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Chapter 10 Image Segmentation
智能视觉理解 实验室
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10.1 Fundamentals Boundary Region of Constant Intensity
Result of Segmentation Result of Segmentation Textured Region Result of Edge Computation
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Horizontal Intensity Profile through the Center
10.2 Point Line and Edge Detection Horizontal Intensity Profile through the Center 𝝏𝒇 𝝏𝒙 = 𝒇 ′ 𝒙 =𝒇 𝒙+𝟏 −𝒇 𝒙 𝝏 𝟐 𝒇 𝝏 𝒙 𝟐 = 𝒇 ′′ 𝒙 =𝒇 𝒙+𝟏 +𝒇(𝒙−𝟏)−𝟐𝒇 𝒙
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A General Spatial Filter Mask
10.2 Point Line and Edge Detection A General Spatial Filter Mask 𝑹= 𝒊=𝟏 𝟗 𝒘 𝒊 𝒛 𝒊
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Point Detection (Laplacian) Mask Result of Convolving the Mask
Detection of Isolated Points Point Detection (Laplacian) Mask Result of Convolving the Mask 𝒈 𝒙,𝒚 = 𝟏 𝐢𝐟 𝑹 𝒙,𝒚 ≥𝑻 𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞 X-ray Image
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Absolute Value of the Laplacian Positive Value of the Laplacian
Line Detection Original Image Laplacian Image Absolute Value of the Laplacian Positive Value of the Laplacian
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Line Detection Line Detection Masks 𝐋𝐢𝐧𝐞 𝒊: 𝑹 𝒊 ≥| 𝑹 𝒋 |
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10.2.3 Line Detection Wire-bond Template Result of +45 Line Detector
Zoomed View of the Top Left Zoomed View of the Bottom Right All Points Whose Value Satisfied 𝒈≥𝑻 All Negative Value set to Zero
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Ideal Representations
Edge Models Ideal Representations Step Edge Ramp Edge Roof Edge
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Edge Models A 1508×1970 image showing (zoomed) actual ramp (bottom, left), step (top, right), and roof edge profiles.
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Horizontal Intensity Profile Separated by an Ideal Vertical Ramp Edge
Edge Models Detail Near the Edge Horizontal Intensity Profile Separated by an Ideal Vertical Ramp Edge
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Ramp Edge Corrupted by Gaussian Noise
First-derivative Second-derivative 𝑵(𝟎,𝟎) Ramp Edge Corrupted by Gaussian Noise 𝑵(𝟎,𝟎.𝟏) 𝑵(𝟎,𝟏) 𝑵(𝟎,𝟏𝟎)
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Gradient Operators 10.2.5 Basic Edge Detection
𝛁𝒇= 𝒈 𝒙 𝒈 𝒚 = 𝝏𝒇 𝝏𝒙 𝝏𝒇 𝝏𝒚 𝛁𝐟 = 𝝏𝒇 𝝏𝒙 𝟐 + 𝝏𝒇 𝝏𝒚 𝟐 𝟏 𝟐 ∞ 𝝏𝒇 𝝏𝒙 + 𝝏𝒇 𝝏𝒚 𝜶 𝒙,𝒚 =𝐭𝐚 𝐧 −𝟏 𝒈 𝒚 𝒈 𝒙
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Basic Edge Detection Using the gradient to determine edge strength and direction at a point.
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One-dimensional Masks
Basic Edge Detection One-dimensional Masks 𝒈 𝒙 = 𝝏𝒇(𝒙,𝒚) 𝝏𝒙 =𝒇 𝒙+𝟏,𝒚 −𝒇(𝒙,𝒚) 𝒈 𝒚 = 𝝏𝒇(𝒙,𝒚) 𝝏𝒚 =𝒇 𝒙,𝒚+𝟏 −𝒇(𝒙,𝒚)
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Roberts Cross Gradients
Basic Edge Detection Roberts Cross Gradients 𝒈 𝒙 = 𝒛 𝟗 − 𝒛 𝟓 𝒈 𝒚 = 𝒛 𝟖 − 𝒛 𝟔 𝛁𝒇 ≈ 𝒛 𝟗 − 𝒛 𝟓 𝟐 + 𝒛 𝟖 − 𝒛 𝟔 𝟐 𝟏 𝟐 𝛁𝒇 ≈ 𝒛 𝟗 − 𝒛 𝟓 +| 𝒛 𝟖 − 𝒛 𝟔 | Prewitt Operators 𝒈 𝒙 = 𝒛 𝟕 + 𝒛 𝟖 + 𝒛 𝟗 −( 𝒛 𝟏 + 𝒛 𝟐 + 𝒛 𝟑 ) 𝒈 𝒚 = 𝒛 𝟑 + 𝒛 𝟔 + 𝒛 𝟗 −( 𝒛 𝟏 + 𝒛 𝟒 + 𝒛 𝟕 ) 𝒗𝒂𝒓𝒊𝒂𝒕𝒊𝒐𝒏 𝒈 𝒙 = 𝒛 𝟕 + 𝟐𝒛 𝟖 + 𝒛 𝟗 −( 𝒛 𝟏 + 𝟐𝒛 𝟐 + 𝒛 𝟑 ) 𝒈 𝒚 = 𝒛 𝟑 + 𝟐𝒛 𝟔 + 𝒛 𝟗 −( 𝒛 𝟏 + 𝟐𝒛 𝟒 + 𝒛 𝟕 )
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Y-Direction X-Direction
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Basic Edge Detection Diagonal Edge 45-Direction -45-Direction
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Basic Edge Detection Using the Sobel Mask Original Image | 𝒈 𝒙 | | 𝒈 𝒚 | 𝒈 𝒙 +| 𝒈 𝒚 |
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Gradient Angle Image 10.2.5 Basic Edge Detection
𝜶 𝒙,𝒚 =𝐭𝐚 𝐧 −𝟏 𝒈 𝒚 𝒈 𝒙
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Smoothed using a 5×5 averaging filter
Basic Edge Detection Smoothed using a 5×5 averaging filter Original Image | 𝒈 𝒙 | | 𝒈 𝒚 | 𝒈 𝒙 +| 𝒈 𝒚 |
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Diagonal Edge Detection
Basic Edge Detection Diagonal Edge Detection Sobel 45 Sobel -45
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Laplacian Masks 10.2.5 Basic Edge Detection
𝛁 𝟐 𝒇= 𝝏 𝟐 𝒇 𝝏 𝒙 𝟐 + 𝝏 𝟐 𝒇 𝝏 𝒚 𝟐 𝟒 𝑵 ′ 𝒔: 𝛁 𝟐 𝒇=𝟒 𝒛 𝟓 − 𝒛 𝟐 + 𝒛 𝟒 + 𝒛 𝟔 + 𝒛 𝟖 𝟖 𝑵 ′ 𝒔: 𝛁 𝟐 𝒇=𝟒 𝒛 𝟓 −( 𝒛 𝟏 + 𝒛 𝟐 + 𝒛 𝟑 + 𝒛 𝟒 + 𝒛 𝟓 + 𝒛 𝟔 + 𝒛 𝟕 + 𝒛 𝟖 + 𝒛 𝟗 )
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Negative of the LoG 10.2.6 Marr-Hildreth Edge Detector
𝑮 𝒙,𝒚 = 𝒆 − 𝒙 𝟐 + 𝒚 𝟐 𝟐 𝝈 𝟐 𝛁 𝟐 𝑮 𝒙,𝒚 = 𝒙 𝟐 + 𝒚 𝟐 −𝟐 𝝈 𝟐 𝝈 𝟒 𝒆 − 𝒙 𝟐 + 𝒚 𝟐 𝟐 𝝈 𝟐
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10.2.6 Marr-Hildreth Edge Detector
Original Image 𝝈=𝟒, 𝒏=𝟐𝟓 Threshold: 0 Threshold: 4
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Gaussian Smoothing Function
Marr-Hildreth Edge Detector Original Image Sobel Gradient Gaussian Smoothing Function Laplacian Mask LoG Threshold LoG Zero Crossings
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Negatives of the LoG (solid) and DoG (dotted)
Marr-Hildreth Edge Detector Negatives of the LoG (solid) and DoG (dotted) Ratio: 1.75:1 Ratio: 1.6:1
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Canny Edge Detector Range of values (in gray) of 𝜶 Two possible orientation of a horizontal edge The angle ranges of the edge normals for the four types of edge directions in a neighborhood.
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10.2.6 Canny Edge Detector Original Image Thresholded Gradient
Marr-Hildreth Canny
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10.2.6 Canny Edge Detector Original Image Thresholded Gradient
Marr-Hildreth Canny
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10.2.7 Edge Linking and Boundary Detection
𝑴 𝒔,𝒕 −𝑴 𝒙,𝒚 ≤𝑬 𝜶 𝒔,𝒕 −𝜶 𝒙,𝒚 ≤𝑨 Original Image Gradient magnitude |𝑮 𝒙 | Morphological thinning |𝑮 𝒚 | 𝐋𝐨𝐠𝐢𝐜𝐚𝐥 𝐎𝐑
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10.2.7 Edge Linking and Boundary Detection
Illustration of the iterative polygonal fit algorithm
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10.2.7 Edge Linking and Boundary Detection
Edge linking using a polygonal approximation
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Morphological Shrinking Morphological cleaning
Gradient Image Majority Filter Morphological Shrinking Morphological cleaning Spur Reduction Skeleton T = 3 T = 6 T = 12 Smoothed Smoothed
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( 𝒙 𝒊 , 𝒚 𝒊 ) & 𝒚=𝒂𝒙+𝒃⇒ 𝒚 𝒊 =𝒂 𝒙 𝒊 +𝒃
Edge Linking and Boundary Detection Global processing using the Hough transform ( 𝒙 𝒊 , 𝒚 𝒊 ) & 𝒚=𝒂𝒙+𝒃⇒ 𝒚 𝒊 =𝒂 𝒙 𝒊 +𝒃 𝒃=− 𝒙 𝒊 𝒂+ 𝒚 𝒊 All ( 𝒙 𝒊 , 𝒚 𝒊 )’s on a line intersect each other at (𝒂,𝒃 )
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10.2.7 Edge Linking and Boundary Detection
Subdivision of the parameter plane for use in the Hough transform
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10.2.7 Edge Linking and Boundary Detection
Geometrical interpretation of the parameters 𝒙 𝒊 𝒄𝒐𝒔𝜽+ 𝒚 𝒊 𝒔𝒊𝒏𝜽=𝝆 xy-plane 𝝆𝜽-plane Accumulator cells
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Image containing five points
Edge Linking and Boundary Detection Image containing five points Parameter space
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10.2.7 Edge Linking and Boundary Detection
Canny Hough parameter space Lines superimposed
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10.3 Thresholding Original H-T Thr-Grad Linked
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𝑭 𝒙,𝒚 >𝑻 𝐭𝐡𝐞𝐧 𝒙,𝒚 𝐢𝐬 𝐛𝐞𝐥𝐨𝐧𝐠 𝐭𝐨 𝐨𝐛𝐣𝐞𝐜𝐭,
10.3 Thresholding 𝑭 𝒙,𝒚 >𝑻 𝐭𝐡𝐞𝐧 𝒙,𝒚 𝐢𝐬 𝐛𝐞𝐥𝐨𝐧𝐠 𝐭𝐨 𝐨𝐛𝐣𝐞𝐜𝐭, 𝐞𝐥𝐬𝐞 𝒙,𝒚 𝐢𝐬 𝐛𝐞𝐨𝐧𝐠 𝐭𝐨 𝐛𝐚𝐜𝐤𝐠𝐫𝐨𝐮𝐧𝐝 Bi-level (T) Multi-level ( 𝐓 𝟏 , 𝐓 𝟐 ,…, 𝐓 𝐧 ) Threshold image: 𝒈 𝒙,𝒚 = 𝟏 𝒇 𝒙,𝒚 >𝑻 𝟎 𝒇 𝒙,𝒚 ≤𝑻
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10.3 Thresholding Thresholds
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10.3 Thresholding Noiseless 𝝁=𝟎 𝝈=𝟏𝟎 𝝁=𝟎 𝝈=𝟓𝟎
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10.3 Thresholding Noised Image Ramp in [0.2, 0.6] Product
![10.3 Thresholding Noised Image Ramp in [0.2, 0.6] Product](http://slideplayer.com/slide/13066226/79/images/44/10.3+Thresholding+Noised+Image+Ramp+in+%5B0.2%2C+0.6%5D+Product.jpg "10.3 Thresholding Noised Image Ramp in [0.2, 0.6] Product")
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10.3.2 Basic Global Thresholding
A Heuristic Approach Initial estimate for T Segment image to 𝑮 𝟏 (>𝑻) and 𝑮 𝟐 (≤𝑻) Compute average value 𝒎 𝟏 and 𝒎 𝟐 in 𝑮 𝟏 and 𝑮 𝟐 𝑻= 𝟏 𝟐 𝒎 𝟏 + 𝒎 𝟐 Repeat 2-4 until small changed
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10.3.2 Basic Global Thresholding
An example of segmentation based on a threshold 𝑻 𝟎 =𝟎 𝑪𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝑻 𝒇 =𝟏𝟐𝟓.𝟒
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10.3.2 Basic Global Thresholding
Local Thr. Mosaic
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10.3.2 Basic Global Thresholding
Good Subimage Bad Subimage Further block-ing
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10.3.3 Optimum Global Thresholding Using Otsu’s Method
𝑷 𝟏 𝒌 = 𝒊=𝟎 𝒌 𝒑 𝒊 , 𝑷 𝟐 𝒌 = 𝒊=𝒌−𝟏 𝑳−𝟏 𝒑 𝒊 =𝟏− 𝑷 𝟏 𝒌 𝒑 𝒊 = 𝒏 𝒊 𝑴𝑵 Mean intensity value: 𝒎 𝟏 𝒌 = 𝟏 𝑷 𝟏 𝒌 𝒊=𝟎 𝒌 𝒊 𝒑 𝒊 , 𝒎 𝟐 𝒌 = 𝟏 𝑷 𝟐 (𝒌) 𝒊=𝒌+𝟏 𝑳−𝟏 𝒊 𝒑 𝒊 𝒎 𝒌 = 𝒊=𝟎 𝒌 𝒊 𝒑 𝒊 , 𝒎 𝑮 = 𝒊=𝟎 𝑳−𝟏 𝒊 𝒑 𝒊 Separability measure: 𝜼 𝒌 = 𝝈 𝑩 𝟐 𝒌 𝝈 𝑮 𝟐 , 𝝈 𝑮 𝟐 = 𝒊=𝟎 𝑳−𝟏 𝟏− 𝒎 𝑮 𝟐 𝒑 𝒊 , 𝝈 𝑩 𝟐 𝒌 = 𝒎 𝑮 𝑷 𝟏 𝒌 −𝒎 𝒌 𝟐 𝑷 𝟏 𝒌 𝟏− 𝑷 𝟏 𝒌
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Basic Global Algorithm
Optimum Global Thresholding Using Otsu’s Method Input Histogram Basic Global Algorithm Ostu’s Method
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10.3.4 Using Image Smoothing to Improve Global Thresholding
Noised Image Ostu’s Method 55 Averaging Mask Ostu’s Method
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10.3.4 Using Image Smoothing to Improve Global Thresholding
Noised Image Ostu’s Method 55 Averaging Mask Ostu’s Method Failed in both cases
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10.3.5 Using Edges to Improve Global Thresholding
Noisy Image Gradient Magnitude Product Ostu’s Method
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Threshold Absolute Laplacian
Using Edges to Improve Global Thresholding Ostu’s Method Threshold Absolute Laplacian Ostu’s Method
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Lower Value to Threshold the Absolute Laplacian
Using Edges to Improve Global Thresholding Lower Value to Threshold the Absolute Laplacian
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10.3.6 Multiple Thresholds Between-class variance:
𝝈 𝑩 𝟐 = 𝒌=𝟏 𝑲 𝑷 𝒌 𝒎 𝒌 − 𝒎 𝑮 𝟐 , 𝑷 𝒌 = 𝒊∈ 𝑪 𝒌 𝒑 𝒊 , 𝒎 𝒌 = 𝟏 𝑷 𝒌 𝒊∈ 𝑪 𝒌 𝒊 𝒑 𝒊 Separability measure: 𝜼 𝒌 𝟏 ∗ , 𝒌 𝟐 ∗ = 𝝈 𝑩 𝟐 𝒌 𝟏 ∗ , 𝒌 𝟐 ∗ 𝝈 𝑮 𝟐 , 𝝈 𝑮 𝟐 𝒌 𝟏 ∗ , 𝒌 𝟐 ∗ = max 𝟎< 𝒌 𝟏 < 𝒌 𝟐 <𝑳−𝟏 𝝈 𝑩 𝟐 ( 𝒌 𝟏 , 𝒌 𝟐 ) Threshold image: 𝒈(𝒙,𝒚)= 𝒂 𝐢𝐟 𝒇 𝒙,𝒚 ≤ 𝒌 𝟏 ∗ 𝒃 𝐢𝐟 𝒌 𝟏 ∗ <𝒇 𝒙,𝒚 ≤ 𝒌 𝟐 ∗ 𝒄 𝐢𝐟 𝒇 𝒙,𝒚 > 𝒌 𝟐 ∗
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Multiple Thresholds Image Segmented into Three Regions Using Dual Ostu Thresholds
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Iterative Global Algorithm
Variable Thresholds Noisy, Shaded Image Iterative Global Algorithm Ostu’s Method Subdivided Ostu’s Method
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Variable Thresholds Histograms of the Six Subimages
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Variable Thresholding Based on Local Image Properties
Variable Thresholds Variable Thresholding Based on Local Image Properties 𝑻 𝒙𝒚 =𝒂 𝝈 𝒙𝒚 +𝒃 𝒎 𝒙𝒚 𝑻 𝒙𝒚 =𝒂 𝝈 𝒙𝒚 +𝒃 𝒎 𝑮 Segmented image 𝒈 𝒙,𝒚 = 𝟏 𝐢𝐟 𝒇 𝒙,𝒚 > 𝑻 𝒙𝒚 𝟎 𝐢𝐟 𝒇 𝒙,𝒚 ≤ 𝑻 𝒙𝒚 Predicts based on the parameters: 𝒈 𝒙,𝒚 = 𝟏 𝐢𝐟 𝑸 𝐥𝐨𝐜𝐚𝐥 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫𝐬 𝐢𝐬 𝐭𝐫𝐮𝐞 𝟎 𝐢𝐟 𝑸 𝐥𝐨𝐜𝐚𝐥 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫𝐬 𝐢𝐬 𝐟𝐚𝐥𝐬𝐞 Q 𝝈 𝒙𝒚 , 𝒎 𝒙𝒚 = 𝐭𝐫𝐮𝐞 𝐢𝐟 𝒇 𝒙,𝒚 >𝒂 𝝈 𝒙𝒚 𝐀𝐍𝐃 𝒇 𝒙,𝒚 >𝒃 𝒎 𝒙𝒚 𝐟𝐚𝐥𝐬𝐞 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
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Dual Thresholding Approach Local Standard Deviations
Variable Thresholds Original Image Dual Thresholding Approach Local Standard Deviations Local Thresholding
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Moving Average: 10.3.7 Variable Thresholds
𝒎 𝒌+𝟏 =𝒎 𝒌 + 𝟏 𝒏 ( 𝒛 𝒌+𝟏 − 𝒛 𝒌−𝒏 ) Ostu’s Method Moving Average
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Variable Thresholds Original Image Ostu’s Method Moving Average
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10.4 Region-Based Segmentation
𝒊=𝟏 𝒏 𝑹 𝒊 𝑹 𝒊 𝐚𝐫𝐞 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝 𝐫𝐞𝐠𝐢𝐨𝐧𝐬 𝐑 𝐢 𝑹 𝒋 =∅ , 𝒊≠𝒋 𝑷 𝑹 𝒊 =𝐓𝐑𝐔𝐄 𝑷 𝑹 𝒊 𝑹 𝒋 =𝐅𝐀𝐋𝐒𝐄
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Smallest Dual Thresholds
Initial Seed Image Final Seed Image Difference Smallest Dual Thresholds Dual Thresholds Region Growing
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10.4 Region-Based Segmentation
Split and Merging
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Limit the Smallest Allowed Quadregion
10.4 Region-Based Segmentation Limit the Smallest Allowed Quadregion 3232 1616 88
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10.5 Segmentation Using Morphological Watersheds
Flooding
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10.5 Segmentation Using Morphological Watersheds
Further Flooding Merging Water Longer Dams Final Watershed
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Dam Construction Stage n-1 Stage n Structuring Element Result
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10.5.3 Water Segmentation Algorithm
Image Gradient Watershed Lines Superimposed on Input
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The Use of Markers Apply the Watershed Segmentation to the Gradient Image
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The Use of Markers Internal Markers and External Markers
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10.6 The Use of Motion in Segmentation
Absolute ADIs Positive ADIs Negative ADIs
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10.6 The Use of Motion in Segmentation
Two Frames in a Sequence Result
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10.6 The Use of Motion in Segmentation
LANDSAT frame
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10.6 The Use of Motion in Segmentation
Intensity plot of the image LANDSAT, with the target circle
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10.6 The Use of Motion in Segmentation
Spectrum of Eq. (10.6-8) showing a peak at 𝑢 1 =3
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10.6 The Use of Motion in Segmentation
Spectrum of Eq. (10.6-9) showing a peak at 𝑢 2 =4
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智能视觉理解 实验室
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