Generative Simplex Mapping: Non-Linear Endmember Extraction and Spectral Unmixing for Hyperspectral Imagery

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this manuscript, the authors propose a new method based on GTM, called the Generative Simplex Mapping (GSM), which can extract endmember spectra and unmix nonlinear mixtures.

Here are some questions and suggestions:

1. The gridded simplex framework proposed in the manuscript is utilized to replace the rectangular framework in GTM. However, both frameworks are grounded on the assumption that the data is uniformly sampled from the embedded manifold in the data space. This assumption may not hold true in real-world scenarios. How do the authors address this potential issue?

2. The adaptive mixing coefficient is used instead of the equal prior probability of GTM to achieve the explanation of the endmember variation phenomenon. However, the adaptive update of the accuracy parameter β actually leads to the endmember variability estimated by the model being a range related to β in each band. How does the author determine the final endmember spectrum with variability?

3. In the experimental section of the manuscript,  the proposed method was compared with three early NMF methods. However, the reviewer believes that the experimental evaluation was not comprehensive and sufficient. Therefore, it is recommended that the authors compare their work with some of the latest and most advanced methods, such as nonlinear unmixing and spectral variability methods, as this, would enhance the contribution of the work in this paper.

Author Response

Thanks, we really appreciate your time! Please see attached file with our detailed replies to the reviewers comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors introduce a new model for non-linear endmember extraction and spectral unmixing of hyperspectral imagery called Generative Simplex Mapping (GSM). This work has certain novelty, detailed suggestions are as follows.

1.     Please reorganize the contributions in Introduction part and highlight the differences with other jobs.

2.     Give more explanations on the functions, variables, and the dimension of variables in equation. For example, what is  in Equation (2)? What is the meaning of  and  in  and  in Equation (3)?

3.     Authors model non-linear mixing by designing an activation function . What is the physical significance of this function?

4.     Only Figure 5 is the result of compared experiments, it is suggested to increase the data set and do more comparative experiments.

5.     The related works should be enhanced. Some recently proposed methods should be investigated, such as GMOGH and Rev-Net.

Author Response

Thanks, we really appreciate your time! Please see attached file with our detailed replies to the reviewers comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1.       The proposed Generative Simplex Mapping (GSM) model offers a fresh approach to non-linear endmember extraction and spectral unmixing in hyperspectral imagery, which is a significant contribution to the field. Highlighting the model’s flexibility to handle both linear and non-linear mixing, as well as its probabilistic nature, is commendable. However, it would be helpful if the authors more clearly articulated how the GSM directly advances the state of the art compared to other existing methods.

2.       The description of the model, especially the non-linear mapping function and the (n−1)-simplex latent space, is interesting but requires more elaboration. Providing additional mathematical explanations or diagrams could help clarify the mechanics of the latent space and the transition between linear and non-linear regimes. Some readers may struggle with the abstract nature of the model description without further illustrative examples.

3.       The comparison with three varieties of non-negative matrix factorization (NMF) on synthetic data is valuable. However, it would strengthen the evaluation if the authors included additional benchmark models, such as other widely used non-linear unmixing algorithms, to more comprehensively demonstrate GSM’s performance advantages. Are there specific scenarios where GSM is expected to significantly outperform standard methods, and if so, could these be highlighted?

4.       The synthetic data experiment and the real-world case study over a pond in North Texas are good choices for demonstrating the capabilities of GSM. Nonetheless, it would be beneficial to include more details on the real dataset, such as the resolution, spectral characteristics, and preprocessing steps, to help assess the generalizability of the method. Additionally, further validation on different types of real-world datasets with varying levels of complexity would strengthen the claims of the model’s robustness.

5.       The probabilistic treatment of spectral variability using a precision parameter is an interesting aspect of the model. However, the authors could provide more details on how this precision parameter is estimated, and how it affects the unmixing results. A deeper discussion of its impact on the overall performance of the GSM model, compared to deterministic approaches, would be valuable.

6.       While the model’s performance is highlighted, the computational cost of GSM relative to other models (e.g., NMF) is not fully discussed. Since hyperspectral data can be large and computationally demanding, it would be useful for the authors to provide an analysis of the algorithm's scalability and its computational requirements, especially for large datasets or real-time applications.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Thanks, we really appreciate your time! Please see attached file with our detailed replies to the reviewers comments.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have responded to the previous questions raised by the reviewers.

Reviewer 2 Report

Comments and Suggestions for Authors

Accept

Reviewer 3 Report

Comments and Suggestions for Authors

The paper has been revised according to my suggestions, and I suggest it can be accepted.