍變點(diǎn)二階導(dǎo)數(shù)為正，在邊緣與亮色一邊相關(guān)的躍變點(diǎn)二階導(dǎo)數(shù)為負(fù)，沿著斜坡和灰度為常數(shù)的區(qū)域?yàn)榱?。在圖10-6(b)中導(dǎo)數(shù)的符號(hào)在從亮到暗的躍變邊緣處取反。 （a） （b）圖10-5 （a）理想的數(shù)字邊緣模型，（b）斜坡數(shù)字邊緣模型。斜坡部分與邊緣的模糊程度成正比圖10-6 (a)由一條垂直邊緣分開(kāi)的兩個(gè)不同區(qū)域,(b)邊界附近的細(xì)節(jié)顯示了一個(gè)灰度級(jí)剖面圖和一階與二階導(dǎo)數(shù)的剖面圖 由這些現(xiàn)象我們可以得到的結(jié)論是:一階導(dǎo)數(shù)可以用于檢測(cè)圖像中的一個(gè)點(diǎn)是否是邊緣的點(diǎn)(也就是判斷一個(gè)點(diǎn)是否在斜坡上)。同樣，二階導(dǎo)數(shù)的符號(hào)可以用于判斷一個(gè)邊緣像素是在邊緣亮的一邊還是暗的一邊。我們注意到圍繞一條邊緣，二階導(dǎo)數(shù)的兩條附加性質(zhì)(1)對(duì)圖像中的每條邊緣二階導(dǎo)數(shù)生成兩個(gè)值(一個(gè)不希望得到的特點(diǎn));(2)一條連接二階導(dǎo)數(shù)正極值和負(fù)極值的虛構(gòu)直線將在邊緣中點(diǎn)附近穿過(guò)零點(diǎn)。將在本節(jié)后面說(shuō)明，二階導(dǎo)數(shù)的這個(gè)過(guò)零點(diǎn)的性質(zhì)對(duì)于確定粗邊線的中心非常有用。 最后，注意到某些邊緣模型利用了在進(jìn)人和離開(kāi)斜坡地方的平滑過(guò)渡(習(xí)題10.5)。然而，我們?cè)诮酉聛?lái)的討論中將得出同樣的結(jié)論。而且，這一點(diǎn)從我們使用局部檢測(cè)進(jìn)行處理就可以很明顯地看出(因此，2.5.2節(jié)中對(duì)于邊緣的局部性質(zhì)進(jìn)行了說(shuō)明)。 盡管到此為止我們的注意力被限制在一維水平剖面線范圍內(nèi)，但同樣的結(jié)論可以應(yīng)用于圖像中的任何方向上。我們僅僅定義了一條與任何需要考察的點(diǎn)所在的邊緣方向相垂直的剖面線，并如前面討論的那樣，對(duì)結(jié)果進(jìn)行了解釋。注：出自Digital Image Processing 2nd Edition . Prentice HallImage SegmentationThe material in the previous chapter began a transition from image processing methods whose input and output are images, to methods in which the inputs are images, but the outputs are attributes extracted from those images (in the sense defined is Section 1.1). Segmentation is another major step in that direction.Segmentation subdivides an image into its constituent regions or objects. The level to which the subdivision is carried depends on the problem being solved. That is, segmentation should stop when the objects of interest in an application have been isolated. For example, in the automated inspection of electronic assemblies, interest lies in analyzing images of the products with the objective of determining the presence or absence of specific anomalies, such as missing components or broken connection paths. There is no point in carrying segmentation past the level of detail required to identify those elements.Segmentation of nontrivial images is one of the most difficult tasks in image processing. Segmentation accuracy determines the eventual success or failure of computerized analysis procedures. For this reason, considerable care should be taken to improve the probability of rugged segmentation. In some situations , such as industrial inspection applications, at least some measure of control over the environment is possible at times. The experienced image processing system designer invariably pays considerable attention to such opportunities. In other applications, such as autonomous target acquisition, the system designer has no control of the environment. Then the usual approach is to focus on selecting the types of sensors most likely to enhance the objects of interest while diminishing the contribution of irrelevant image detail. A good example is the use of infrared imaging by the military to detect objects with strong heat signatures , such as equipment and troops in motion.Image segmentation algorithms generally are based on one of two basic properties of intensity values: discontinuity and similarity. In the first category, the approach is to partition an image based on abrupt changes in intensity, such as edges in an image. The principal approaches in the second category are based on partitioning an image into regions that are similar according to a set of predefined criteria. Thresholding, region growing, and region splitting and merging are examples of methods in this category.In this chapter we discuss a number of approaches in the two categories just mentioned. We begin the development with methods suitable for detecting gray level discontinuities such as points, lines, and edges. Edge detection in particular has been a staple of segmentation algorithms for many years. In addition to edge detection per se, we also discuss methods for connecting edge segments and for assembling edges into region boundaries. The discussion on edge detection is followed by the introduction of various thresholding techniques . Thresholding also is a fundamental approach to segmentation that enjoys a significant degree of popularity, especially in applications where speed is an important factor. The discussion on thresholding is followed by the development of several region-oriented segmentation approaches. We then discuss a morphological approach to segmentation called watershed segmentation. This approach is particularly attractive because it combines several of the positive attributes of segmentation based on the techniques presented in the first part of the chapter. We conclude the chapter with a discussion on the use of motion cues for image segmentation.10.1Detection of DiscontinuitiesIn this section we present several techniques for detecting the three basic types of gray-level discontinuities in a digital image: points, lines, and edges. The most common way to look for discontinuities is to run a mask through the image in the manner described in Section 3.5. For the 3 x 3 mask shown in Fig. 10.1, this procedure involves computing the sum of products of the coefficients with the gray levels contained in the region encompassed by the mask. That is. with reference to Eq. (3.5-3). the response of the mask at anv point in the image is given by（10.1.1）FIGURE 10.1 A general 3 x 3 mask.where z; is the gray level of the pixel associated with mask coefficient Wi. As usual, the response of the mask is defined with respect to its center location. The details for implementing mask operations are discussed in Section 3.5.10.1.1 Point DetectionThe detection of isolated points in an image. is straightforward in principle. Using the mask shown in Fig. 10.2(a), we say that a point has been detected at the location on which the mask is centered if|R| T (10.1.2)where T is a nonnegative threshold and R is given by Eq. (10.1-1). Basically,this formulation measures the weighted differences between the center point and its neighbors. The idea is that an isolated point (a point whose gray level is significantly different from its background and which is located in a homogeneous or nearly homogeneous area) will be quite different from its surroundings, and thus be easily detectable by this type of mask. Note that the mask in Fig. 10.2(a) is the same as the mask shown in Fig. 3.39(d) in connection with Laplacian operations. However, the emphasis here is strictly on the detection of points. That is, the only differences that are considered of interest are those large enough (as determined by T, to be considered isolated points. Note that the mask coefficients sum to zero, indicating that the mask response will be zero in areas of constant gray level.-1-1-1-18-1-1-1-1（a）（b） （c） （d）FIGURE 10.2(a) Pointdetection mask. (b) X-ray image of a turbine blade with a porosity. (c) Result of point detection. (d) Result of using Eq. (10.1-2).(Original image courtesy of X-TEK Systems Ltd.)EXAMPLE 10.1:Detection of isolated points in an image.We illustrate segmentation of isolated points from an image with the aid of Fig. 10.2(6), which shows an X-ray image of a jet-engine turbine blade with a porosity in the upper, right quadrant of the image. There is a single black pixel embedded within the porosity. Figure 10.2(c) is the result of applying the point detector mask to the X-ray image, and Fig. 10.2(d) shows the result of using Eq. (10.1.2) with T equal to 90% of the highest absolute pixel value of the image in Fig. 10.2(c). (Threshold selection is discussed in detail in Section 10.3) The single pixel is clearly visible in this image (the pixel was enlarged manually so that it would be visible after printing). This -type of detection process is rather specialized because it is based on single-pixel discontinuities that have a homogeneous background in the area of the detector mask. When this condition is not satisfied, other methods discussed in this chapter are more suitable for detecting gray-level discontinuities.10.1.2 Line DetectionThe next level of complexity is line detection. Consider the masks shown in Fig. 10.3. If the first mask were moved around an image, it would respond more strongly to lines (one pixel thick) oriented horizontally. With a constant background, the maximum response would result when the line passed through the middle row of the mask. This is easily verified by sketching a simple array of 1s with a line of a different gray level (say, 5s) running horizontally through the array. A similar experiment would reveal that the second mask in Fig. 10.3 responds best to lines oriented at +450; the third mask to vertical lines; and the fourth mask to lines in the -450 direction . These directions can be established also by noting that the preferred direction of each mask is weighted with a larger coefficient (i.e., 2) than other possible directions. Note that the coefficients in each mask sum to zero, indicating a zero response from the masks in areas of constant gray level.-12-1-122-1-1-1-12-12-1-12-1-1-1-1-12-12-12-12-1-12-1-1-1-12 Horizontal +45 Vertical -45FIGURE 10.3 Line masks.Let R1, R2, R3, and R4 denote the responses of the masks in Fig. 10.3, from left to right, where the Rs are given by Eq. (10.1-1). Suppose that the four masks are run individually through an image. If, at a certain point in the image, |Ri| |Rj|, for all j i, that point is said to be more likely associated with a line in the direction of mask i. For example, if at a point in the image, |Ri|Rj|, for j = 2, 3. 4, that particular point is said to be more likely associated with a horizontal line. Alternatively, we may be interested in detecting lines in a specified direction. In this case, we would use the mask associated with that direction and threshold its output, as in Eq . (10.1.2). In other words, if we are interested in detecting all the lines in an image in the direction defined by a given mask, we simply run the mask through the image and threshold the absolute value of the result. The points that are left are the strongest responses, which, for lines one pixel thick, correspond closest to the direction defined by the mask. The following example illustrates this procedure.EXAMPLE 10.2:Detection of lines in a specified directionFigure 10.4(a) shows a digitized (binary) portion of a wire-bond mask for an electronic circuit. Suppose that we are interested in finding all the lines that are one pixel thick and are oriented at-45. For this purpose, we use the last mask shown in Fig. 10.3.The absolute value of the result is shown in Fig. 10.4(b). Note that all vertical and horizontal components of the image were eliminated, and that the components of the original image that tend toward a -45 direction（a） （b） （c）FIGURE 10.4 Illustration of line detection (a) Binary wirebond mask.(b) Absolute value of result after processing with -45 line detector. (c) Result of thresholding image. (b) produced the strongest responses in Fig. 10.4(b).In order to determine which lines best fit the mask, we simply threshold this image. The result of using a threshold equal to the maximum value in the image is shown in Fig. 10.4(c).The maximum value is a good choice for a threshold in applications such as this because the input image is binary and we are looking for the strongest responses. Figure 10.4(c) shows in white all points that passed the threshold test. In this case, the procedure extracted the only line segment that was one pixel thick and oriented at -450 (the other component of the image oriented in this direction in the top, left quadrant is not one pixel thick). The isolated points shown in Fig. 10.4(c) are points that also had similarly strong responses to the mask. In the original image, these points and their immediate neighbors are oriented in such as way that the mask produced a maximum response at those isolated locations. These isolated points can be detected using the mask in Fig. 10.2(a) and then deleted, or they could be deleted using morphological erosion, as discussed in the last chapter.10.1.3 Edge DetectionAlthough point and line detection certainly are important in any discussion on segmentation, edge detection is by far the most common approach for detecting meaningful discontinuities in gray level. In this section we discuss approaches for implementing first- and second-order digital derivatives for the detection of edges in an image. We introduced these derivatives in Section 3.7 in the context of image enhancement. The focus in this section is on their properties for edge detection. Some of the concepts previously introduced are restated briefly here for the sake continuity in the discussion.Basic formulation Edges were introduced informally in Section 3.7.1. In this section we look at the concept of a digital edge a little closer. Intuitively, an edge is a set of connected pixels that lie on the boundary between two regions. However, we already went through some length in Section 2.5.2 to explain the difference between an edge and a boundary. Fundamentally, as we shall see shortly, an edge is a local concept whe