WAVELETS: MATHEMATICAL TOOLS FOR IMAGE ANALYSIS

Mathematics is very human—we all enjoy using our brains, thinking, making connections, and checking to see if our conclusions are true. For me, mathematics is also a way to experience creativity, which is what makes me happy. In this article, I want to share this delight with you, through the story of wavelets. Wavelets are mathematical tools that my colleagues and I developed, which are very useful in analyzing images and other signals like medical scans or audio files. I hope that, by the end of this article, you will see that mathematics is much more interesting and diverse than memorizing formulas and performing calculations. It is a creative endeavor that advances society and can make our lives richer and more beautiful.


WHAT ARE WAVELETS?
Wavelets are mathematical functions that help us process and analyze images and other kinds of signals [ ], including medical recordings of the body's activities and even ripples in space itself, created by the strongest astronomical events in the universe!(To learn more about wavelets and signal processing, see this video).In the 's, mathematicians, engineers, and physicists all studied wavelets simultaneously.This was an exciting time, as it was clear that each field was looking at one part of a whole concept that would be extremely useful across multiple fields.
As their name suggests, wavelets are little waves that can come in various forms (Figure A).Wavelets are finite, meaning that their value is non-zero only in a specific area in space.Every wavelet can be stretched or squished to make it narrower or broader and can also be shifted to the left or right along the x-axis (Figure B), in order Once we have a signal or an image that we want to analyze, we can figure out which wavelets are most similar to the patterns in the signal (Figure ).Importantly, we can adjust the scale to "zoom in" to see smaller details of the signal (squish the wavelet), or to "zoom out" to get a broader view (stretch the wavelet).Then, using some mathematical tricks, we can use wavelets to either get certain information out of the signal or image, like whether it contains a specific shape or pattern we are looking for, or compress that signal or image-meaning to save

COMPRESS (DATA)
Save only the necessary part of the data.
the image using only a small part of the original memory space, which still gives us enough information for our purpose.Using wavelets to detect patterns.We often use wavelets to analyze patterns in signals and images.In this artistic representation, the man is using wavelets to study the wave pattern in the painting The Great Wave o Kanagawa by Katsushika Hokusai.As you can see on the left, the man has frames with di erent wavelets and he is searching for the wavelet that would best suit the shape of the wave in the image.B).The number represents the color black, and represents white.All the numbers in between are various hues of gray (if the image is colorful, then we use three numbers to represent the three components of the color-red, green, and blue).Computers represent all information using only two states: " " and " ".Each " " or " " unit is called a bit, and each bit can represent the two states.When we combine two bits, since each of them can independently represent two states, both of them together can represent = states.If we want to represent numbers (from to ), we need a system of bits ( = ).A typical rectangular digital image is formed of rows, each having pixels, so × = , pixels overall.Each pixel is represented by bits, so we need , × = , , Daubechies Wavelets bits, or units of information, to represent only one typical greyscale image (For colored images, we need three times as many)!That is a huge amount of information, especially if we think about movies, in which we have images per second, each represented by millions of bits.We must often find ways to decrease the amount of computer memory needed to store images, without losing important information.To do so, we can try to compress the image and reduce the number of pixels stored in memory.A simple way of doing this is to take every two neighboring pixels in an image, calculate the average number of their colors, and save only one pixel with that average color number (Figure C).By doing this, we decrease the number of pixels of an image by a factor of two horizontally (for neighboring pixels in every row) and by a factor of two vertically (for neighboring pixels in every column).We can repeat this process many times and produce a very small image.

WAVELETS AS TOOLS FOR IMAGE PROCESSING
When we do this averaging, we lose important information that was contained in the original image-information that might make a reconstructed image blurrier.If we do not want to lose that information, we must ensure that we can reconstruct the original image from the averaged image.To do that, we also save the di erences between the original pixels that we averaged, because it is possible to find the original numbers from the average and the di erence of the two numbers.It turns out that we can represent these averages and di erences using wavelets!If we take the averages and di erences only between two near neighbors, we get a staircase-like wavelet (Haar wavelet in Figure A).Often, we want to combine larger groups of neighboring pixels, which means each averaged pixel will contain more information.When we do that, we get other wavelets that are smoother (Daubechies wavelet in Figure A).
Wavelets can show us the most important parts in our image, which will be areas of the image where "something is happening," like image compression standard [ , ]. JPEG is widely used in the compression of high-resolution images, in many applications on the internet and for digital movies.

WAVELETS IN INTERESTING PLACES
Wavelets are a powerful tool for analyzing signals and images.In principle, they can be used in any situation for which signal analysis is required, including in the fields of astronomy [for automatic selection of the best-focused images of astronomical objects [ ]]; forensics [for detecting manipulations of digital audio files, images, and videos [ ]]; and medicine [for detecting heart diseases that cause irregular blood supply to the heart [ ]].With time, as we install more and more sensors in our environments and collect an increasing amount of data that we want to store, I expect that wavelets will have an even bigger role in many applications.
I am currently working with colleagues on using wavelets for interesting and perhaps quite surprising applications.In one of my projects, for example, I am working with art historians.We use wavelets to analyze paintings and understand their history.Specifically, we are trying to get the best possible view of hidden layers, called underpaintings, that are beneath the visible paintings (Figure A).After we have identified an underpainting using x-rays, we use wavelets to "expose" the details of the underpainting and reconstruct it as accurately as possible.We can also use wavelets for other applications To end, I would like to share with you an important lesson I learned from working on mathematical problems.To solve a mathematical problem, you must try to solve it over and over again, approaching it from multiple "sides."This is similar to sports, in which you have to work hard and make an e ort to get better-but the fact that it is hard does not mean it is impossible.So, keep trying and keep using your creativity to find novel, beautiful ways to understand the things around you, and be sure to share your insights with others.

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Figure

Figure
Figure As you may know, images are represented in computers as sets of numbers (Figure A).In a grayscale image, the color of each pixel PIXEL The smallest unit of color (tiny circle) in digital images.is represented by one number, often in the range of -(Figure matched to the analyzed signal.You can think of wavelets as building blocks with which we can reconstruct and analyze signals and images.
DaubechiesWavelets to be Wavelets can give us information about general trends in the values of pixels in our image, and about the horizontal, vertical, and diagonal details or sharp changes in color.Wavelets are better at analyzing sharp changes in images (and other signals) compared to other common signal-analysis methods.Wavelets let us identify and save only important features of the image, where there are large changes, and discard information about uniform areas.The information we save about the important features is enough to allow us to reconstruct a fairly accurate version of the original image.Saving as little as % or even % of the original data can often be enough to reconstruct a pretty good version of the original image (see this video and check whether you can notice the di erences between the various compressions).I developed wavelets that are particularly useful for image analysis.While most scientists looked at known wavelets and tried to change them to make them useful for image analysis, I first looked at the image-analysis task at hand and asked: which properties should wavelets have to be useful for this task?Based on these properties, my colleagues and I constructed a completely new family of wavelets [ ].We then showed how wavelets can be used for image processing and compression [ ].Our suggestion turned out to be very useful, and it was applied in a method that is called the JPEG JPEG such as studying and removing (or generating) cracks in digital images of paintings (FigureB) and rejuvenating aged colors.