hadamard matrix  (Hamdard Laboratories)

Journal: Scientific Reports

Article Title: Super Sub-Nyquist Single-Pixel Imaging by Total Variation Ascending Ordering of the Hadamard Basis

doi: 10.1038/s41598-020-66371-5

Figure Lengend Snippet: Image reconstructions with different subsets of normal Hadamard masks. ( a ) The measured signal intensity distribution of the image using normal order Hadamard. ( b,c ) are the measured signal intensity distribution in real value ( y ) descending order and absolute value (| y |) descending order. ( d,i ) are the ground truth of the phantom image and natural image. ( e,f,j,k ) give the reconstructed images at sampling ratio of 15% and 40% based on normal order Hadamard. ( g,h,l–m ) give the reconstructed images of the sparse image and natural image at sampling ratios of 15% and 40% based on absolute value descending order Hadamard.

Article Snippet: We define the sum of total variation of each row of a Hadamard matrix as: 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T{V}_{i}=\sum \sqrt{{({h}_{i}{D}_{x})}^{2}+{({h}_{i}{D}_{y})}^{2}}$$\end{document} T V i = ∑ ( h i D x ) 2 + ( h i D y ) 2 where D x and D y is the discretized gradient operators, which are N × N sparse diagonal matrices, for the variation in x direction and y direction respectively; h i is the i-th row of normal Hamdard matrix; TV is the sum of variation.

Techniques: Sampling

Journal: Scientific Reports

Article Title: Super Sub-Nyquist Single-Pixel Imaging by Total Variation Ascending Ordering of the Hadamard Basis

doi: 10.1038/s41598-020-66371-5

Figure Lengend Snippet: Different Hadamard ordering example. ( a–e ) are the normal order, TV order, TG order, CC order and Paley order of 16 × 16 Hadamard matrices and their mask sequence orders respectively, each mask is one row of Hadamard matrix which are reshaped into a 4 × 4 2D matrix.

Article Snippet: We define the sum of total variation of each row of a Hadamard matrix as: 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T{V}_{i}=\sum \sqrt{{({h}_{i}{D}_{x})}^{2}+{({h}_{i}{D}_{y})}^{2}}$$\end{document} T V i = ∑ ( h i D x ) 2 + ( h i D y ) 2 where D x and D y is the discretized gradient operators, which are N × N sparse diagonal matrices, for the variation in x direction and y direction respectively; h i is the i-th row of normal Hamdard matrix; TV is the sum of variation.

Techniques: Sequencing

Journal: Scientific Reports

Article Title: Super Sub-Nyquist Single-Pixel Imaging by Total Variation Ascending Ordering of the Hadamard Basis

doi: 10.1038/s41598-020-66371-5

Figure Lengend Snippet: Total variation of the proposed four reordered Hadamard matrices comparison. The blue, orange, red and black dots are the total variation value distribution of CC order, TG order, TV order and Paley Hadamard matrices.

Article Snippet: We define the sum of total variation of each row of a Hadamard matrix as: 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T{V}_{i}=\sum \sqrt{{({h}_{i}{D}_{x})}^{2}+{({h}_{i}{D}_{y})}^{2}}$$\end{document} T V i = ∑ ( h i D x ) 2 + ( h i D y ) 2 where D x and D y is the discretized gradient operators, which are N × N sparse diagonal matrices, for the variation in x direction and y direction respectively; h i is the i-th row of normal Hamdard matrix; TV is the sum of variation.

Techniques: Comparison

Journal: Scientific Reports

Article Title: Super Sub-Nyquist Single-Pixel Imaging by Total Variation Ascending Ordering of the Hadamard Basis

doi: 10.1038/s41598-020-66371-5

Figure Lengend Snippet: Image reconstruction with different fractions of four Hadamard basis. ( a ) is the 12.5% of full measured signal intensity sorted in descending order according to its absolute value. ( b–e ) are the measured signals intensity using subset of natural order, Paley order, CC order, TG order and TV order Hadamard basis respectively. ( f ) is the ground truth image. ( g–j ) are the reconstructed images using subset of Paley order, CC order, TG order and TV order Hadamard basis respectively.

Article Snippet: We define the sum of total variation of each row of a Hadamard matrix as: 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T{V}_{i}=\sum \sqrt{{({h}_{i}{D}_{x})}^{2}+{({h}_{i}{D}_{y})}^{2}}$$\end{document} T V i = ∑ ( h i D x ) 2 + ( h i D y ) 2 where D x and D y is the discretized gradient operators, which are N × N sparse diagonal matrices, for the variation in x direction and y direction respectively; h i is the i-th row of normal Hamdard matrix; TV is the sum of variation.

Techniques: