In this paper, for the first time in literature, we introduce one-sided weighted inverses and extend the notions of one-sided inverses, outer inverses and inverses along given elements. Although our results are new and in the matrix case, we decided to present them in tensor space with reshape operator. For this purpose, a left and right $\left(\mathcal{M,N}\right)$-weighted $\left(\mathcal{B,C}\right)$-inverse and the $\left(\mathcal{M,N}\right)$-weighted $\left(\mathcal{B,C}\right)$-inverse of a tensor are defined. Additionally, necessary and sufficient conditions for the existence of these new inverses are presented. We describe the sets of all left (or right) $\left(\mathcal{M,N}\right)$-weighted $\left(\mathcal{B,C}\right)$-inverses of a given tensor. As consequences of these results, we consider the one-sided $\left(\mathcal{B,C}\right)$-inverse, $\left(\mathcal{B,C}\right)$-inverse, one-sided inverse along a tensor and inverse along a tensor. Further, we introduce a $\left(\mathcal{M,N}\right)$-weighted $(\mathcal{B},\mathcal{C})$-outer inverse and a $\mathcal{W}$-weighted $(\mathcal{B},\mathcal{C})$-outer inverse of tensors with a few characterizations. Then, corresponding algorithms for computing various types of outer inverses of tensors are proposed, implemented and tested. The prowess of the proposed inverses are demonstrated for finding the solution of Poisson problem and the restoration of 3D color images.

在本文中，我们首次引入单面加权逆，并沿给定元素扩展了单面逆，外部逆和逆的概念。尽管我们的结果是新的并且是在矩阵情况下，但我们还是决定使用重整算子将它们呈现在张量空间中。为此，一个左右$（\mathcal{M，N}）$加权 $（\mathcal{公元前}）$-逆和 $（\mathcal{M，N}）$加权 $（\mathcal{公元前}）$-定义张量的倒数。此外，为这些新逆的存在提供了必要和充分的条件。我们描述所有左（或右）的集合$（\mathcal{M，N}）$加权 $（\mathcal{公元前}）$-给定张量的逆。作为这些结果的结果，我们认为单方面$（\mathcal{公元前}）$-逆， $（\mathcal{公元前}）$-逆，沿张量的单侧逆和沿张量的逆。此外，我们介绍了$（\mathcal{M，N}）$加权 $（\mathcal{乙}，\mathcal{C}）$-外逆和 $\mathcal{w\; ^}$加权 $（\mathcal{乙}，\mathcal{C}）$张量的外部逆定理。然后，提出，实现和测试了用于计算各种类型的张量的外部逆的相应算法。为了找到泊松问题的解决方案和3D彩色图像的恢复，证明了所提出的逆的能力。