在遙測領域當中，解混是一項至關重要的技術，尤其是處理影像的像素混和問題。本研究將專注於欠定型多光譜解混。本文旨在探索訊號源位於源凸包內部之欠定型多光譜解混的可行性。為此，我們將凸幾何引入作為輔助工具，根據幾何意義對多光譜影像設計了凸限制，並將篩選過的多光譜數據餵入一個多光譜解混框架，稱為「稜鏡啟發的多光譜端元萃取(PRIME)」。首先，多光譜影像受到空間解析度的限制會導致像素混和問題的產生，這意味著一個像素可能會涵蓋大片區域，因此會包含多個物質，所以對多光譜影像進行解混是必然的。然而，多光譜影像具有豐富的頻譜資訊，但是僅涵蓋數個至十幾個特定的波段，所以在進行多光譜解混時，當頻帶數小於純物質數的時候就會面臨所謂的欠定問題。為了解決欠定問題，我們提出了PRIME，其中的核心方法—量子虛擬稜鏡，對頻譜進行超解析，使得頻譜的數量大於純物質數。這個創新的量子神經網路解決了傳統多光譜解混方法沒有考慮到的欠定問題。我們並沒有因此而滿足，相反的我們想更深入的探討欠定問題之下的幾何意義。我們提出了兩個凸限制，證明在幾何空間上若多光譜的純物質位於源凸包內部，依然能夠準確估計出純物質的指紋以及對應的豐度圖。我們成功在三個不同的地區解混出源凸包內部的純物質，證明了訊號源位於源凸包內部之欠定型多光譜解混的可行性和有效性。

In the field of remote sensing, unmixing is a critical technology, especially for dealing with mixed-pixel phenomenon issues in images. We will focus on underdetermined multispectral unmixing (MU). In this thesis, we aim to prove the feasibility of underdetermined MU with some interior sources inside the source convex hull. To this end, we introduce convex geometry as an auxiliary tool. According to the convex geometry, a convex constraint is designed for multispectral images (MSIs), and the filtered mutlispectral data was fed into a MU framework termed prism-inspired multispectral endmember extraction (PRIME). Firstly, MSIs are limited by spatial resolution, which can lead to mixed-pixel phenomenon issues. This means that a single pixel may cover a large area and contain multiple substances. Hence, it is necessary to unmix MSIs. However, MSIs contain rich spectral information, but only cover a few to a dozen specific bands. Therefore, when performing MU, if the number of bands is less than the number of substances, we will face the so-called underdetermined problem. To solve the underdetermined problem, we propoesd the PRIME model. The core of the framework, virtual quantum prism, super-resolution on the spectrum of MSIs. This innovative quantum neural network (QNN) solves the underdetermined problem that traditional MU methods do not consider. We are not satisfied with this, on the contrary we want to investigate the geometric meaning behind the underdetermined problem. We proposed two convex constraints to prove that, in geometric space, if the substances of MSIs is located inside the source convex hull, it is still possible to accurately estimate the signature of the substance and the corresponding abundance map. We successfully separated substances within the source convex hull in three different regions, demonstrating the feasibility and effectiveness of underdetermined MU with some interior sources inside the source convex hull.

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