[SOLVED] AMATH482 Assignment5-Background Subtraction in Video Streams

Use the Dynamic Mode Decomposition method on the video clips ski drop.mov and monte carlo.mov containing a foreground and background object and separate the video stream to both the foreground video and a background.

The DMD spectrum of frequencies can be used to subtract background modes. Specifically, assume that ω_p, where p ∈ {1,2,…,`}, satisfies kω_pk ≈ 0, and that kω_jk ∀ j 6= p is bounded away from zero. Thus,

XDMD =               bpϕpeωpt                   +            Xbjϕjeωjt                                                                                                     (1)

| {z }

Background Video

Foreground Video

Assuming that X ∈ R^n×m, then a proper DMD reconstruction should also produce X_DMD ∈ R^n×m. However, each term of the DMD reconstruction is complex: b_jϕ_j exp(ω_jt) ∈ C^n×m ∀j, though they sum to a real-valued matrix. This poses a problem when separating the DMD terms into approximate low-rank and sparse reconstructions because real-valued outputs are desired and knowing how to handle the complex elements can make a significant difference in the accuracy of the results. Consider calculating the DMD’s approximate low-rank reconstruction according to

XLow-Rank_DMD  = b_p^ϕ_peω^pt.

Since it should be true that

X = XLow-Rank_DMD + XSparse_DMD ,

then the DMD’s approximate sparse reconstruction,

XSparse_DMD = Xb_j^ϕ_jeω^jt,

j6=p

can be calculated with real-valued elements only as follows…

SparseLow-Rank

XDMDXDMD                                 ,

where | · | yields the modulus of each element within the matrix. However, this may result in X^Sparse_DMD having negative values in some of its elements, which would not make sense in terms of having negative pixel intensities. These residual negative values can be put into a n×m matrix R and then be added back into X^Low-Rank_DMD as follows:

XLow-Rank_DMD XLow-Rank_DMD      

XSparse_DMD ← XSparse_DMD − R

This way the magnitudes of the complex values from the DMD reconstruction are accounted for, while maintaining the important constraints that

X = XLow-Rank_DMD + XSparse_DMD ,

so that none of the pixel intensities are below zero, and ensuring that the approximate low-rank and sparse DMD reconstructions are real-valued. This method seems to work well empirically.

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