Hello and welcome back, in this segment we will discuss homomorphic filtering. It is a concept developed in the 60s by Oppenheim Schaefer in Stockholm, according to which a nonlinear mapping is first applied to a signal, then linear filtering is performed with this new domain. And finally, the signal is transformed back to the original domain by applying the inverse transformation. The concept is general, and can be applied to the processing of any signal. For example, speech, audio and, of course, images. This concept is utilized here with the objective of decreasing the dynamic range while increasing the local contrast of an image. Based on a model of an image, as the product of a slowly varying illumination term, and the fast varying reflectance term, after taking the logarithm of the image, the two terms are now additive. They can then be separated by LSI filtering. Then, in the frequency domain, the illumination term is reduced while the reflectance term is boosted, exponentiation is then performed, the inverse of the log, so as to bring the image back to its original domain. We will explain all these steps in this segment and also show an imaging sample. So, let us proceed with this topic. And, I hope you find it interesting and exciting. The interesting concept of homomorphic filtering, as applied specifically to an image enhancement task, is presented next. We'll introduce homomorphic filtering in the context of solving a specific enhancement problem that we will explain right away. However, it should become clear with homomorphic filtering is a more general framework that can be used to solve various problems. So, the problem at hand is that we have a image with a large dynamic range, such as a natural scene on a bright day. And, such an image is recorded on a, on a medium with a small dynamic range. And the result is that there is a significant contrast reduction, especially in the dark and bright regions. So, therefore, the objective is, given an image, how can we reduce the dynamic range while increasing the local contrast. We model given image x as a product of two terms, the illumination term i n1 and 2, which is assumed to be slowly varying. A low frequency term and contributes mainly to the dynamic range. And this is multiplied by the reflectance, which represents the details and is therefore rapidly varying, and is the primary contributor to the local contract. So, with this model, the problem we set out to solve is to decrease the dynamic range, therefore decrease the term i n1 and 2, while increasing the local contrast as expressed by the reflectance. Clearly the difficulty here, the challenge is that I have the product of these two terms, the illumination with the reflectance and I would, therefore, like to separate them before being able to process each one separately according to my objectives. [BLANK_AUDIO] Homomorphic processing comes to our rescue. The main idea behind homomorphic processing is to take the original signal, the original image, and through the logarithmic operation, map it onto another domain. Then we perform our processing, our desired processing in that other domain and after we obtain the signal, we want to undo the operation of the logarithm, therefore, the end result, that operation is the exponentiation. So, I exponentiate here the signal and it brings me here to the native] domain. So, based on the model we used for the image as the product of the illumination term with the reflectance term. If I take the log of the image, it will give me the sum of the logs. So, it's the log of the illumination, the first term here, plus the log of the reflectance. This first term is a low-frequency term, and my objective is to decrease it while the log of the reflectance is a high mid-frequency term, and my objective is to increase it. So, I want to decrease the dynamic range here while I want to increase the local contrast. So, starting with the original image, taking the log, then we take the output and low bus filter it, this will isolate the first component. The log of the illumination. I want to decrease it, therefore I multiply it with a gamma one less than one. At the other hand, I high pass the signal y1. This will let the second component, the high frequency component, go through. That's the log of the reflectance. I want to ampl, amplify that, therefore, I multiply by gamma 2 greater than 1. I add up the two terms. I just exponentiate them, and come back to the original domain. So, for this particular, again, image model is the product of the illumination with the reflectance. Homomorphic processing is the natural framework to be applied. And that would allow us to achieve our objectives, which is to decrease the dynamic range while increasing the local contrast. So, here's again, the block diagram of the overall system, and summarizing y2 n1 n2 as the output of the filtering operation here, so it's approximately equal to gamma 1, the log of the illumination plus gamma 2 the log of the reflectance. And after exponentiation, my output, I call here x0 is the illumination component raised to the gamma 1 power, times the reflectance component raised to the gamma 2 power and gamma 1 is less than 1 while gamma 2 is greater than 1. So, here is the overall filtering operation results in really filtering y1 here by a filter which is readily symmetric and has the shape, low frequencies multiplied by gamma 1, while high frequencies multiplied by a gamma 2. So, we are doing our filtering in the log intensity domain, which is also referred to as a perceptual domain. Since it reflects some of the properties of the human visual system, according to which intensities are modified at the peripheral level using some type of non linearity such as the logarithmic operation. Here's an image weu would like to enhance through homomorphic filtering. The local contrast seems to be low in the dark regions, for example here, maybe back here as well. So, here's a result we obtained by applying the procedure I just described. Actually, the values were gamma 1 equals to 1 and gamma 2 equals to 2. And one can see that indeed the local contrast has increased in a region like this, maybe back here, but also the numbers seem to be sharper on the face of the clock.
