Data-driven and physics-constrained acoustic holography based on optimizer unrolling

Open Access

Table 1.

Number and dimensionality of the learnable complex-valued parameters $θ \in ℂ$ $\theta \in {\mathbb {C}}$ of each VN layer t.

Network Layers t (0≤t≤10)	Filter Kernels / Weights per Layer t	Activation Weights per Layer t
Initializer Network $Q$ ${\frak {Q}}$ (t=0)	$u^{1} : N_{Q} \times 1 \times d_{u} \times d_{u} :$ $u^1: N_{\cal {{Q}}} \times 1 \times d_{u} \times d_{u}:$ 32×1×5×5	$w_{Q}^{1, 2} : N_{Q} \times N_{w} \times 2 :$ ${{\mathbf {w}}}^{1,2}_{{\cal {{Q}}}}: N_{\cal {{Q}}} \times N_w \times 2:$ 32×64×2
	$u^{2} : N_{Q} \times N_{Q} \times d_{u} \times d_{u} :$ $u^2: N_{\cal {{Q}}} \times N_{\cal {{Q}}} \times d_{u} \times d_{u}:$ 32×32×5×5	$b_{Q}^{1, 2} : N_{Q} \times 2 : 32 \times 2$ ${{\mathbf {b}}}_{\cal {{Q}}}^{1,2}: N_{\cal {{Q}}} \times 2: 32 \times 2$
	$u^{3} : 1 \times N_{Q} \times d_{u} \times d_{u} :$ $u^3: 1 \times N_{\cal {{Q}}} \times d_{u} \times d_{u}:$ 1×32×5×5	$w_{Q}^{3} : N_{w} : 64, b_{Q}^{3} : 1$ ${{\mathbf {w}}}^{3}_{{\cal {{Q}}}}:N_w: 64, {{\mathbf {b}}}_{\cal {{Q}}}^{3}: 1$

	=27200	=4225
Mismatch Network $D^{t}$ ${\frak {D}}^t$ + Least-Squares Gradient Step (1≤t≤10)	$B^{t} \Leftrightarrow b^{t} : N_{D} \times 1 \times d_{b} \times d_{b} :$ ${{\mathbf {B}}}^t \Leftrightarrow b^t: N_{\cal {{D}}} \times 1 \times d_{b} \times d_{b}:$ 8×1×5×5	$w_{D}^{t} : N_{D} \times N_{w} :$ ${{\mathbf {w}}}^t_{\cal {{D}}}: N_{\cal {{D}}} \times N_w:$ 8×64
	$A^{t} \Leftrightarrow a^{t} : N_{D} \times N_{D} \times d_{a} \times d_{a} :$ ${{\mathbf {A}}}^t \Leftrightarrow a^t: N_{\cal {{D}}} \times N_{\cal {{D}}} \times d_{a} \times d_{a}:$ 8×8×3×3	$b_{D}^{t} : N_{D} : 8$ ${{\mathbf {b}}}^t_{\cal {{D}}}: N_{\cal {{D}}}: 8$
	τ^t:1


	=776	=520
Regularizer Network $R^{t}$ ${\frak {R}}^t$ (1≤t≤10)	K^t⇔k^t:N_ℛ×1×d_k×d_k: 32×1×5×5	$w_{R}^{t} : N_{R} \times N_{w} :$ ${{\mathbf {w}}}^t_{\cal {{R}}}: N_{\cal {{R}}} \times N_w:$ 32×64
	L^t⇔l^t:N_ℛ×N_ℛ×d_l×d_l: 32×32×3×3	$b_{R}^{t} : N_{R} : 32$ ${{\mathbf {b}}}^t_{\cal {{R}}}: N_{\cal {{R}}}: 32$

	=10016	=2080
Momentum Network $M^{t}$ ${\frak {M}}^t$ (2≤t≤10)	$η^{t} : N_{D} \times (t - 1) :$ $\eta ^t: N_{\cal {{D}}} \times (t - 1):$ 9×(t−1)
	γ^t:N_ℛ×(t−1): 32×(t−1)

	=(32+9)×(t−1)

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