If I ask you to find Waldo in an image just by giving you a template of Waldo, how would you search for him? One simple way is by performing template matching. We can use the template as a filter and do sum of squared differences abbreviated as SSD based matching, or cross-correlation based matching to do that. Wherever the matching score crosses the threshold, we say we detected the template. What if the scale of the Waldo image or the scale of the original image has changed. Would you still be able to find Waldo using the template matching technique? This is one of the key motivations for a multi-resolution representation of an image. Multi-Resolution Image Processing can be used to accelerate coarse-to-fine search algorithms, to look for an object or patents at different scales, and to perform multi-resolution blending operations. To begin, let us look at the Gaussian pyramid representation of an image. When trying to resize an image into half, we saw earlier in this course that sampling alternate rows and columns is not a good strategy, which resulted in aliasing. This phenomenon creates the necessity for a good sampling strategy. The answer is in the Nyquist-Shannon sampling theorem. In the field of digital signal processing, the Nyquist-Shannon sampling theorem is a fundamental bridge between analog and digital signals. This theorem states that we need to sample at a frequency, which is two times more than the maximum frequency in the signal. Let us look at this concept by visualizing what kind of output we get while sampling this sinusoid at different rates. We observe that any sampling frequency that is less than twice the frequency of the input signal results in a signal that nowhere resembles the original one. In other words, the essence of the original signal is lost. This becomes problematic in the case of images with too much of detail, which corresponds to the presence of several high frequencies. To retain the essence of the original image, we want to get rid of the high frequencies in the image that are causing the aliasing problem. To do that, we can apply Gaussian smoothing on the image before resizing it. The next image pyramid representation we will discuss is called as Laplacian pyramid. Watch out for this word Laplacian as it crops up multiple times across the course in several contexts. To compute the Laplacian pyramid, we interpolate a Gaussian image in the lower level to obtain a reconstructed low pass version of the original image. Then, we subtract this low pass version from the original to yield a Laplacian image, which is stored in the memory for further processing. One of the most engaging and fun applications of the Laplacian pyramid is the creation of blended composite images. It can also be used to compress the images. Talking about compression, let us look into the most famous image extension of all, the jpeg. Check the format of pictures on your phone. Most likely they are jpeg's. They created a compression format, which is a discrete cosine transform based technique. It works very well for photographs of natural scenery giving over 24 of compression ratio with a very minor trade off in terms of perceivable loss of visual detail. The loss is much more pronounced if you try to compress computer generated graphics, text online drawings. The amount of this loss can be varied by adjusting compression parameters that determine image quality and speed of decoding. Let us briefly look at the image compression scheme of jpeg. There are applications that may require lossless compression like, medical imaging, High resolution and high fidelity color imaging. The Joint Photographic Experts Group came up with a discrete wavelet based technique called as JPEG2000. This technique can achieve higher compression ratios without creating the block artifacts of the conventional JPEG standard. Let us look at the visual that shows JPEG2000 wavelet based compression in action. Now that you've seen how wavelet based compression looks like, let us dive deeper into the wavelet based image processing techniques. While pyramids are used extensively in computer vision applications, some people use wavelet decomposition as an alternative. Wavelets are filters that localize a signal in both space and frequency domain. They are defined over a hierarchy of scales. Wavelets provide a smooth way to decompose a signal into frequency components and are closely related to the pyramids. Wavelets were originally developed in the applied math and signal processing communities and we're introduced to the computer vision community in early 90's. Wavelets are widely used in computer graphics community to perform multi-resolution geometry processing. In computer vision, wavelets are used for de-noising and multi-scale oriented filtering. Since both image pyramids and wavelets decompose an image into multi-resolution descriptions that are localized both in space and frequency, how do they differ? The common answer is that traditional pyramids are over complete. That is, they use more pixels than the original image to represent the decomposition. On the other hand, wavelets provide a tight frame. That is, they keep the size of the decomposition the same as the original image.