Fixed-Point Arithmetic of ℓ2-Norm Approximation for 2-Tuple Arrays with Rotated ℓ2-Norm Evaluation

This paper proposes a high-precision fast approximation method for the ℓ
₂
-norm evaluation of 2-tuple arrays by means of a rotated ℓ
₁
-norm evaluation with fixed-point arithmetic. A considerable number of calculations for 2-tuple ℓ
₂
-norm are frequently required in several signal processing applications such as image restoration with isotropic total variation and complex ℓ
₁
-norm regularization. Typical embedded applications prefer parallel processing, constant scaling, and fixed-point arithmetic compared with serial processing, variable multiplication, and floating-point arithmetic. To achieve a hardware-friendly calculation, square and square root operations should be adequately approximated. However, several existing techniques are challenged with respect to approximations with all three preferable features. Thus, in this paper, a hardware-friendly approximation algorithm is proposed. The proposed method uses the fact that the upper bound of the surface of a first-order rotational cone traces a second-order cone, i.e., ℓ
₂
-cone, requests less variable multiplication, and can easily be implemented in parallel with fixed-point arithmetic. To verify the effectiveness of the proposed method, an image restoration performance and software/hardware co-design report are evaluated.