Image Enhancement contd. An example of low pass filters is:

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1 Image Enhancement contd. An example of low pass filters is: We saw: unsharp masking is just a method to emphasize high spatial frequencies. We get a similar effect using high pass filters (for instance, taking the original and adding a high pass filtered version to it). Examples of high pass filters are:

2 Dither Here we assume that the original image has all possible intensity values (meaning there are no gaps in the histogram). Goal: We would like to obtain a processed image, which also uses all possible intensity values, such that it leads to a constant histogram (all intensity values are equally likely, as was the goal for the histogram equalization). This should also avoid artificial intensity steps in the processed image. Problem: From using the histogram equalization, we might get wide gaps in the histogram of the processed image, resulting from steep rises in the processing function g. This means that the intensity values within these gaps are not present in the processed image. This can then also lead to

3 visible artificial intensity steps, like the steps in the sky area in the example image. Approach: We use Dither. This means we redistribute the pixels belonging to a certain intensity, next to a gap in the histogram. We define a neighbourhood around this intensity value, for instance the center of the gap could be the boundary between 2 neighbourhoods. Then we take the pixels belonging to this intensity value, and redistribute them over the just defined neighbourhood of intensities, using a random number generator to equally cover the neighbourhood. This figure shows the principle of using dither. Assume we have an original image with pixel intensity I(x,y), and a histogram mapping function g(i), then we obtain the processed image P(x,y) with dither as

4 P(x, y) = g I(x, y) + d(i) Where d(i) is the dither, the random number whose range depends on the size of the gaps around intensity value I. Obersve that we designed the mapping function g(i) such that we obtain a constant histogram after equalization. This means that large lines in the histogram (many pixels at that intensity), automatically have a correspondingly wide gap around it, such that they can be distributed evenly in that neighbourhood so that we obtain a constant height in the resulting histogram. Example: Assume we have a part of the histogram which looks like: 9 pixels at intensity 231, zero pixels at intensities 232 and 233, 9 pixels at intensity 234, and again zero pixels at 235 and 236. The we would draw the boundaries around intensity 234 such that the neighbourhood is 233, 234, 235. Each time we see a pixel of the (histogram equalized) image P with intensity 235, we add an equally distributed random number of the set -1,0,+1 to the intensity. After processing the entire image (all pixels), all intensities of that neighbourhood should have about 3 pixels! This means for each intensity I in the histogram equalized processed image we have a random function d(i), whose range of random numbers depends on the size of the neighbourhood (the gap width) in the histogram at that intensity I. For non-symmetric neigbourhoods these random numbers could have the range of -1,0,1,2, for instance, still

5 euqally distributed. In this way we turn a tower of pixels into a flat rectangle of pixels in the histogram. Histgram of an histigram equalized image Figure showing the histogram of a histogram equalized image. Observe that high lines in this histogram have large neighbourhoods, small lines have small neighbourhoods, such that the number of pixels per intensity step is constant in each neighbourhood. Homomorphic Processing Consider an image with bright sunlight: there are areas where there is a lot of lighting, and there are areas in the shadow, where there is only very little light. This means we have a very high dynamic range of light intensities. The eye can process a very high dynamic range, because it has a non-

6 linear (log) pre-processing, and also something like an adaptive gain control. But common displays don t have such a high dynamic range, so that they cannot display such images faithfully. Similar, common cameras don t have a high dynamic range as the eye. But the main limiting factor are the displays. Assume, we have an image with a fairly high dynamic intensity range, and we would like to display it on a display with a more limited dynamic range, such that we can still see details in the dark and the bright areas. How can we do it? A possible approach would be to apply a non-linear display function, a suitable Gamma, as described above. This would increase the lower intensities on the display, to make them more visible. But this has limits with very high dynamic ranges, the contrasts for the dark and bright parts of an image might not become high enough using just the Gamma method. What else could we do? We can take a look at the image formation process. A simple model is the separate the lighting effects (the illumination, i(n 1, n 2 )) from the reflection effects r(n 1, n 2 ). The latter show the actual information about an object. The resulting image f(n 1, n 2 ) is: f(n 1, n 2 ) = i(n 1, n 2 )r(n 1, n 2 ) We can now assume that the illumination i(n 1, n 2 ) only varies slowly over an image, which means we get mostly low

7 spatial frequencies for it, whereas the reflectance r contains the fine details of the image, and hence has stronger high frequency components. To display this image on our low dynamic range display, the approach now is to reduce the effect of the illumination i and increase the effect of the reflectance r. This would bring the resulting image closer to a diffuse illumination. To do this, we first need to separate these two factors. To apply linear systems for the enhancement of the reflectance r, we first need to turn the multiplication of i and r into a sum. We obtain this by applying the Logarithm: log f(n 1, n 2 ) = log i(n 1, n 2 ) + log r(n 1, n 2 ) Now we can separate the reflectance r from the illumination i by linear filtering. We apply low pass filtering to obtain the illuminance component, and high pass filtering to obtain the reflectance component. Then we can apply different factors to these two components. We apply a factor bigger than 1 for the reflectance, and a factor smaller than 1 for the illuminance. After that, we can add those modified two components again, and apply an exponential factor to get our enhanced image. The block diagram of this processing can be seen in the following picture:

8 Observe: This can also be seen as unsharp masking in the logarithmic domain. An example of this processing can be seen in following picture:

9 Observe: The effect of this process is an unequal treatment of different areas of the image, unlike for instance the Gamma processing, which applies the same function to all areas of an image. The effect here is to apply different functions to dark areas that to bright areas in an image, which can be seen as an adaptive processing. Noise Smoothing Assume we have a noisy image, for instance from camera noise. How can we reduce the noise?

10 Noise is primarily wideband in character, unlike images which are mostly low pass in character. To reduce the total noise power, we can hence apply a low pass filter to the image, since the high frequencies of the noisy image contain mostly noise. The disadvantage of this method is, that also the high frequency components of the image are lost, which might not hav much power in the signal, but which are still important to maintain sharp edges in the image. Hence the low pass filtered version looks blurred, more unsharp. A different approach is the so-called Median Filtering. This is especially efficient for the so-called Salt and Pepper noise, which consist mainly of outliers in our picture, which are a few pixels which have a big error (for instance from bit errors). Definition of Median: this is the value of a sequence of numbers, where half of the values are larger than the median, and half of the values are smaller than the median. (Unlike the average, which is the sum divided by the number of samples). Example: Take a sequence over values: 1,3,4,6,7. Here, 4 is the median, which is independent of the highest or lowest value in the sequence (outliers), unlike the average. The average is 21/5=4.2. The median filter looks at a window over the image, and replaces the value in the center of the window with the

11 median over that window. This window is the shifted, sample by sample, over the entire image. Crucial here is the size of the window, for the strength of the median filtering. This has the advantage that it maintains edges, it is still sharp. Examples of the effect: (Pictures from: Jae. S. Lim, Two-Dimensional Signal and Image Processing)

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