# Latitude-Adaptive Integer Bit Allocation for Quantization of Omnidirectional Images
## 2 Related Work  
Sampling density correction can be divided into two main types, down-sampling and adaptive quantization. In the case of down-sampling methods, [PERSON] et al. [20] introduced a variable smoothing approach that employs Gaussian smoothing filters on the upper and lower regions of \(360^{\circ}\) video under ERP. Similarly, [PERSON] et al. [21] proposed a strategy to divide panoramic images under ERP into several down-sampled strips based on latitude. Note that in both approaches [20, 21], down-sampling becomes progressively harsher in the two vertical directions towards the top and bottom borders. Additionally, [PERSON] et al. [27] presented a scheme that employs rhombus-shaped latitude down-sampling followed by pixel rearrangement. Nevertheless, the preprocessing approaches may not have direct control over the bitrate for panoramic image coding. Consequently, compressed panoramic images may struggle to regain their original quality at higher rates.  
When it comes to adaptive quantization, rate-distortion optimization (RDO) methods have been adapted for omnidirectional videos and images. [PERSON] et al. [22] proposed optimizing panoramic video encoding by maximizing S-PSNR [28] at a given bitrate. [PERSON] et al. [24] suggested incorporating the weight value of WS-PSNR [26] at the center of each block as a scalar for the Lagrangian multiplier \(\lambda\) in RDO. However, the RDO methods mainly relied on the HEVC technique and did not give out the formulation for integer bit allocation. Apart from RDO methods, [PERSON] et al. [10] proposed a method to adapt the typical quantization tables of JPEG according to frequency shift, which was easy to implement but constrained by the JPEG table. Both the RDO methods and the methods proposed by [PERSON] et al. used the quantization techniques designed for 2D images instead of for omnidirectional images.  
Integer bit allocation performs as a pivotal technique in image transform coding, playing a crucial role in the quantization of transform coefficients. This method significantly contributes to the coding process by efficiently managing the allocation of bits. [PERSON] et al. [29] used the high-resolution quantization approximations and found the solution to the ORBA problem for the transform coders under mean squared error (MSE). However, the practical solution for quantizing the transform coefficient has to be an NNI value. [PERSON] [30] improved the ORBA algorithm and proposed an NNIBA algorithm under MSE. Bit allocation approaches have been widely used in image processing. [PERSON] et al. [25] proposed a greedy algorithm under MSE measure and obtained a lookup table for the quantization table elements via non-linear regression analysis, thus facilitating the reduction in the additional side information requirement. [PERSON] et al. [31] conducted an in-depth study focusing on the joint design of graph signal sampling along with quantization for graph signal compression in a task-based quantization manner. They proposed a joint design of the sampling and recovery mechanisms for a fixed quantization mapping and presentedan iterative algorithm for dividing the available bit budget. Nevertheless, none of these methods were designed for omnidirectional images. Besides, neither [PERSON] et al. nor [PERSON] et al. took the distribution of coefficients into consideration, and their methods cannot be directly applied to 360\({}^{\circ}\) images.