Segment 47 Group 2 Activity Sanmit

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Segment 47 Class Activity

1. The Fundamental Equation

The equation can be shown to hold true by calculating the (i,j)th element of matrix A, and showing that it holds true for both sides of the equation. Since the derivations are long, I only show the final result:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A_{i,j})_s = s_s u_{is}v_{js}}

There are s such matrices (one for each singular value) which are summed together to produce the final A matrix.

2. Low rank approximations for image compression:

Most of the code was straightforward to modify, so we only show the resulting graphs and images:


SanmitSeg47-1.png

SanmitSeg47Cacti.png

SanmitSeg47-2.png

SanmitSeg47ryb.png

SanmitSeg47-3.png

SanmitSeg47rybtwist.png

As you can see, twisting the ryb image resulted in many more significant singular values, and hence required more values to create a good approximation.