## 1 Introduction
[PERSON] et al. proposed a method for acoustic contrast control in a wave domain for three dimension cases. The three-dimensional sound field is represented by spherical harmonic decomposition, and the sound energy of the interested region is calculated. The three-dimensional multizone sound field is reconstructed by using a planar circular loudspeaker array instead of a spherical one. Experiments show that this method is better than the ACC method in high-frequency acoustic contrast performance [11]. The time-domain ACC method (TACC) is favored because it can optimize the entire bandwidth by one step, but the disadvantage of TACC is that the frequency response is not uneven. In order to explain this problem theoretically, [PERSON] et al. constructed a progressively equivalent form of TACC in frequency domain, and demonstrated that TACC has an inclination to fetch frequency constituents of the largest contrast [12]. By considering the signal feature and human auditory system, [PERSON] et al. introduced a sound zone control frame called perception VAST. The experiments suggest that the perception VAST method performs better than the ACC and PM methods in terms of the perceptive index of STOI and PESQ [13]. Then, in order to make the leakage error under a fixed acoustic contrast as imperceptible as possible, they extended this work to an adaptive case and proposed a time-domain sound region control method using a variable span tradeoff filter [14]. To reduce computational complexity, they then proposed two VAST framework-based approaches (narrowband approach and broadband approach) in the discrete Fourier transform domain [15]. [PERSON] et al. proposed a personal audio effect controller whose effects change linearly with the customer's input. The weight trajectories of loudspeakers are modeled as continuous functions based on piecewise linear approximations of the performance variation. The experimental results show that the controller can regulate the system performance linearly [16]. [PERSON] et al. introduced a sound field control method in a time domain based on sound pressure and particle velocity for single-zone sound field reconstruction. In this method, the eigenvalue-decomposition-based way and the conjugate-gradient way are used to decrease the complexity of computing [17]. In order to reduce the requirements on microphone placement geometry and a control sound field in the whole region, [PERSON] et al. proposed a two-region 2D sound field reproduction approach based on equivalent source and the ACC method. The goal of this method is to maximize the acoustic energy difference between the two regions [18]. Additionally, [PERSON] et al. introduced another two-region sound field reconstruction method based on equivalent source and the PM and ACC methods. Compared with the traditional method, sound field control in a dark region by [PERSON]'s method is not limited to several points at the boundary of the dark zone, but extends to the whole dark zone [19]. The number and location of loudspeakers have important influence on the reconstruction effect in multizone sound field reconstruction. When people perform multizone sound field reconstruction, people often arrange the loudspeakers in a circular, linear, and curved fashion, but these arrangements are obtained by experience. To solve the problem, [PERSON] et al. introduced a method to select the required number of loudspeaker locations from candidate loudspeaker locations iteratively [20]. Then, enlightened by the theory in [21], [PERSON] et al. proposed an evolutionary optimization approach based on a candidate location set to optimize loudspeaker array placement [22]. Different from the iterative method, this method eliminates one loudspeaker from the loudspeaker candidate set at each iteration. [PERSON] et al. investigated the feasibility of using multiparameter array loudspeakers to remotely generate a quiet zone in a free sound field.