In this paper, we introduce an efficient algorithm for generating specific Hadamard rows, addressing the memory demands of pre-computing the entire matrix. Leveraging Sylvester's recursive construction, our method generates the required $i$-th row on demand, significantly reducing computational resources. The algorithm uses the Kronecker product to construct the desired row from the binary representation of the index, without creating the full matrix. This approach is particularly useful for single-pixel imaging systems that need only one row at a time.

翻译：本文提出一种高效算法，用于生成特定哈达玛矩阵行，以解决预计算整个矩阵所需的内存需求。该方法基于西尔维斯特递归构造原理，能够按需生成所需的第$i$行，显著降低计算资源消耗。该算法利用克罗内克积，通过索引的二进制表示直接构建目标行，而无需生成完整矩阵。此方法特别适用于每次仅需单行数据的单像素成像系统。