We propose and analyze a new class of unbalanced weak optimal transport (OT) problems with total variation penalties, motivated by spatial resource allocation tasks. Unlike classical OT, our framework accommodates general unbalanced nonnegative measures and incorporates cost objectives that directly capture operational trade-offs between transport cost and supply–demand mismatch. In the general setting, we establish the existence of optimal solutions and a dual formulation. We then focus on the semi-discrete setting, where one measure is discrete and the other is absolutely continuous, a structure relevant to applications such as service area partitioning for facilities like schools or medical stations. Exploiting a tessellation-based structure, we derive the corresponding explicit optimality conditions. We further address a quantization problem that jointly optimizes the locations and weights of discrete support points, applicable to facility location tasks such as the cost-efficient deployment of battery swap stations or e-commerce warehouses, informed by demand-side data. The dual-tessellation structure also yields explicit gradient expressions, enabling efficient numerical optimization in finite dimensions.

Local contrast methods are efficient approaches for infrared (IR) small target detection. However, since its baseline is obtained locally, a local contrast method can work well only when the target is locally prominent, which is not always the case when the background is very complex. In this paper, a new detection framework named true global contrast measure (TGCM) is proposed, in which its baseline is obtained by global decomposition, so that it can handle situations where the target is not locally salient. For example, when it is adjacent to some bright background. The complex edges / corners and noises that are easily lost during decomposition will be suppressed through some simple but ingenious design: A Gaussian filtering operation is adopted to suppress noises, and a local max dilation is utilized on the separated baseline to maintain as much edge / corner information as possible. The residual information during decomposition will be used as a weighting function to further suppress clutters. Experiments on both real and simulated IR images demonstrated the effectiveness of the proposed method. Compared to some existing algorithms, it has advantages in target enhancement and background suppression. Especially, it can achieve good performance in certain scenarios where the target is not locally prominent.

Local contrast has been proven to be effective in infrared (IR) small target detection, but existing algorithms still have some drawbacks. In this letter, a ratio-difference (RD) local feature contrast (LFC) method is proposed. First, a new nested window is designed, and the local gray contrast is selected as the features of inner and outer windows to investigate the characteristics of a local area. Then, the RD LFC is calculated. At last, a simple but useful weighting function is proposed. Experiments show that the proposed algorithm can achieve better performance in terms of detection rate and false alarm rate and has good robustness against noise.

In this letter, a weighted strengthened local contrast measure (WSLCM) algorithm for infrared (IR) small target detection is proposed, it consists of two modules, the strengthened local contrast measure (SLCM), and the weighting function. In the SLCM calculation, the ideas of matched filter and background estimation are adopted to enhance true target and suppress complex background, then both ratio and difference operations are used to calculate the SLCM. In the weighting function definition, three components are considered: the characteristics of the target, the characteristics of the background, and the difference between them. Especially, an improved regional intensity level (IRIL) algorithm is proposed to evaluate the complexity of a cell, thus it can suppress random noises better. Experiments on some real IR images show that the proposed WSLCM can achieve a better detection performance under complex background.

Local contrast has been proved efficient for infrared (IR) small-target detection. However, current algorithms do not enhance true target purposefully before local contrast calculation and may easily be disturbed by noises. In this letter, a new detection framework named multiscale tri-layer local contrast measure (TLLCM) is proposed. First, a tri-layer filtering window is proposed, and it consists of a core layer, a reserve layer, and a surrounding layer. The idea of a matched filter is adopted, and a Gaussian filtering will be performed on the core layer to enhance true target purposefully according to the target shape. Then, the multiscale TLLCM of the central pixel of the window will be calculated between the enhanced core and the surrounding local background. Finally, the target can be extracted by an adaptive threshold. Experimental results show that the proposed method can achieve better detection performance than some existing algorithms.

Local contrast has been proven as an efficient method for infrared (IR) small target detection, but existing local contrast algorithms just directly choosing the neighboring area of a current position as the reference when calculating the local contrast of the current position, which may bring an inaccurate result. Meanwhile, existing algorithms are either ratio form or difference form, they cannot effectively enhance true target and suppress all the types of complex backgrounds simultaneously. In this letter, a new local contrast scheme that introduces the adaptive background estimation is proposed to provide a more accurate reference, and the multidirectional 2-D least mean square (MDTDLMS) algorithm that is more suitable for small target detection is presented. Then, a new ratio-difference joint local contrast measure (RDLCM) is proposed between raw IR image and the MDTDLMS result to enhance true small target and suppress all the types of complex backgrounds simultaneously. Experimental results show that the proposed MDTDLMS-RDLCM algorithm can achieve a good detection performance for different types of backgrounds and targets.

Infrared (IR) small target detection with high detection rate, low false alarm rate, and high detection speed has a significant value, but it is usually very difficult since the small targets are usually very dim and may be easily drowned in different types of interferences. Current algorithms cannot effectively enhance real targets and suppress all the types of interferences simultaneously. In this letter, a multiscale detection algorithm utilizing the relative local contrast measure (RLCM) is proposed. It has a simple structure: first, the multiscale RLCM is calculated for each pixel of the raw IR image to enhance real targets and suppress all the types of interferences simultaneously; then, an adaptive threshold is applied to extract real targets. Experimental results show that the proposed algorithm can deal with different sizes of small targets under complex backgrounds and has a better effectiveness and robustness against existing algorithms. Besides, the proposed algorithm has the potential of parallel processing, which is very useful for improving the detection speed.

Infrared (IR) small target detection with high detection rate, low false alarm rate, and multiscale detection ability is a challenging task since raw IR images usually have low contrast and complex background. In recent years, robust human visual system (HVS) properties have been introduced into the IR small target detection field. However, existing algorithms based on HVS, such as difference of Gaussians (DoG) filters, are sensitive to not only real small targets but also background edges, which results in a high false alarm rate. In this letter, the difference of Gabor (DoGb) filters is proposed and improved (IDoGb), which is an extension of DoG but is sensitive to orientations and can better suppress the complex background edges, then achieves a lower false alarm rate. In addition, multiscale detection can be also achieved. Experimental results show that the IDoGb filter produces less false alarms at the same detection rate, while consuming only about 0.1 s for a single frame.

Robust human visual system (HVS) properties can effectively improve the infrared (IR) small target detection capabilities, such as detection rate, false alarm rate, speed, etc. However, current algorithms based on HVS usually improve one or two of the aforementioned detection capabilities while sacrificing the others. In this letter, a robust IR small target detection algorithm based on HVS is proposed to pursue good performance in detection rate, false alarm rate, and speed simultaneously. First, an HVS size-adaptation process is used, and the IR image after preprocessing is divided into subblocks to improve detection speed. Then, based on HVS contrast mechanism, the improved local contrast measure, which can improve detection rate and reduce false alarm rate, is proposed to calculate the saliency map, and a threshold operation along with a rapid traversal mechanism based on HVS attention shift mechanism is used to get the target subblocks quickly. Experimental results show the proposed algorithm has good robustness and efficiency for real IR small target detection applications.