Salient object detection in HSI using MEV-SFS and saliency optimization

Abstract

The existing methods in salient object detection (SOD) in hyperspectral images (HSI) have used different priors like center prior, boundary prior to procure cues to find the salient object. These methods fail, if the salient object is slightly touching the boundary. So, we extrapolate boundary connectivity, a measure to check if the object touches the boundary. The salient object is obtained by using background and foreground cues, which are calculated using boundary connectivity and contrast map, respectively. Also, to reduce the information redundancy and hence time complexity, we select top three most informative bands using different feature selection and feature extraction algorithms. The proposed algorithm is tested on HS-SOD dataset. It is observed that the proposed algorithm performs better than the state-of-the-art techniques in almost all the metrics, such as Precision (0.57), Recall (0.46), $f_1$ score (0.51), CC (0.43), NSS (2.13), and MAE (0.09). In addition, we performed a comparative analysis of four different feature selection (MEV-SFS, OPBS) and feature extraction (PCA, MNF) algorithms in the context of SOD in HSI. We observed that feature selection algorithms are computationally efficient with OPBS and MEV-SFS taking about 7.98 and 8.34 s on average to reduce the feature space, respectively.

Introduction

Object detection recognizes the semantic category and then estimates the spatial position of the object. Salient object detection (SOD) is the detection of the most noticeable part in the image. Mathematically, it is defined as the sparse entries in certain transformed feature space, where background is a low-rank matrix. In Hyperspectral Images (HSI), saliency detection has been achieved using different priors like center prior, boundary prior, etc. They are used to procure cues to detect the object. The individual saliency maps are merged into a single saliency map. The saliency map is further processed to get the most salient object. The salient object detection is demonstrated in Fig. 1. The first row shows HSI rendered sRGB images, and second row shows the corresponding salient object or ground truth. SOD has been explored heavily in low-dimensional images. Approaches like [1] use a guided smoothing technique to enhance the regions with most saliency. By utilizing a deep saliency model with guided non-local blocks, the approach accurately detects salient regions in input images [1]. However, these approaches have not been extrapolated to HSI. These approaches primarily focus on de-noising and may inadvertently blur spectral details critical for SOD in HSI. SOD has a wide range of applications like autonomous driving, robotics, medical image analysis, surveillance, remote sensing, etc.

Fig. 1 [Images not available. See PDF.]

Salient object detection in HSI, the first row shows the HSI rendered sRGB images and second row are the corresponding salient objects in the image

Hyperspectral imaging sensors accumulate information from a broad portion of electromagnetic spectrum. The hyperspectral cameras capture visible (0.4–0.7 $\mu$m) to short-, mid-, and long-wave infrared region (2.4 $\mu$m). The wavelength captured is divided into large number of spectral bands [2]. HSI is visualized as a three-dimensional cube with spatial and spectral dimensions along the axis as demonstrated in Fig. 2. The traditional sensors capture information in only three bands in visual spectrum. Hence miss out on the additional non-visual but useful information. The additional information helps us to identify the objects more precisely and exquisitely. The advances in sensor technologies are making data more readily available, building a promise for the future uses of this technology in image analysis tasks. HSI provides valuable auxiliary information enabling fine detection of objects. The low-dimensional images like RGB [1, 3, 4, 5, 6, 7–8] grayscale, etc do not differentiate between two different objects with same color. HSI have been used in the field of biomedical imaging [9], remote sensing [10], quality analysis [11], defense and security-based applications, etc.

Fig. 2 [Images not available. See PDF.]

Hyperspectral image with spectral band frames and reflectance curves [12]

HSI is very expensive to store and process. In addition, the higher spectral resolution induces redundancy across bands. So, it becomes imperative to reduce the dimensionality of HSI. Dimension reduction can be achieved by either feature extraction or feature selection. Feature extraction methods like PCA [13, 14–15], MNF [16] etc. extract new feature set from original features using some functional mapping. Since these methods extract new features, they lose the original physical information. However, in feature selection, subset of original features is selected and indigenous information is preserved. There have been very few studies of the above two categories in the context of SOD in HSI. This paper attempts to compare the two categories by taking two algorithms from each category and perform salient object detection using the features generated.

The spectral information-based approaches use spectral information only. They are effective in detecting spectral saliency. The inclusion of spatial information will improve the results. So, we use Simple Linear Iterative Clustering (SLIC) [17] to make superpixels.

Some researchers have shown that different priors like center prior, boundary prior [18, 19–20] can be useful to detect the salient object. The center prior-based methods [18, 19] assume that the salient object lies near the center. So, the pixels toward center get higher saliency values. These methods have difficulty in detecting objects that lie further from center. This prior gets an added advantage in the images taken under human supervision, where main focus, i.e., the salient part generally lies near center. On the contrary, Boundary prior considers the boundary pixels as non-salient. This method presumes the salient object would not lie on the boundary. This method fails to detect the object if it slightly touches the boundary. In remote sensing images the probability of an object lying near the boundary is higher. So we used boundary connectivity, a measure to evaluate the object connectivity with boundary to generate saliency cues [21].

The scarcity of labeled data and sparsity of high-dimensional data has hindered the growth of Deep Learning (DL) and HSI until now [22]. The high dimensionality and correlation of hyperspectral data complicate parameter estimation and information extraction. Additionally, limitations in ground truth datasets and dataset shift hinder the development and evaluation of models [23]. There have been attempts to use self-supervised Convolution Neural Networks (CNN) [24, 25] to extract features. The CNN model is updated based on cross-entropy loss, and saliency is computed by performing manifold ranking on the features learned by the task. The extracted features are used to detect the final object by applying manifold ranking [24]. These approaches need lots of data to train properly. In the absence of large standard object detection datasets, the object detection research is skewed toward the non-learning-based approach. Moreover, the heuristic integration of individual saliency maps in general saliency detection methods may sometimes give sub-optimal results. So, an optimization-based approach [21] is used to integrate the saliency maps. Subsequently, this paper attempts to achieve a following set of objectives:

• The research in salient object detection in HSI has been mostly concerned with improving the results. There has been no attempt to include the processing time in the results. This paper is the first attempt to take the processing time into account and attempt to reduce it. To reduce the processing time, we extrapolated the saliency optimization framework to HSI as a deterministic approach. Henceforth, reducing the processing time.

• Since HSI is high dimensional, so the feature reduction has a very important role to play as the research in the field advances. So, this paper attempts to analyze different feature reduction algorithms in the context of salient object detection.

• This paper also attempts to compare different feature selection and feature extraction algorithms computational efficiency.

The rest of this paper is organized as follows. In Sect. 2, related literature is reviewed. The proposed method is discussed in Sect. 3, and finally the results are given and discussed in Sect. 4.

Literature review

Saliency detection was formulated by Itti and koch [26]. They introduced the saliency detection based on the visual attention model of early primates. It was designed for RGB images originally but has been extrapolated to HSI as well [27]. The direct extrapolation of Itti’s model to HSI [28] failed to detect the spectral saliency. Yan cao et al. used PCA to calculate top three components to replace the color components used by Itti. These methods use center surround difference to calculate saliency and hence may get stuck in local minima. Also, these methods use only spectral saliency and hence they would detect spectral saliency only.

To detect the spectrally similar but spatially different objects, some methods have used both spectral and spatial information [18, 19, 29]. It improves the detection but increases the computational complexity of the algorithm. Also, to avoid the effects of spectral variation gradient-based methods have been proposed [29]. The gradient helps to reduce the spectral variation leading to the improved detection performance especially in scenes with uneven illumination and shadows. Super-pixels are obtained by using SLIC algorithm [17]. The saliency score of each region is calculated using central prior and local contrast. The region with most distinctive reflectance property is produced as salient [18]. Zhang et al. [19] uses spatial and spectral information to reduce the effect of weaker cues. The inclusion of regional information helps us to eradicate the weak cues and smooth out noises. This method is insensitive to high contrast edges. In comparison, Liang et al. [27] performs better as it calculates angle similarity, which is independant of the spectral variation.

Imamoglu et al. [24] uses self-supervised CNN to extract the features and applied manifold ranking on the extracted features to detect the final salient object. The CNN extracts 64 features and applies manifold ranking on the extracted feature set. The CNN-based approaches need lots of data to train properly, which is scarce in case of HSI. The data scarcity has also affected the usage of deep learning for hyperspectral object detection tasks. So, we used a non-training-based approach to perform salient object detection.

Proposed method

The proposed method operates in two stages I). Feature reduction II). Saliency detection. The full architecture of the proposed method is given in Fig. 3. The detailed explanation of the various stages is given below:

Feature reduction In feature reduction, we select top most informative bands using different feature selection and feature extraction algorithms. It helps us to reduce the computational complexity and produces better results as-well. We drafted four different feature reduction algorithms with two from each feature selection (MEV-SFS [30], OPBS [31]) and feature extraction (PCA [13], MNF [16]) categories. Different feature reduction algorithms retain the original features or extract the new ones based on certain criteria. MEV-SFS selects the features based on their contribution to ellipsoid volume. The features that maximize the determinant of the selected bands covariance matrix i.e. ellipsoid volume are selected as optimal/final features. OPBS selects features based on their orthogonal projections. The largest orthogonal projection designates the feature to be selected. PCA retains the features which enhances the variance in the transformed feature space. MNF performs noise whitening before calculating the principal components.

Fig. 3 [Images not available. See PDF.]

The overall architecture of the proposed method

The resultant top three features obtained are given as input to the saliency detection algorithm and the results corresponding to all four algorithms were taken. The experimental evaluation indicates that MNF [16] produces better results in terms of precision, but MEV-SFS [32] is less complex computationally and gives almost equal $f_1$ score. The working of MEV-SFS [30] is given in Algorithm 1. The internal comparison and a detailed discussion of the algorithms is given in Sect. 4.2.1.

Saliency detection SOD is performed using background and foreground cues calculated using Eqs. (7) and (8), respectively [21]. These cues are integrated using an optimization framework as given in Eq. (9). To reduce the computational complexity, the pixels are initially grouped by performing segmentation using SLIC [17] on the reduced feature set. The SLIC [17] assists in reducing the complexity by reducing the number of individual units to be processed further. An undirected weighted graph is constructed with super pixels as nodes. The background and foreground cues are calculated using the undirected weighted graph. The euclidean distance $(d_{\text{cor}})$ between average pixel values of each super pixels is taken as an edge weight. Also, the minimum distance between two super pixels $(dis{t}_m(S_i,S_j))$ is defined as sum total of weights aggregated along the shortest path between the two superpixels as shown in Eq. (3).

3

$$\begin{array}{c}\hfill {\mathit{dist}}_m(S_i,S_j)=min\sum _{k=i}^j{d}_{\text{cor}}(S_k,S_{k+1}),\end{array}$$where k is the total number of superpixels between i and j.

The boundary connectivity $(B_{\text{con}})$ is calculated as shown in Eq. (4). The boundary connectivity assists us in calculating the background cue ($W_i^{\mathit{bg}}$) as shown in Eq. (7). The background cue value is close to 1, if boundary connectivity is large indicating that the super pixel belongs to background. If the boundary connectivity is small, the super pixel could be salient.

4

$$\begin{array}{cc}\hfill & B_{\text{con}}=\frac{L_b(s)}{\sqrt{A_s}}\hfill \end{array}$$

5

$$\begin{array}{cc}\hfill & L_b(s)=A_s·\mathit{Bnd}(S_i);\mathit{Bnd}(S_i)=\left\{\begin{array}{cc}1,\hfill & \text{if}{S}_i\in B,\hfill \\ 0,\hfill & \text{Otherwise}\hfill \end{array}\right)\hfill \end{array}$$

6

$$\begin{array}{cc}\hfill & A_s=\sum _{j=1}^N{e}^{\left(\frac{-dis{t}_m^2\left(S_j,,,S_{j+1}\right)}{2{\sigma }_c^2}\right)};{\sigma }_c=10\hfill \end{array}$$where B is the set of superpixels containing boundary pixels. The optimal value for ${\sigma }_c$ was found empirically and equals to 10.

7

$$\begin{array}{c}\hfill W_i^{\mathit{bg}}=\left\{\begin{array}{cc}1-exp\left(\frac{-B_{\text{con}}^2(S_i)}{2{\sigma }_{\mathit{bd}}^2}\right),\hfill & \text{if}{B}_{\text{con}}\text{is; large}\hfill \\ 0,\hfill & \text{Otherwise}\hfill \end{array}\right)\end{array}$$The parameter ${\sigma }_{\mathit{bd}}\in [0.5,2]$ with optimal value for our experimentation equal to 0.75. $W_i^{\mathit{bg}}\in [1,0]$; ’1’ indicates that the pixel belongs to background and ’0’ indicates that it belongs to foreground. The effectiveness of boundary connectivity is demonstrated in Sect. 4.2.

The foreground cue $W_i^{\mathit{fg}}$ is calculated using contrast map and background weight as shown in Eq. (8). The contrast map is defined as the addition of color distance to all other super pixels, weighted by position distance. $d_{\text{loc}}$ is location distance and is calculated as euclidean distance between average value of pixel coordinates of each superpixel.

8

$$\begin{array}{cc}\hfill W_i^{\mathit{fg}}=& \sum _{j=1}^n{d}_{\text{cor}}(S_i,S_j)exp\left(\frac{-d_{\text{loc}}^2(S_i,S_j)}{2{\sigma }_{\text{loc}}^2}\right)W_i^{\mathit{bg}};\hfill \\ \hfill & \text{where}{\sigma }_{\text{loc}}=0.25.\hfill \end{array}$$Finally, the foreground and background cues are integrated using an optimization framework as shown in Eq. (9):

9

$$\begin{array}{c}\hfill min\left(\sum _{i=1}^n,W_i^{\mathit{bg}},G_i^2,+,\sum _{i=1}^n,W_i^{\mathit{fg}},{(G_i-1)}^2,+,\sum _{i,j},W_{\mathit{ij}},{(G_i-G_j)}^2\right)\end{array}$$where $G_i$ is the saliency value of ith segment

10

$$\begin{array}{c}\hfill w_{\mathit{ij}}=exp\left(-,\frac{d_{\text{cor}}^2(S_i,S_j)}{2{\sigma }_c^2}\right);\end{array}$$where i and j are two adjacent superpixels.

Equation (9) converges by taking smaller value $G_i$ for high probable background pixels and higher value for high probable foreground pixels.

Results and discussion

Dataset and metrics

Dataset

One of the biggest challenges in HSI has been the lack of standard dataset for SOD. Different classification datasets have been used by researchers to evaluate SOD algorithms. However, HS-SOD [24] was made available recently and is now considered as the benchmark dataset for saliency detection in HSI. We evaluate our model using the same dataset. It contains a total of 60 hyperspectral images along with the corresponding ground truths and representative sRGB images. The images have been taken using the wavelength range 380–780 nm. The dataset contains images of different contrasts, sizes and locations. The camera used to capture the images is NH-AIK.

Table 1. A quantitative comparison of different feature reduction algorithm w.r.t salient object detection

Algorithm	Precision$\uparrow$	Recall $\uparrow$	$f_1$ score $\uparrow$	NSS $\uparrow$	CC $\uparrow$	MAE $\downarrow$
[30]	0.57	0.46	0.51	2.13	0.43	0.09
OPBS [30]	0.56	0.48	0.50	2.10	0.43	0.08
PCA [13]	0.52	0.41	0.42	1.67	0.31	0.09
MNF [16]	0.60	0.45	0.53	2.13	0.43	0.09

Metrics

Different metrics evaluate different aspects of saliency, so we used six different metrics to evaluate the performance of our algorithm. We have used both location based (Normalized Scanpath Saliency (NSS)) and distribution-based (Correlation Coefficient (CC)) metrics to evaluate the effects of false positives and false negatives [33]. In addition, we calculate Precision, Recall and $f_1$ score as-well. We also calculated Mean Absolute Error (MAE) to illustrate our models insensitivity to non-salient pixels. Considering $S_i$ and $G_i$ as the predicted saliency and ground truth values, respectively, the metrics used are as follows:

• Precision: It is the proportion of correctly identified salient pixels to the total number of pixels predicted as salient. Mathematically, it can be expressed as follows:

  11

  $$\begin{array}{c}\hfill P=\frac{|S_i\cap G_i|}{S_i}\end{array}$$

• Recall: It is the proportion of correctly detected salient pixels to the total number of salient pixels. It can be expressed as follows:

  12

  $$\begin{array}{c}\hfill R=\frac{|S_i\cap G_i|}{G_i}\end{array}$$

• $f_1$ score: It is the weighted harmonic mean of precision and recall. The weight $\beta$ was set to 0.3 to increase the priority for precision and is defined as below:

  13

  $$\begin{array}{c}\hfill f_1\text{score}=\frac{(1+{\beta }^2)\times \text{Precision}\times \text{Recall}}{{\beta }^2\times \text{Precision}+\text{Recall}}\end{array}$$

• Normalized Scanpath Saliency (NSS): It calculates the resemblance between ground truth and saliency maps. It becomes sensitive to false positives if there are more false positives. The positive NSS scores show the correspondence and negative shows anti-correspondence as defined below:

  14

  $$\begin{array}{c}\hfill \begin{array}{c}\hfill \text{NSS}(S,G)=\frac{1}{N}\sum _{i=1}^w\sum _{j=1}^h{\overline{S}}_{\mathit{ij}}\times G_{\mathit{ij}};\\ \hfill N=\sum _{i=1}^w\sum _{j=1}^h{G}_{\mathit{ij}}\text{and}\overline{S}=\frac{S-\mu (S)}{\sigma (S)}\end{array}\end{array}$$

• Correlation Coefficient: It measures the linear correlation between saliency map and ground truth and is defined as:

  15

  $$\begin{array}{c}\hfill \text{CC}=\frac{\sigma (S,G)}{\sigma (S)\times \sigma (G)}\end{array}$$

• Mean absolute Error (MAE): MAE is sensitive to pixels that are being correctly detected as non-salient, i.e true negatives and is defined as below:

  16

  $$\begin{array}{c}\hfill \text{MAE}=\sum _{i=1}^N\frac{|S_i-G_i|}{N}\end{array}$$

Table 2. Time taken by each feature reduction algorithm in seconds

Algorithm	Feature reduction time (In seconds)	Saliency detection time (In seconds)	Total time (In seconds)
MNF [16]	15.15	4.33	19.48
PCA [13]	13.08	4.51	17.59
OPBS [30]	7.98	4.75	12.73
MEV-SFS [30]	8.34	4.61	12.95

Discussion

This section discusses the results in two subsections i.e. feature reduction (Sect. 4.2.1) and saliency detection (Sect. 4.2.2). The feature reduction subsection includes the comparison of different feature selection and feature extraction algorithms with respect to SOD. The saliency detection section compares the SOD algorithm with state-of-the-art methods.

Table 3. Comparison of saliency detection-based algorithm in HSI performed on HS-SOD dataset

Model	Precision $\uparrow$	Recall $\uparrow$	$F_1$ score $\uparrow$	NSS $\uparrow$	CC $\uparrow$	MAE $\downarrow$
Itti [26]	0.19	0.23	0.35	1.36	0.35	0.25
SED [28]	0.34	0.12	0.34	1.35	0.28	0.18
SAD [28]	0.15	0.23	0.26	1.18	0.30	0.23
GS [28]	0.27	0.23	0.30	1.56	0.34	0.22
SED-OCM-GS [28]	0.27	0.22	0.33	1.59	0.37	0.18
SED-OCM-SAD [28]	0.15	0.23	0.26	1.53	0.39	0.21
SGC [18]	0.23	0.28	0.19	1.47	0.50	0.21
HS-MR [34]	0.43	0.36	0.36	1.27	0.35	0.18
Proposed	0.57	0.46	0.51	2.13	0.43	0.09

Feature reduction

In feature reduction, we used two feature selection such as MEV-SFS [30], OPBS [30] and two feature extraction algorithms such as MNF [16], PCA [13] to reduce the feature space. We included both the feature selection algorithms, where the final features preserve the original information and feature extraction algorithms that use different components (PCA [13], MNF [16]) and hence does not necessarily preserve the original information. In our experimental evaluation, we found that in feature extraction methods MNF produces better results with respect to the saliency detection. Whereas, PCA produced the poorest results. The MEV-SFS and MNF produce compatible results in terms of $f_1$ score (0.51, 0.53), NSS (2.13, 2.13), CC (0.43, 0.43) and MAE (0.09, 0.09). The quantitative comparison of saliency detection using four selected algorithms is given in Table 1. In addition, MEV-SFS is temporally efficient. It takes an average of 12.95 s to detect an object, which is about 33% less as compared to MNF which takes an average of 19.48 s. The time comparison of different feature reduction algorithm is given in Table 2. We also observed that feature selection algorithms require lower processing time than feature extraction algorithms. Since, feature extraction algorithms requires the data to be transformed to another feature space inducing an additional computational cost.

Fig. 4 [Images not available. See PDF.]

Time comparison (in seconds) between different feature reduction algorithms w.r.t salient object detection

The variation in the results of the feature extraction algorithms maybe explained by the arbitrary nature of transformed feature space. The feature space obtained after applying the transformation function may not be optimal always. MNF performs the noise whitening first, leading to reduction of noise in the final features selected. While as PCA equates variance and information, which is not true for images like HSI, which are highly sensitive to noise. OPBS band selection algorithms performed well with respect to PCA. We found out feature selection algorithms are stable as compared to feature extraction algorithms. This can be explained as feature extraction is non arbitrary with respect to the transformed feature space. The final components may not always contain enough relevant information. A graphical representation of time taken by different feature selection and feature extraction algorithms is shown in Fig. 4. The experiments were performed on Intel Xeon(R) [email protected] with OS type: Ubuntu (64 bit) and 32 GB RAM.

Fig. 5 [Images not available. See PDF.]

Visual sample of salient detection in HSI using different methods. (I) sRGB (II) GT (III). Itti [26] (IV). SED [28] (V). SAD [28] (VI). SG [28] (VII). SED-SAD [28] (VIII). SED-SG [28] (IX). HSISO [25] (X). $HSIS{O}_{\mathit{spa}}$ [25] (XI). Proposed Method

Fig. 6 [Images not available. See PDF.]

A visual sample illustrating the effectiveness of boundary connectivity

Saliency detection

In terms of saliency detection, we observed that our method performs well with respect to state-of-the-art as can be seen from Table 3. Since, we have discussed and compared the feature reduction methods in Sect. 4.2.1.

This section compares results by taking MEV-SFS-based saliency optimization. OPBS is computationally more efficient as compared to MEV-SFS. It takes about 7.98 s on average to select the top three features, while it takes about 8.3 s in case of MEV-SFS. However, we finalized MEV-SFS as our choice for feature selection for its slight improvement in the results. It gave better results on all metrics in comparison with the state-of-the-art. Our model gave higher precision indicating that our model does not produce much false positives. It also produces higher CC and NSS indicating that our model is able to suppress both false positive and false negative fairly well. Moreover, the lower MAE confirms that the model makes very less individual prediction errors.

The proposed method being deterministic doesn’t need training. It also has an added advantage of low temporal complexity with whole method taking 12.11 s to complete on average. In addition, our method performs better than deep learning-based feature extraction methods [24, 25]. Moreover, deep learning-based algorithms require lots of data to train and test, which is not available in case of HSI. A sample visual representation of proposed method and other state-of-the-art algorithms is given in Fig. 5.

Our method generates noticeable saliency maps compared to other already existing methods. It produces better results in almost all metrics used to analyze SOD in HSI. Among the drafted feature reduction algorithms, we found out MEV-SFS is the most efficient algorithm in terms of $f_1$ score, NSS, CC and MAE. While PCA produces the least precision, recall and $f_1$ score. Since MEV-SFS is a feature selection algorithm, it preserves the original features. In addition, the feature selection algorithms do not transform the original feature space, that makes them computationally less complex.

Fig. 7 [Images not available. See PDF.]

Visual representation of final $f_1$ score obtained corresponding to different feature selection and feature extraction algorithms and other state-of-the-art

Fig. 8 [Images not available. See PDF.]

Visual representation of final NSS obtained corresponding to different feature selection and feature extraction algorithms and other state-of-the-art

Fig. 9 [Images not available. See PDF.]

Visual representation of final MAE obtained corresponding to different feature selection and feature extraction algorithms and other state-of-the-art

Moreover, MNF also produces good results but it is computationally more complex than MEV-SFS. The graphical representation of $f_1$ score, NSS and MAE in SOD are given in Figs. 7, 8, and 9. In addition, the proposed algorithm is effective in detecting the object even if it slightly touches the boundary. The effectiveness of the boundary connectivity is demonstrated in Fig. 6.

Conclusion

In this paper, we present a technique for salient object detection using feature reduction and saliency optimization. To improve accuracy of SOD, we used boundary connectivity and illustrated its effectiveness. Moreover, we analyzed different feature selection and feature extraction algorithms. We observed that feature selection methods were computationally less complex. MNF and MEV-SFS gave comparable results but, MNF is computationally less efficient. The experimental results on HS-SOD dataset confirm that the proposed method outperforms other state-of-the-art techniques by achieving better Precision, Recall, $f_1$ score, NSS, CC and lower MAE. The different metrics indicate that our method produces better results and is also somewhat resistant to background noise as-well. For future work, we can use other cues under the optimization framework to evaluate if the results can be improved.

Data availability

The dataset used in this paper is available in [HS-SOD] repository and can be accessed at: [https://github.com/gistairc/HS-SOD]. The code will be made available based on a reasonable request to corresponding author ([email protected]).

Declarations

Conflict of interest

The authors declare they have no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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