Sparse fast DCT for vectors with one-block support

Abstract

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the input vector x ∈ R N , N = 2 J−1 , with short support of length m from its discrete cosine transformThe resulting algorithm has a runtime of O m log m log 2N m and requires O m log 2N m samples of x II .In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector y ∈ R 2N with reflected block support of block length m from y with the same runtime and sampling complexities as our DCT algorithm.

Real sparse fast DCT for vectors with short support

“…, b} ⊂ N 0 of integers. We say that a vector x = x (J) = (x k ) N −1 k=0 ∈ R N has a short support, or one-block support, S Note that, unlike in [3], we do not allow a periodic support in this paper. The interval S (J) := I µ (J) ,ν (J) is called the support interval, µ (J) the first support index and ν (J) the last support index of x.…”

“…The algorithm presented in this paper generalizes ideas introduced in [3,[12][13][14] for reconstructing a vector x ∈ R N , N = 2 J , with short support of length M , M -sparse support or reflected two-block support with block length M from its DFT. In these papers the sought-after vector x is recovered iteratively from its 2 j -length periodizations x (j) , where x (J) := x and x (j) is obtained by adding the first and second half of x (j+1) .…”

“…, J − 1}, where 2 L ≥ 2M , our new algorithm is based on efficiently and iteratively recovering x (j+1) from x II using that x (j) is known. Note that, unlike the DCT reconstruction algorithm for vectors with one-block support in [3], which uses a closely related DFT reconstruction and hence complex arithmetic, our algorithm only employs real arithmetic, as it utilizes real factorizations of cosine matrices. This approach requires some observations about the support of x (j+1) if the support of x (j) is given, which are summarized in Section 2.…”

“…In this paper we want to find a deterministic algorithm for reconstructing x ∈ R N with short support of length m from its discrete cosine transform of type II, x II , if an upper bound M ≥ m is known and using, unlike in [3], only real arithmetic. In order to do so we adapt techniques used in [3,[12][13][14] for the FFT reconstruction of vectors with short support to the real DCT setting. There exist several different factorizations of the orthogonal matrix C II n , but the following one, see Lemma 2.2 in [11], has proven to be particularly useful in our case.…”

“…In [3], where the recovery of a vector x ∈ R N with short support of length m from x II based on (1) is studied, the short support of x and the resulting symmetric reflected block support of y are exploited. The algorithm in [3] x II ∈ R N , N = 2 J . Thus it performs better than general sparse FFT methods, as it is specifically tailored to the occurring support structure.…”