#
Infrared Non-Destructive Testing via Semi-Nonnegative Matrix Factorization^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Testing Specimen and Experimental Setup

#### 2.2. Methodology

**n**data vectors in its columns correspond to

**n**vectorized infrared images, $\mathit{X}\in {\mathit{R}}^{\mathit{p}\times \mathit{n}},$and $H\in {\mathit{R}}^{\mathit{n}\times \mathit{k}}$. This represents the singular value decomposition (SVD), where there are no limitations on the signs of

**B**and

**H**, also non constrain for input matrix

**X**. NMF assumes that matrices

**X, B**, and

**H**are not negative, but when the data matrix is unconstrained (i.e., when mixed signs exist among the matrices) NMF is referred to be called Semi-NMF, which

**H**is restricted to be non-negative but there is no restriction for matrix

**B**[11]. This provides freedom to basis matrix (similar to PCT) which is restricted by a coefficient matrix. The NMF directly associates with clustering [1,3], Semi-NMF can be modified from this perspective, where if Semi-NMF performs grouping (similar to K-means clustering) on input infrared data of

**X**, the

**B**and

**H**can be represented by a cluster centroids matrix $\mathit{B}=\left({\mathit{\beta}}_{\mathbf{1}},{\mathit{\beta}}_{\mathbf{2}},\dots ,{\mathit{\beta}}_{\mathit{k}}\right)$, and by cluster indicators ; ${\mathit{\eta}}_{\mathit{i}\mathit{k}}=\mathbf{1}$ if ${\mathit{x}}_{\mathit{i}}$ belonging to cluster ${\mathit{c}}_{\mathit{k}}$; ${\mathit{\eta}}_{\mathit{i}\mathit{k}}=\mathbf{0}o$therwise. This can be shown similar to a clustering objective function as:

## 3. Results and Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Evaluation of the qualitative accuracy of Semi-NMF (

**h.**,

**i.**) versus the state-of-the-art approaches (

**a.**–

**g.**) in thermography.

Method | Computational Load (s) | IoU (%) |
---|---|---|

CCIPCT | 0.04 | 0.2 |

PCT | 0.25 | 0.91 |

NMF | 0.56 | 0.95 |

Standard-NMF-gd | 14.81 | 0.87 |

Standard-NMF-nnls | 45.18 | 0.83 |

Sparse-PCT | 18.31 | 0.4 |

Sparse-NMF | 39.52 | 0.93 |

SemiNMF-Ruls | 27.56 | 0.53 |

SemiNMF-nnls | 27.5 | 0.86 |

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**MDPI and ACS Style**

Yousefi, B.; Ibarra-Castanedo, C.; Maldague, X.P.V.
Infrared Non-Destructive Testing via Semi-Nonnegative Matrix Factorization. *Proceedings* **2019**, *27*, 13.
https://doi.org/10.3390/proceedings2019027013

**AMA Style**

Yousefi B, Ibarra-Castanedo C, Maldague XPV.
Infrared Non-Destructive Testing via Semi-Nonnegative Matrix Factorization. *Proceedings*. 2019; 27(1):13.
https://doi.org/10.3390/proceedings2019027013

**Chicago/Turabian Style**

Yousefi, Bardia, Clemente Ibarra-Castanedo, and Xavier P.V. Maldague.
2019. "Infrared Non-Destructive Testing via Semi-Nonnegative Matrix Factorization" *Proceedings* 27, no. 1: 13.
https://doi.org/10.3390/proceedings2019027013