built-in matlab function ssim  (MathWorks Inc)

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: ( a ) A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5 \times 5$$\end{document} 5 × 5 image domain P lies in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}^2$$\end{document} Z 2 , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {x}}}$$\end{document} x is a specified point in P . ( b ) A grayscale image is defined on the image domain P , it has pixel values 0, 1, 2, and 3. ( c ) A binary image is defined on the image domain, where the pixels in the image with 0 are the black pixels and 1 for the white pixels. ( d )–( f ) Depictions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \times 3$$\end{document} 3 × 3 structuring elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B = \{ 0,1,2 \} \times \{ 0,1,2 \}$$\end{document} B = { 0 , 1 , 2 } × { 0 , 1 , 2 } , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-B = \{ 0,-1,-2 \} \times \{ 0,-1,-2 \}$$\end{document} - B = { 0 , - 1 , - 2 } × { 0 , - 1 , - 2 } , and (symmetric) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = \{ -1,0,1 \} \times \{ -1,0,1 \}$$\end{document} C = { - 1 , 0 , 1 } × { - 1 , 0 , 1 } with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {0}=(0,0)$$\end{document} 0 = ( 0 , 0 ) marked. Blue regions in ( g )–( i ) are the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathbf {x}}}+ B) \cap P, ({{\mathbf {x}}}- B) \cap P$$\end{document} ( x + B ) ∩ P , ( x - B ) ∩ P , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathbf {x}}}+ C) \cap P$$\end{document} ( x + C ) ∩ P , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathbf {x}}}- C) \cap P$$\end{document} ( x - C ) ∩ P . Rows ( j )–( m ) and ( n )–( q ) are illustrations for erosion, dilation, opening, and closing operations.

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques:

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: Illustration of erosion ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document} ε ), dilation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} δ ), opening ( O ), and closing ( C ) with respect to the structuring element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_6$$\end{document} B 6 which is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7\times 7$$\end{document} 7 × 7 square defined in .

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques:

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: Opening filtration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{{\mathscr {O}}}_i := O_{{\mathfrak {B}}_i}(f)^{-1}(0)$$\end{document} X i O : = O B i ( f ) - 1 ( 0 ) (row 2), closing filtration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i := X^{{\mathscr {C}}}_{-(5 - i)} := C_{{\mathfrak {B}}_i}(f)^{-1}(0)$$\end{document} Y i : = X - ( 5 - i ) C : = C B i ( f ) - 1 ( 0 ) (row 3), and their relevant persistence diagrams (row 4). ( a ) Original binary image g ; ( b ) noised image f ; ( c ) the 1-dimensional holes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} X 0 that have death \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$< 6$$\end{document} < 6 ; ( d ) the connected components in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} X 0 that have birth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$> 1$$\end{document} > 1 ; ( e )–( j ) binary representations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{{\mathscr {O}}}_i := O_{{\mathfrak {B}}_i}(f)^{-1}(0)$$\end{document} X i O : = O B i ( f ) - 1 ( 0 ) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0, 1, \ldots , 5$$\end{document} i = 0 , 1 , … , 5 ; ( k )–( p ) binary representations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i := X^{{\mathscr {C}}}_{-(5 - i)} := C_{{\mathfrak {B}}_i}(f)^{-1}(0)$$\end{document} Y i : = X - ( 5 - i ) C : = C B i ( f ) - 1 ( 0 ) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0, 1, \ldots , 5$$\end{document} i = 0 , 1 , … , 5 . The structuring elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {B}}_0, {\mathfrak {B}}_1, ..., {\mathfrak {B}}_5$$\end{document} B 0 , B 1 , . . . , B 5 used here are the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_0, B_2, B_4, B_6, B_8$$\end{document} B 0 , B 2 , B 4 , B 6 , B 8 , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{10}$$\end{document} B in . ( q ) Persistence diagram \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_1[\{X^{{\mathscr {O}}}_i\}_{i=0}^{5}]$$\end{document} P 1 [ { X i O } i = 0 5 ] . ( r ) Persistence diagram \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_0[\{Y_i\}_{i=0}^{5}]$$\end{document} P 0 [ { Y i } i = 0 5 ] . By diagrams ( q ) and ( r ), for the image in ( b ), the values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_o$$\end{document} i o and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_c$$\end{document} i c derived from the first iteration in Algorithm 1 are 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(5 + 1) - 3 = 3$$\end{document} ( 5 + 1 ) - 3 = 3 respectively.

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques: Filtration, Derivative Assay

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: Top panel: Schematic illustration of steps in Algorithm 1 in the multiparameter filtration that would produce the alternating sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2, -2, 1, -1)$$\end{document} ( 2 , - 2 , 1 , - 1 ) . Red dotted lines highlight a bifiltration layer, and blue dotted lines highlight a different bifiltration layer. The black solid line represents the path and selections made by Algorithm 1. Bottom panel: An application. ( a ) Ground truth binary image. ( b ) Ground truth with salt and pepper noise with noisy density 0.4. ( c ) Denoised image produced by Algorithm 1. The parameters used for Algorithm 1 are MaxIter=10 and Sizetol=5. The resulting alternating opening/closing sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {u}}}$$\end{document} u of ( c ) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-2,4,-1,3,-1,2,-1,1,-1)$$\end{document} ( - 2 , 4 , - 1 , 3 , - 1 , 2 , - 1 , 1 , - 1 ) .

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques: Filtration, Sequencing, Produced

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: A 2-parameter filtration using structuring elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_i$$\end{document} B i defined in . The notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{{\mathscr {O}},{\mathscr {C}}}_{(-i,j)}$$\end{document} X ( - i , j ) O , C denotes the set of black pixels in the image \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C_{B_i} \circ O_{B_j})(f)$$\end{document} ( C B i ∘ O B j ) ( f ) where f is the left-top image in the 2-parameter filtration. The formal definition can be found in Definition .

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques: Filtration

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: Mean (solid) plus standard deviation (dashed) curves of IOU, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (\beta _0)$$\end{document} log ( β 0 ) , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (\beta _1)$$\end{document} log ( β 1 ) , PSNR, and SSIM scores for 100 trials at each computed salt and pepper noise density. Images in Fig. are the representatives of images for obtaining the results. See also Supplementary Table .

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques: Standard Deviation

Journal: Scientific Reports

Article Title: A multi-parameter persistence framework for mathematical morphology

doi: 10.1038/s41598-022-09464-7

Figure Lengend Snippet: First row: ( a ) An example of image with larger spatial scale salt and pepper noise. ( b ) Corresponding denoised images produced by CNN with Median Layers. ( c ) Corresponding denoised images produced by Algorithm 1. Second row: The table of evaluation scores IOU, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0$$\end{document} β 0 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1$$\end{document} β 1 , PSNR, and SSIM of CNN with median layers and Algorithm 1. For each noise type, 100 images were formed and tested. All scores are recorded by mean ± standard deviation for the 100 trials. The ground truth of the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta _0, \beta _1)$$\end{document} ( β 0 , β 1 ) is (6, 5).

Article Snippet: As mentioned above, we use Matlab built-in function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {ssim}$$\end{document} ssim with default parameters.

Techniques: Produced, Standard Deviation