Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment
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We study the problem of efficiently summarizing a short video into several keyframes, leveraging recent progress in fast graph sampling. Specifically, we first construct a similarity path graph (SPG) 𝒢, represented by graph Laplacian matrix 𝐋, where the similarities between adjacent frames are encoded as positive edge weights. We show that maximizing the smallest eigenvalue λ_min(𝐁) of a coefficient matrix 𝐁 = diag(𝐚) + μ𝐋, where 𝐚 is the binary keyframe selection vector, is equivalent to minimizing a worst-case signal reconstruction error. We prove that, after partitioning 𝒢 into Q sub-graphs {𝒢^q}^Q_q=1, the smallest Gershgorin circle theorem (GCT) lower bound of Q corresponding coefficient matrices – min_q λ^-_min(𝐁^q) – is a lower bound for λ_min(𝐁). This inspires a fast graph sampling algorithm to iteratively partition 𝒢 into Q sub-graphs using Q samples (keyframes), while maximizing λ^-_min(𝐁^q) for each sub-graph 𝒢^q. Experimental results show that our algorithm achieves comparable video summarization performance as state-of-the-art methods, at a substantially reduced complexity.
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