# Joint probability distribution of two images

• atrus_ovis
In summary, the conversation discusses the concept of joint probability distributions in the context of image processing. It explains that the joint probability is calculated by dividing the number of pixels with a specific value in image A and the same value in the corresponding pixel in image B by the total number of pixels. The conversation also clarifies that this calculation is done for all possible pixel value combinations, not just specific ones. Finally, it mentions that there are multiple ways to define joint distributions for images.
atrus_ovis
I'm posting this here, as i feel it's more probability-related than image processing.

I'm reading this lecture pdf.
At end of page 1 , beginning of page 2 it says:
The spatial information obviously required for a registration method is provided by the
definition of a joint probability distribution that depends simultaneously on image A and B.

The conventional expression for it is pA,B (a, b). It is calculated as the number of times out
of the total number of pixels N that a pixel in A contains the value a and the same pixel
that is, the pixel in the same image position, in B contains the value b; this number of pixels
is then divided by the total number of pixels to give the joint probability of a, b.

Is this a bit vague, or am i missing something?
Since we are calculating frequency of the value a in the pixels of image A, when was the information about any position introduced?

Or does it mean that for *any* pixel in image A that contains the value a, check if the pixel(s) in image B at the corresponding position have the value b?

If this is too specific or too dependent on image processing,can you provide some material to build my intuition about joint probability distributions?

Thank you.

Let's say the images are
$$A = \begin{pmatrix} 1&1& 2 \\ 2&7&1 \end{pmatrix}$$
and
$$B = \begin{pmatrix} 1&2&2 \\ 2&1&2 \end{pmatrix}$$

My interpretation of that passage is that

$$p_{A,B}(1,1) = \frac{1}{6}$$
$$p_{A,B}(1,2) = \frac{2}{6}$$
$$p_{A,B}(1,7) = \frac{0}{6}$$

$$p_{A,B}(2,1) = \frac{0}{6}$$
$$p_{A,B}(2,2) = \frac{2}{6}$$
$$p_{A,B}(2,7) = \frac{0}{6}$$

$$p_{A,B}(7,1) = \frac{1}{6}$$
$$p_{A,B}(7,2) = \frac{0}{6}$$
$$p_{A,B}(7,7) = \frac{0}{6}$$

Although there are (3)(3) = 9 possible ordered pairs of pixel values to consider, the total match-ups will only be 6.

As to whether we should call this "the" joint distribution of two images - that is merely the terminology that those authors wish to use. There are many many ways to take properties of images and define joint distributions.

## What is a joint probability distribution of two images?

A joint probability distribution of two images is a mathematical representation of the likelihood of two specific events occurring together. It shows the probability of both images appearing at the same time or in the same context.

## How is a joint probability distribution of two images calculated?

The joint probability distribution of two images is calculated by multiplying the individual probabilities of each image occurring. It can also be calculated by dividing the number of times the two images appear together by the total number of times both images appear.

## What is the purpose of a joint probability distribution of two images?

The purpose of a joint probability distribution of two images is to understand the relationship between two events and their likelihood of occurring together. It can also be used to make predictions about future occurrences of the two images.

## How does a joint probability distribution of two images differ from a single image probability distribution?

A joint probability distribution of two images considers the probability of two events occurring together, while a single image probability distribution only looks at the probability of one event occurring. Additionally, a joint probability distribution takes into account the relationship between the two images, while a single image probability distribution does not.

## Can a joint probability distribution of two images be used to analyze more than two images at a time?

Yes, a joint probability distribution can be used to analyze any number of images at a time. However, as the number of images increases, the complexity of the calculations also increases. It is important to carefully select which images to analyze together to ensure meaningful results.

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