introduction image processing >introduction & image … · local adaptive thresholding....
TRANSCRIPT
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IMAGE PROCESSING>AUTOMATIC THRESHOLDING
UTRECHT UNIVERSITYRONALD POPPE
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OUTLINEHistogram-based thresholding
Local adaptive thresholding
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HISTOGRAM-BASED THRESHOLDING
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AUTOMATIC THRESHOLDINGWe have discussed operations on binary images
• What we need is a way to convert grayscale to binary
Requires setting a threshold
• Domain- and image-dependent: there is no single best threshold• Manual setting is not ideal (takes time, is subjective)
We focus on determining the threshold automatically
• Information used only from the grayscale image
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AUTOMATIC THRESHOLDING2
Two options:
• Global thresholding: single optimal threshold• Local thresholding: threshold per pixel
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GLOBAL THRESHOLDING
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GLOBAL THRESHOLDINGGoal: Find a single threshold q that is applied to each pixel of grayscale image I:
• Divides the image into two disjoint sets C0 (background) and C1 (foreground)
•
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GLOBAL THRESHOLDING2
Essential information present in histogram
We can identify two types of global thresholding methods:
• Shape-based: look for peaks, valleys, etc.• Statistical: use means, variance, etc.
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GLOBAL THRESHOLDING3
We consider histograms h(g) of grayscale image I with N pixels and Kpossible intensity values 0 ≤ 𝑔𝑔 ≤ 𝐾𝐾
We can calculate the mean and variance from the histogram:
• Mean:
• Variance:
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GLOBAL THRESHOLDING4
If we apply threshold q, we obtain two disjoint sets with number of pixels:
•
Each of these sets is a histogram and we can again calculate their means:
•
•
Relation to overall mean:
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GLOBAL THRESHOLDING5
Variance per set can also be easily calculated
• No trivial relation to overall variance
We want to find a threshold q to split the values into background and foreground
To set threshold q, we can simply:
• Set q = mean(I)• Set q = median(I)• Set q to be the average of the minimum and maximum:
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GLOBAL THRESHOLDING6
These measures are typically suboptimal:
• For median, equal number of pixels in foreground and background are assumed• For the mid-range technique, extreme pixel values largely affect the result
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GLOBAL THRESHOLDING7
Mean, median, mid-range:
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GLOBAL THRESHOLDING8
If we know in advance which percentage b of the pixels is background, we can determine q as:
•
• Typically, we don’t know this percentage
We now discuss specific threshold finding algorithms
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ISODATA ALGORITHMAssumptions:
• Image consists of two distributions: background and foreground• Variances of the two distributions are assumed equal
Initially, q is set to the mean or median of histogram h(g)
Iterative steps:
• The means of the foreground and background are calculated• q is repositioned to the average• Repeat until convergence (q doesn’t change between iterations)
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ISODATA ALGORITHM2
Simple algorithm, but suffers when number of pixels between classes is biased
Example:
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OTSU’S METHODAssumption:
• Image consists of two distributions: background and foreground
Goal is to find q such that:
1. The variances of each distribution are minimal (within)2. The distance between the means is maximal (between)
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OTSU’S METHOD2
The within-class variance:
•
• With P0(q) and P1(q) the class probabilities:
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OTSU’S METHOD2
Between-class variance:
•
The total variance is the sum of within- and between-class variance:
•
Since total variance is constant for a given image, we can either minimize within or maximize between:
• Maximizing between is easier (means are easier to calculate than variances)
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OTSU’S METHOD3
Construct table of between-class variances as a function of q
• Use a for-loop to find optimal value for q
In red, between-class variance
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MAXIMUM ENTROPY
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MAXIMUM ENTROPYWe define the probability of an intensity value g to occur as:
• p(g) = p(I(u,v) = g)• Termed prior (a priori) probabilities
Probability distribution is the vector of probabilities for all possible values:
• (p(0), p(1), … , p(K-1))
Estimate of the probability distribution comes from the histogram:
•
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MAXIMUM ENTROPY2
We define the cumulative distribution function as:
•
• Here, P(0) = p(0) and P(K – 1) = 1
We define entropy as:
•
• logb(x) is the logarithm of x to the base b
g=1
K-1
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MAXIMUM ENTROPY3
Entropy is a measure of the “surprise” of the values
When there is only one intensity value:
• p(g) = 0 for all g expect for one value• H(I) = 0 so entropy is minimal (because logb(1)=0)
When all intensities have the same probability (1/K):
• Entropy is maximum, H(I) = log(K)
Entropy is therefore always in range [0, log(K)]
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MAXIMUM ENTROPY4
Entropy can be used as a criterion for threshold selection
Given threshold q, the probability distributions for C0 and C1 are:
•
• with
• P0(q) and P1(q) are the cumulative probabilities for the pixels in the background and foreground
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MAXIMUM ENTROPY5
Given a threshold q, we can calculate the entropy within each partition:
•
•
The overall entropy is the sum of both: H01(q) = H0(q) + H1(q)
Since we aim for the maximum entropy, we search for q that maximizes H01
• Algorithm for determining q is presented in the book
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MAXIMUM ENTROPY6
Background H0 (green), foreground H1 (blue) and overall H01 (red) entropy
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MAXIMUM ENTROPY7
Maximum entropy algorithm is relatively fast O(K)
• Calculations based on image histogram
Extensions make use of local structure of the image (see later algorithms)
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QUESTIONS?
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MINIMUM ERROR THRESHOLDINGGoal of minimum error thresholding is to optimally fit Gaussians to a probability distribution
• Derived from image histogram
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MINIMUM ERROR THRESHOLDING2
Idea: each pixel originates from either foreground (C1) or background (C0) class
• Both classes are modeled as Gaussian distribution (with μ and σ2)
We now need to determine for each pixel value x whether it belongs to C0 or C1
Probability that value x belongs to background is p(x | C0)
• Probability that value x is observed given that it is background• Requires that we know what is background
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MINIMUM ERROR THRESHOLDING3
We are interested in the reverse problem: P(C0 | x) and P(C1 | x)
• For each x, select the class with highest probability: foreground or background
We can calculate these using Bayes’ theorem:
•
• 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙∗𝑝𝑝𝑝𝑝𝑙𝑙𝑙𝑙𝑝𝑝𝑙𝑙𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒𝑙𝑙
• Posterior: probability that we are interested in• Prior: general knowledge how often Cj occurs• Likelihood: conditional probability• Evidence: what we can observe only “scales” result
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MINIMUM ERROR THRESHOLDING4
𝑝𝑝 𝐴𝐴 𝐵𝐵 =𝑝𝑝 𝐵𝐵 𝐴𝐴 𝑝𝑝(𝐴𝐴)
𝑝𝑝(𝐵𝐵)
Example:
• A = it is raining (3 days per week)• B = streets are wet (4 days per week)• 𝑝𝑝 𝐵𝐵 𝐴𝐴 = 0.9, 90% chance on wet streets when it is raining• Calculate the probability it was raining when the streets are wet:
• 𝑝𝑝 𝐴𝐴 𝐵𝐵 = 𝑝𝑝 𝐵𝐵 𝐴𝐴 𝑝𝑝(𝐴𝐴)𝑝𝑝(𝐵𝐵)
=0.9∗3747
= 2740
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MINIMUM ERROR THRESHOLDING5
We can thus formulate the Bayes’ decision rule:
•
• Also called minimum error criterion
If we model P(x | C0) and P(x | C0) as Gaussian distributions, the (scaled) probability distribution becomes:
•
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MINIMUM ERROR THRESHOLDING6
With some work, we can derive the measure of potential error of classifying value x as class Cj:
•
We can thus make a decision for value x as follows:
•
Assumption is that classes are distributed normally, and parameters are well estimated
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MINIMUM ERROR THRESHOLDING7
When we apply a threshold q, all values g ≤ q are considered background
Criterion function e(q) of this classification is:
•
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MINIMUM ERROR THRESHOLDING8
Our goal is now to find the q that minimizes e(q)
• Requires the prior probabilities and the means and variances of both classes
Prior probabilities are obtained by “counting values”:
•
• Means and variances also calculated from histogram
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MINIMUM ERROR THRESHOLDING9
Again, algorithm is relatively fast O(K) due to calculations on image histogram
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QUESTIONS?
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LOCAL ADAPTIVE THRESHOLDING
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LOCAL ADAPTIVE THRESHOLDINGOften, a single threshold is not optimal due to local changes in intensity in the image
Local methods take into account the pixel location (u,v) and have varying thresholds Q(u,v) per pixel
• We discuss two methods that differ in how to derive Q from an input image
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BERNSENS’ METHODFind a dynamic threshold based on the minimum and maximum pixel value of a neighborhood R(u,v) around (u,v)
•
The threshold is then determined to be the mid-range value:
•• Only when Imax and Imin differ sufficiently• When (Imax(u,v) – Imin(u,v)) < cmin, all pixels are assigned to the background
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BERNSENS’ METHOD2
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BERNSENS’ METHOD3
Less predictable results with smoothly varying backgrounds
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BERNSENS’ METHOD4
Clear relation with morphological filters:
• Min and max filter: erosion and dilation
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NIBLACK’S METHODNiblack’s method estimates Q(u,v) as a function of local intensity average μR(u,v) and standard deviation σR(u,v):
•• Q(u,v) is set to the mean intensity plus K times the standard deviation
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NIBLACK’S METHOD2
In even areas, σR(u,v) is small and the threshold is close to the mean intensity
• Can be solved by adding a constant:
Original formulation assumes dark background
• Inverse formulation used when background is lighter
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NIBLACK’S METHOD3
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NIBLACK’S METHOD4
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QUESTIONS?
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ASSIGNMENT 3
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ASSIGNMENT 3Object detection/recognition
• Pick an object class in a given context• Search at least 10 images with variation and 10 images with distractions
Two/three phases:
1. Pre-processing2. Object detection3. Refinement (only phase in which you are allowed to use color!)
Hand in report, code and images
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NEXT LECTURE
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NEXT LECTURENext lecture is about:
• Comparing Images (Book II, Chapter 11)• Wednesday October 24, 15:15 - 17:00 (RUPPERT-PAARS)
Deadline for Assignment 3 (Shape detection):
• November 11, 23:00• Walk-in session Friday 26-10 and 2-11, 9:00-10:45
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CONTENTS OF THIS LECTURE
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CONTENTS OF THIS LECTUREAdvanced Methods (book III)
• Chapter 2: Automatic Thresholding (not 2.3 and 2.4)