Efficient finite impulse response filters in massively-parallel recursive systems

Abstract

This paper presents strategies to massively parallelize complete recursive systems. Each algorithm handles systems with feedforward and feedback coefficients allowing to compute high-complexity filtering operators. The final algorithm is linear in time and memory, exposes a high number of parallel tasks, and it is implemented on graphics processing units, i.e. GPUs. The key to the final algorithm is the derivation of closed-form formulas to combine both non-recursive and recursive linear filters, based on an efficient state-of-the-art block-based strategy. Applications to early vision are considered in this work, hence the GPU implementation runs on images computing an approximation of the Gaussian filter and its first and second derivatives. Finally, comparison results are given showing that this work outperforms prior state-of-the-art algorithms, enabling it to achieve real-time image filtering on ultra-high-definition videos.

Appendix: Linear filter matrices

The $r\times r$ matrices $A_{ C }$ and $A_{ F }$ correspond to all multiply-add operations on (1) and (2), grouping the $b[k]$ and $a[k]$ feedforward and feedback coefficients when applying the $C_1$ and $F$ filter on an $r$ vector. The $b\times r$ matrices $A_{ CP }$ and $A_{ FP }$ are the result of $C_1(I,{\mathbf{0}})$ and $F(I,{\mathbf{0}})$, where $I$ is an $r\times r$ identity matrix and $\mathbf {0}$ is an $b\times r$ matrix of zeros. To illustrate, for $r=2$ we have

$$\begin{aligned} A_{ CP }&= \left[ \begin{array}{ll} b[2]&b[1] \\ 0&b[2] \\ 0&0 \\ \vdots&\vdots \end{array}\right]\end{aligned}$$

(13)

$$\begin{aligned} A_{ FP }&= \left[\begin{array}{ll} -a[2]&-a[1] \\ a[1]\,a[2]&a[1]^2 - a[2] \\ -a[1]^2\,a[2] + a[2]^2&-a[1]^3 + 2\,a[1]\,a[2] \\ \vdots&\vdots \end{array}\right] \end{aligned}$$

(14)

and the $r\times r$ matrix $T(A_{ FP })$ contains the last $r$ rows of $A_{ FP }$. Following, the $b\times b$ matrices $A_{ C_1\!B }$ and $A_{ FB }$ is the result of $C_1(\mathbf {0},I)$ and $F(\mathbf {0},I)$, where $I$ is an $b\times b$ identity matrix and $\mathbf {0}$ is an $r\times b$ matrix of zeros. To illustrate,

$$\begin{aligned} A_{ C_1\!B }&= \left[\begin{array}{llll} b[0]&0&0&\cdots \\ b[1]&b[0]&0&\cdots \\ b[2]&b[1]&b[0]&\cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right]\end{aligned}$$

(15)

$$\begin{aligned} A_{ FB }&= \left[\begin{array}{llll} 1&0&0&\cdots \\ -a[1]&1&0&\cdots \\ a[1]^2 - a[2]&-a[1]&1&\cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right] \end{aligned},$$

(16)

and the $r\times b$ matrix $T(A_{ FB })$ contains the last $r$ rows of $A_{ FB }$. Finally, the reverse filter matrices $A_{ CE }$ from $C_2({\mathbf {0}},I)$, $A_{ RE }$ from $R({\mathbf {0}},I)$, $A_{ C_2\!B }$ from $C_2(I,{\mathbf{0}})$, $A_{ RB }$ from $R(I,{\mathbf{0}})$ and all their row-processing counterparts ($\,\,\cdot ^{{\scriptscriptstyle \mathcal {T}}}$) are analogous.