Update proj1.html

Author: cjxthecoder

project-1/proj1.html

@@ -77,8 +77,12 @@ <h2>Image Pyramid</h2>
 <p>
 Unfortunately, because each crop has a height of <i>(7W / 8)</i> &times; <i>(5H / 24)</i>, the total number of NCC computations for each alignment is <i>((W - 7W / 8) + 1)</i> &times; <i>((H/3 - 5H / 24) + 1)</i> = <i>(W / 8 + 1)</i> &times; <i>(H / 8 + 1)</i> = <i>O(HW)</i>. Since each NCC computation requires <i>O(HW)</i> operations, aligning an image of dimensions <i>W</i> &times; <i>H</i> takes <i>O((HW)<sup>2</sup>)</i> time using the naive search above. Because the .tif files are about 9 times bigger than the .jpg files in both dimensions, performing the same search on these files will take more than 6500x more times longer to compute (hours instead of seconds). Even if the width is fixed, it would still require more than 9<sup>3</sup> &approx; 720x more time on .tif files. A more efficient method is required.<br>
 <br>
-Instead of searching over the entire image, we can scale down the image, find the best shift at a much smaller size, and scale the result back to its original size. Once the most accurate displacement (<i>x</i><sub>lowest</sub>, <i>y</i><sub>lowest</sub>) is calculated for the lowest-sized image, we can perform another search at the image 2x the size. Only this time, we start at (<i>2x</i><sub>lowest</sub>, <i>2y</i><sub>lowest</sub>), and only search over a window of [-2, 2] for pixel corrections. Once the best displacement at the layer below the lowest scale is computed, we can multiply the result again by 2 and pass it to the image in the next layer below. The limited window is used because each downscaled coordinate could have only come from a total of 4 different coordinates, so the best coordinate on the higher scale is limited to [-1, 1] of the interpolated coordinate. An additional pixel in the search range is used to further reduce the effect of noise on the best placement at the lowest scale. To further optimize this approach without storing all downscale images at once, we can modify the best displacement function to have a recursive call.<br>
-<br>
+Instead of searching over the entire image, we can scale down the image, find the best shift at a much smaller size, and scale the result back to its original size. Once the most accurate displacement (<i>x</i><sub>lowest</sub>, <i>y</i><sub>lowest</sub>) is calculated for the lowest-sized image, we can perform another search at the image 2x the size. Only this time, we start at (<i>2x</i><sub>lowest</sub>, <i>2y</i><sub>lowest</sub>), and only search over a window of [-2, 2] for pixel corrections. Once the best displacement at the layer below the lowest scale is computed, we can multiply the result again by 2 and pass it to the image in the next layer below. The limited window is used because each downscaled coordinate could have only come from a total of 4 different coordinates, so the best coordinate on the higher scale is limited to [-1, 1] of the interpolated coordinate. An additional pixel in the search range is used to further reduce the effect of noise on the best placement at the lowest scale. To further optimize this approach without storing all downscale images at once, we can modify the best displacement function to have a recursive call.
+</p>
+<div align="center">
+<img src="images/Image_pyramid_svg.png" alt="Image_pyramid_svg.png" width="25%">
+<figcaption>Source: <a href="https://en.wikipedia.org/wiki/Pyramid_%28image_processing%29">Wikipedia</a></figcaption>
+</div>
 Now, we will only calculate the best displacement if the given width is below a certain threshold. For images above this threshold, we can first downscale the image by 2x, pass it back to the function, and the returned shifts are scaled up by 2x to return the best shifts on the input image. This means that the best shifts in the base case (<i>x</i><sub>lowest</sub>, <i>y</i><sub>lowest</sub>) will be scaled up and used in the previous recursive call, which is the scaled image 1 call above. This will continue until we return to the top-level call, and at that point, the returned shifts will be within 1 or 2 pixels of the best overall displacement. The last thing to keep in mind is to downscale the cropping box as well, which is simple to do since it is computed on the full image, and one can simply divide its coordinates by 2 for each recursive call. In practice, setting W<sub>min</sub> = 72 gives the best tradeoff between the search size and the number of rescales. With these optimizations in place, the computing time is now much faster:
 
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