Fig. 1. Performing lower-diagonal-upper decomposition on a 16-order naturally ordered Hadamard matrix, the first, second, third and fourth matrices are the lower triangular matrix, diagonal matrix, upper triangular matrix, and 16-order naturally ordered Hadamard matrix, respectively, the black dots indicate matrix multiplication

Fig. 2. Performing Russian dolls sorting on 16-order naturally ordered Hadamard basis patterns^[47]

Fig. 3. Generation process of the origami ordering. (a) Rules for generating origami ordering; (b) the mask results obtained according to Fig. 3 (a)

Fig. 4. Performing cake-cutting sorting on 16-order naturally ordered Hadamard basis patterns^[49]

Fig. 5. Generation process of multi-resolution pipeline ordered Hadamard basis patterns^[55]. (a) The process of reshaping the rows of naturally ordered Hadamard matrix into the Hadamard basis patterns, (a1)‒(a4) the processes of reshaping the rows of naturally ordered Hadamard matrix of size 1×64, 4×64, 16×64, and 64×64 pixels into the Hadamard basis patterns, respectively; (b) the process of generating multi-resolution pipeline ordered Hadamard basis patterns by applying the Hadamard pipeline encoder

Fig. 6. Performing two-dimensional discrete Fourier transform on 16-order Hadamard basis patterns, the grid sub-figures are the Hadamard basis patterns and the corresponding Fourier spectrum images are given below them^[59]

Fig. 7. 16-order gray-level co-occurrence matrix contrast (ascending inertia) ordered Hadamard matrix, the row indices of this matrix correspond to the row numbers of naturally ordered Hadamard matrix^[59]

Fig. 8. Zigzag alignment. (a) The traversal order of Zigzag; (b) the big square matrix consisting of the Hadamard basis patterns according to the row and column indexes of each pattern; (c) the Hadamard basis patterns extracted from Fig. 8 (b) by using Zigzag traversal

Fig. 9. Snake alignment^[62,64]. (a) The snake traversal order; (b) (c) the big square matrix and the snake extraction strategy of the Hadamard basis patterns, respectively

Fig. 10. Single-pixel imaging based on the diffusion arrangement from visual fovea^[66]. (a) a uniform 32×32 grid; (b) 1024 Hadamard basis patterns acquired by reformatting the rows of 1024-order Hadamard matrix onto the uniform grid shown in Fig. 10 (a); (c) the experimentally reconstructed image of a cat by using the spatially uniformly aligned patterns; (d) spatial variable resolution pixel grid, containing 1024 pixel cells of varying sizes, the pixels in the focus area follow a Cartesian grid, while the pixels in the periphery of the focus follow a circular polar grid; (e) examples of 1024 Hadamard basis patterns reshaped onto the spatially variant grid shown in (d); (f) the reconstructed result of the identical scene using the spatially variant patterns shown in Fig. 10 (e)

Fig. 11. Comparison of 16-order Hadamard matrices by using different spatially optimized orders. (a)‒(f) 16-order naturally, sequency, Russian dolls, origami, cake-cutting, multi-resolution pipeline ordered Hadamard matrices, respectively

Table 1. Hadamard basis pattern optimized ordering and within-pattern pixel alignment methods