Perhaps an example will make this concept clearer. Further on, this CDF is multiplied by levels, to find the new pixel intensities, which are mapped into old values, and your histogram is equalized.A cumulative distribution function, \(F(x)\), gives the probability that the random variable \(X\) is less than or equal to \(x\):īy analogy, this concept is very similar to the cumulative relative frequency.Ī cumulative distribution is the sum of the probabilities of all values qualifying as “less than or equal” to the specified value. And CDF gives us the cumulative sum of these values. So PMF helps us calculating the probability of each pixel value in an image. Since in histogram equalization, we have to equalize all the pixel values of an image. In the histogram equalization, the first and the second step are PMF and CDF. PMF and CDF are both use in histogram equalization as it is described in the beginning of this tutorial. Histogram equalization is used for enhancing the contrast of the images. Histogram equalization is discussed in the next tutorial but a brief introduction of histogram equalization is given below. PMF and CDF usage in histogram equalization Histogram equalization The third value of PMF is added in the second value of CDF, that gives 110/110 which is equal to 1.Īnd also now, the function is growing monotonically which is necessary condition for histogram equalization. The second value of PMF is added in the first value and placed over 128. Now as you can see from the graph above, that the first value of PMF remain as it is.
Here is the CDF of the above PMF function. We will simply keep the first value as it is, and then in the 2nd value, we will add the first one and so on. Since this histogram is not increasing monotonically, so will make it grow monotonically. Consider the histogram shown above which shows PMF. It is a function that calculates the cumulative sum of all the values that are calculated by PMF. What is CDF?ĬDF stands for cumulative distributive function. So in order to increase it monotonically, we will calculate its CDF. So the PMF of the above histogram is this.Īnother important thing to note in the above histogram is that it is not monotonically increasing. Now if we have to calculate its PMF, we will simple look at the count of each bar from vertical axis and then divide it by total count. The above histogram shows frequency of gray level values for an 8 bits per pixel image. Note that the sum of the count must be equal to total number of values. After count they can either be represented in a histogram, or in a table like this below. Now if we were to calculate the PMF of this matrix, here how we are going to do it.Īt first, we will take the first value in the matrix, and then we will count, how much time this value appears in the whole matrix. Then we will take another example in which we will calculate PMF from the histogram. Also, note that the CDF is defined for all x R. Note that the subscript X indicates that this is the CDF of the random variable X. First from a matrix, because in the next tutorial, we have to calculate the PMF from a matrix, and an image is nothing more then a two dimensional matrix. Use probability plots to see your data and visually check model assumptions Probability plots are simple visual ways of summarizing reliability data by plotting CDF estimates versus time using a log-log scale. The cumulative distribution function (CDF) of random variable X is defined as.
We will calculate PMF from two different ways. As it name suggest, it gives the probability of each number in the data set or you can say that it basically gives the count or frequency of each element. PMF stands for probability mass function.
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So if you dont know how to calculate PMF and CDF, you can not apply histogram equalization on your image. It is because these two concepts of PMF and CDF are going to be used in the next tutorial of Histogram equalization. Now the question that should arise in your mind, is that why are we studying probability. PMF and CDF both terms belongs to probability and statistics.
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