**1. Introduction**

Nowadays, we are living in a period of big remote sensing data [1] because numerous Earth observation sensors provide a huge amount of remote sensing data for our lives; therefore, the development of fast and accurate content-based image retrieval (CBIR) methods is becoming increasingly important in the remote sensing community. In order to make better use of big data, machine learning methods are essential [2]. In 2007, Salakhutdinov and Hinton [3,4] proposed a hash learning method in the field of machine learning. Since then, the hashing method has been widely studied and applied in the fields of computer vision, information retrieval, pattern recognition, data mining, etc. [2]. Hash learning methods convert high-dimensional data into the form of binary codes through machine learning methods. At the same time, the transformed binary codes retain the neighboring relationships in the original high-dimensional space. In recent years, hash learning methods have rapidly developed into a research hotspot in the field of machine learning and big data.

Traditionally, the representation of remote sensing images (RSIs) is described by a real number vector with thousands of dimensions. Traditional remote sensing image retrieval methods usually describe images by using real vectors with thousands of dimensions. Each dimension can be stored in computer memory by floating-point data with four bytes, which may lead to the following issues: (1) The storage of a large-scale dataset requires many hard disks; (2) exhaustively searching for relevant images in a large-scale dataset is computationally expensive. When a 4096-dimensional feature of the fully connected

**Citation:** Liu, N.; Mou, H.; Tang, J.; Wan, L.; Li, Q.; Yuan, Y. Fully Connected Hashing Neural Networks for Indexing Large-Scale Remote Sensing Images. *Mathematics* **2022**, *10*, 4716. https://doi.org/ 10.3390/math10244716

Academic Editor: Danilo Costarelli

Received: 22 September 2022 Accepted: 6 December 2022 Published: 12 December 2022

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layer in a deep network is expressed and stored, it takes 4096 × 4 bytes of storage space. Since one byte is equal to eight bits, the storage space of a 4096-dimensional real vector is 4096 × 4 × 8 bits. In contrast, when hash learning is used to map deep features, supposing that the deep features are mapped to 64 bits through hashing coding, the storage space used is eight bytes. In this case, in comparison with the storage space of 4096 × 4 × 8 bits, the hash learning method can greatly reduce the hard disk storage space of data and greatly improve the computational efficiency of image retrieval. To address the above issue, hashing-based approximate nearest neighbor search, which is a highly time-efficient search with a low storage space, is becoming a popular retrieval technique due to the emergence of big data. Hash mapping [5] represents an image as binary codes that contain a small number of bits, such as 32 bits (4 bytes), thereby significantly helping in the reduction of the amount of memory required for storage.
