Infrared Small Target Detection via tensor $L_{2,1}$ norm minimization and ASSTV regularization: A Novel Tensor Recovery Approach

In this paper, we have compiled a comprehensive list of symbols and definitions essential for reference, as presented in TABLE I. In order to effectively articulate our model, it is imperative to introduce fundamental definitions and theorems. Primarily, we emphasize a critically significant operation associated with tensor product and matrix product— the Discrete Fourier Transform (DFT). Here, we denote $\hat{\mathbf{v}}$ as an integral part of this discussion.

$\hat{\mathbf{v}}=\mathbf{F_{n}\mathbf{v}}$

where $\mathbf{F_{n}}$ is DFT matrix denoted as

$\mathbf{F}_{n}=\begin{bmatrix}1&1&1&\cdots&1\\ 1&\omega&\omega^{2}&\cdots&\omega^{n-1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^{n-1}&\omega^{2\left(n-1\right)}&\cdots&\omega^{\left(n-1\right)\left(n-1\right)}\end{bmatrix}\in\mathbb{C}^{n\times n}$,

where $\omega=e^{-\frac{2\pi i}{n}}$ and $i$ is the imaginary unit. So we can learn that $\frac{\mathbf{F}_{n}}{\sqrt{n}}$ is orthogonal, i.e.,

By providing a proof, we can easily obtain the following conclusions:

where $\otimes$ denotes the Kronecker product and $\frac{\mathbf{F}_{{n}_{3}}^{-1}\otimes\mathbf{I}_{{n}_{2}}}{\sqrt{n_{3}}}$ is orthogonal. Moreover, $\bar{\mathbf{A}}$ and $\text{bcirc}\left(\mathcal{A}\right)$ are defined as follows.

$\bar{\mathbf{A}}=\text{bdiag}\left(\hat{\mathcal{A}}\right)=\begin{bmatrix}\hat{\mathcal{A}}^{\left(1\right)}&&&\\ &\hat{\mathcal{A}}^{\left(2\right)}&&\\ &&\ddots&\\ &&&\hat{\mathcal{A}}^{\left(n_{3}\right)}\end{bmatrix}$,

$\text{bcirc}\left(\mathcal{A}\right)=\begin{bmatrix}\mathcal{A}^{\left(1\right)}&\mathcal{A}^{\left(n_{3}\right)}&\cdots&\mathcal{A}^{\left(2\right)}\\ \mathcal{A}^{\left(2\right)}&\mathcal{A}^{\left(1\right)}&\cdots&\mathcal{A}^{\left(3\right)}\\ \vdots&\ddots&\ddots&\vdots\\ \mathcal{A}^{\left(n_{3}\right)}&\mathcal{A}^{\left(n_{3}-1\right)}&\cdots&\mathcal{A}^{\left(1\right)}\end{bmatrix}$ .

Here, we provide the definition of the T-product: Let $\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ and $\mathcal{B}\in\mathbb{R}^{n_{2}\times l\times n_{3}}$. Then the t-product $\mathcal{A}\ast\mathcal{B}$ is the following tensor of size $n_{1}\times l\times n_{3}$:

where $\text{unfold}\left(\cdot\right)$ and $\text{fold}\left(\cdot\right)$ are the unfold operator map and fold operator map of $\mathcal{A}$ respectively, i.e.,

$\text{unfold}\left(\mathcal{A}\right)=\begin{bmatrix}\mathcal{A}^{\left(1\right)}\\ \mathcal{A}^{\left(2\right)}\\ \vdots\\ \mathcal{A}^{\left(n_{3}\right)}\end{bmatrix}\in\mathbb{R}^{n_{1}n_{3}\times n_{2}}$,

$\text{fold}\left(\text{unfold}\left(\mathcal{A}\right)\right)=\mathcal{A}$.

The most important component in $L_{2,1}$-norm [23] is the tensor QR decomposition, which is an approximation of SVD. Let $\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$. Then it can be factored as

where $\mathcal{Q}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}$ is orthogonal, and $\mathcal{R}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is analogous to the upper triangular matrix.

The TV regularizaiton [17, 18, 19] is widely applied in infrared small target detection, as it allows us to better utilize the information in the images.By incorporating the TV regularizaiton, we can maintain the smoothness of the image during the processes of image recovery and denoising, thereby eliminating potential artifacts. Typically, the TV regularizaiton is applied to matrices as shown in Equation (5).

However, this approach neglects the temporal information. Hence, we use a new method called asymmetric spatial-temporal total variation regularizaiton method to model both spatial and temporal continuity. There are two main reasons for choosing ASSTV regularizaiton method:

• •

  It facilitates the preservation of image details and better noise elimination.

• •

  By introducing a new parameter $\delta$, it allows us to assign different weights to the temporal and spatial components, enabling more flexible tensor recovery.

Now, we provide the definition of ASSTV, as shown in Equation (6):

Where $D_{h}$, $D_{v}$, and $D_{z}$ are the horizontal, vertical, and temporal difference operators, respectively. $\delta$ denotes a positive constant, which is used to control the contribution in the temporal dimension. Here are the definitions of the three operators:

In the realm of matrix recovery, a commonly employed approach involves solving the nuclear norm minimization problem through Singular Value Decomposition (SVD) to disentangle noise and restore images. When applying the SVD to a single channel of an RGB image, the singular values can be observed, as depicted in Fig. 1. However, the SVD is computationally demanding and often requires a substantial amount of algorithmic execution time to converge. Consequently, we adopt a novel tensor recovery method based on tensor tri-factorization to address the $L_{2,1}$ norm minimization problem. This approach aims to achieve image recovery with reduced computational complexity.

First, we will introduce the matrix version. For the problem (10), there exists an optimal solution (11).

However, computing the entire matrix $Z$ requires significant computational resources and time. Moreover, in most cases, smaller singular values in the matrix are primarily used to reconstruct noise information within the matrix. Therefore, it is unnecessary to compute them during the process of matrix recovery. Hence, it aims to find a method that only computes the first few larger singular values and their corresponding singular vectors, accelerating the overall computation process.

This problem is essentially finding an approximate rank-$r$ approximation of the matrix $\mathbf{Z}$, meaning solving the problem (12).

Where $\mathbf{L}\in\mathbb{R}^{m\times r}$, $\mathbf{D}\in\mathbb{R}^{r\times r}$, $\mathbf{R}\in\mathbb{R}^{r\times n}$, and $r$ represents the rank of matrix $\mathbf{Z}$ ($r\in(0,n]$). $\mathbf{L}$ and $\mathbf{R}$ are column orthogonal and row orthogonal matrices respectively, while $\mathbf{D}$ is a regular square matrix. $\varepsilon$ is a positive error threshold. For problem (12), we can use the iterative method to solve it.

We initialize three matrices as follows: $\mathbf{L}_{1}=\text{eye}(m,r)$, $\mathbf{D}_{1}=\text{eye}(r,r)$, and $\mathbf{R}_{1}=\text{eye}(r,n)$. Then, we iterate to update $\mathbf{L}$, $\mathbf{D}$, and $\mathbf{R}$ using the QR decomposition. First, fixing $\mathbf{R}$, we have:

Next, fixing $\mathbf{L}$, we obtain:

Finally:

After multiple iterations, the tri-factorization of $\mathbf{Z}$ is completed.

Next, we utilize the obtained tri-factorization matrices to optimize our problem. First, we have $\mathbf{L}\mathbf{D}\mathbf{R}\approx Z$, and then we establish the following $L_{2,1}$ minimization model:

Since we know that both $\mathbf{L}$ and $\mathbf{R}$ are orthogonal, Equation (13) simplifies to Equation (14).

Let $\mathbf{C}=\mathbf{L}^{T}\mathbf{Y}\mathbf{R}^{T}$, then we have Equation (15).

We observe that this problem, resembling (10), can be solved using the contraction operator from (11), as shown in Equation (16).

Similarly, for the tensor ${\cal Z}$, we need to solve the following optimization problem:

We need to perform tri-factorization on the tensor ${\cal Z}$ and find three tensors such that ${{\cal L}{\cal D}{\cal R}\approx\cal Z}$, where ${\cal L}\in\mathbb{R}^{n_{1}\times r\times n_{3}}$ and ${\cal R}\in\mathbb{R}^{r\times n_{2}\times n_{3}}$ are orthogonal, and ${\cal D}\in\mathbb{R}^{r\times r\times n_{3}}$ is a tensor.

We obtain ${\cal L}$, ${\cal D}$, and ${\cal R}$ by solving the following optimization problem:

Then, we can utilize iterative method to obtain ${\cal L}$, ${\cal D}$, and ${\cal R}$. The final solution to the problem (17) is transformed into solving the following problem:

Simplifying (19) leads to Equation (20).

Let ${{{\cal L}^{*}}\left({{\cal B}-\frac{y}{\mu}}\right){{\cal R}^{*}}}={{\cal D}_{T}}$. The optimal solution for this modified problem is:

The complete algorithmic process is illustrated in Algorithm 1.

By combining the $L_{2,1}$ norm and ASSTV regularization, we present the definition of our model as follows:

$s.t.\quad\mathcal{K}=\mathcal{B}+\mathcal{T}+\mathcal{N},$

By utilizing the ASSTV formulation (6), we can rewrite (23) as follows:

$s.t.\quad\mathcal{K}=\mathcal{B}+\mathcal{T}+\mathcal{N},$

Our model employs the $L_{2,1}$ norm, which significantly enhances algorithm efficiency. Additionally, we utilize tensor low-rank approximation to better assess the background, and the use of ASSTV allows for more flexible recovery of tensor.

In this section, we utilize the Alternating Direction Method of Multipliers (ADMM) method to solve (23). First, we modify (23) to the following form:

$s.t.\quad{\cal K}={\cal B}+{\cal T}+{\cal N},{\cal Z}={\cal B},{{\cal V}_{1}}={{\cal K}_{h}}{\cal B},{{\cal V}_{2}}={{\cal K}_{v}}{\cal B},{{\cal V}_{3}}={{\cal K}_{z}}{\cal B}$

Next, we present the augmented Lagrangian formulation of (25):

where $\mu$ represent a positive penalty scalar and $y_{1}$, $y_{2}$, $y_{3}$, $y_{4}$, $y_{5}$ represent the Lagrangian multiplier. We will use ADMM algorithm to solve (26), which includes $\cal Z$, $\cal B$, $\cal T$, $V_{1}$, $V_{2}$, $V_{3}$ and $\cal N$. Then we will alternately update the variable as:

1. 1.

   We update the value of ${\cal Z}$ using the following equation:

   $${{\cal Z}^{k+1}}=\mathop{\arg\min}\limits_{Z}{\left\|{\cal Z}\right\|_{2,1}}+\frac{{{\mu^{k}}}}{2}\left\|{{\cal Z}-{\cal B}+\frac{{y_{2}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}$$

   (27)

   For more details about the update of ${\cal Z}$, please refer to Algorithm 1. In the experiment, one iteration produced promising results, so we only performed one.

2. 2.

   We update the value of ${\cal B}$ using the following equation:

   $$\begin{array}[]{l}{{\cal B}^{k+1}}=\frac{{{\mu^{k}}}}{2}\left({\left\|{{\cal K}-{\cal B}-{{\cal T}^{k}}-{{\cal N}^{k}}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}}\right.\\ +\left\|{{{\cal Z}^{k+1}}-{\cal B}+\frac{{y_{2}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}+\left\|{{\cal V}_{1}^{k}-{{\cal K}_{h}}{\cal B}+\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\\ \left.{+\left\|{{\cal V}_{2}^{k}-{{\cal K}_{v}}{\cal B}+\frac{{y_{4}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}+\left\|{{\cal V}_{3}^{k}-{{\cal K}_{z}}{\cal B}+\frac{{y_{5}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}}\right)\end{array}$$

   (28)

   To solve (28), we can utilize the following system of linear equations:

   $$\left({2I+\Delta}\right){{\cal B}^{k+1}}={{L}^{k}_{A}}+{\theta_{1}}+{\theta_{2}}+{\theta_{3}}$$

   (29)

   where $\Delta={\cal K}_{h}^{T}{{\cal K}_{h}}+{\cal K}_{v}^{T}{{\cal K}_{v}}+{\cal K}_{z}^{T}{{\cal K}_{z}},{{\cal L}^{k}}={\cal K}-{{\cal T}^{k}}-{{\cal N}^{k}}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}+{\cal Z}+\frac{{y_{2}^{k}}}{{{\mu^{k}}}},{\theta_{1}}={\cal K}_{h}^{T}\left({{\cal V}_{1}^{k}+\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right),{\theta_{2}}={\cal K}_{v}^{T}\left({{\cal V}_{2}^{k}+\frac{{y_{4}^{k}}}{{{\mu^{k}}}}}\right),{\theta_{3}}={\cal K}_{z}^{T}\left({{\cal V}_{3}^{k}+\frac{{y_{5}^{k}}}{{{\mu^{k}}}}}\right)$ ,and T is the matrix transpose. By considering convolutions along two spatial directions and one temporal direction, we obtain a new Equation (30) as follows:

   $${{\cal B}^{k+1}}={F^{-1}}\left({\frac{{F\left({{{\cal L}^{k}}+{\theta_{1}}+{\theta_{2}}+{\theta_{3}}}\right)}}{{2+{{\sum{{}_{i\in\left\{{h,v,z}\right\}}F\left({{{\cal K}_{i}}}\right)}}^{H}}F\left({{{\cal K}_{i}}}\right)}}}\right)$$

   (30)

3. 3.

   We update the value of ${\cal T}$ using the following equation:

   $$\begin{split}{{\cal T}^{k+1}}&=\mathop{\arg\min}\limits_{T}{\lambda_{s}}{\left\|{\cal T}\right\|_{1}}\\ &+\frac{{{\mu^{k}}}}{2}\left\|{{\cal K}-{{\cal B}^{k+1}}-{\cal T}-{{\cal N}^{k}}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\end{split}$$

   (31)

   The closedform solution of (31) can be obtained by resorting to the element-wise shrinkage operator, that is:

   $${{\cal T}^{k+1}}={\cal T}{h_{{\lambda_{1}}\left({{\mu^{k}}}\right)}}^{-1}\left({{\cal K}-{{\cal B}^{k+1}}-{{\cal N}^{k}}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}}\right)$$

   (32)

4. 4.

   We update the values of ${\cal V}_{1}$, ${\cal V}_{2}$, and ${\cal V}_{3}$ using the following equations:

   $$\left\{\begin{array}[]{l}\begin{split}V_{1}^{k+1}&=\mathop{\arg\min}\limits_{V1}{\lambda_{tv}}{\left\|{{V_{1}}}\right\|_{1}}\\ &+\frac{{{\mu^{k}}}}{2}\left\|{V_{1}^{k}-{{\cal K}_{h}}{{\cal B}^{k+1}}+\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\\ V_{2}^{k+1}&=\mathop{\arg\min}\limits_{V2}{\lambda_{tv}}{\left\|{{V_{2}}}\right\|_{1}}\\ &+\frac{{{\mu^{k}}}}{2}\left\|{V_{2}^{k}-{{\cal K}_{v}}{{\cal B}^{k+1}}+\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\\ V_{3}^{k+1}&=\mathop{\arg\min}\limits_{V3}\delta{\lambda_{tv}}{\left\|{{V_{3}}}\right\|_{1}}\\ &+\frac{{{\mu^{k}}}}{2}\left\|{V_{3}^{k}-{{\cal K}_{z}}{{\cal B}^{k+1}}+\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\\ \end{split}\end{array}\right.$$

   (33)

   Here, we can also utilize the element-wise shrinkage operator to solve the above problem. The Equation is as follows:

   $$\left\{\begin{array}[]{l}V_{1}^{k+1}={\cal T}{h_{{\lambda_{tv}}\left({{\mu^{k}}}\right)}}^{-1}\left({{{\cal K}_{h}}{{\cal B}^{k+1}}-\frac{{y_{3}^{k}}}{{{\mu^{k}}}}}\right)\\ V_{2}^{k+1}={\cal T}{h_{{\lambda_{tv}}\left({{\mu^{k}}}\right)}}^{-1}\left({{{\cal K}_{v}}{{\cal B}^{k+1}}-\frac{{y_{4}^{k}}}{{{\mu^{k}}}}}\right)\\ V_{3}^{k+1}={\cal T}{h_{\delta{\lambda_{tv}}\left({{\mu^{k}}}\right)}}^{-1}\left({{{\cal K}_{z}}{{\cal B}^{k+1}}-\frac{{y_{5}^{k}}}{{{\mu^{k}}}}}\right)\end{array}\right.$$

   (34)

5. 5.

   We update the value of ${{\cal N}^{k+1}}$ using the following equation:

   $$\begin{split}{{\cal N}^{k+1}}&=\mathop{\arg\min}\limits_{\cal N}{\lambda_{3}}\left\|{\cal N}\right\|_{F}^{2}\\ &+\frac{{{\mu^{k}}}}{2}\left\|{{\cal K}-{{\cal B}^{k+1}}-{{\cal T}^{k+1}}-{\cal N}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}\end{split}$$

   (35)

   Sequence	Frames	Image Size	Average SCR	Target Descriptions	Background Descriptions
   1	120	$256\times 256$	5.45	Far range, single target	ground
   2	120	$256\times 256$	5.11	Near to far, single target	ground
   3	120	$256\times 256$	6.33	Near to far, single target	ground
   4	120	$256\times 256$	6.07	Far to near, single target	ground
   5	120	$256\times 256$	5.20	Far distance, single target	ground-sky boundary
   6	120	$256\times 256$	1.98	Target near to far, single target, faint target	ground

   Methods	Acronyms	Parameter settings
   Total Variation Regularization and Principal Component Pursuit [25]	TV-PCP	$\lambda_{1}=0.005$, $\lambda_{2}=\frac{1}{\sqrt{max(M,N)}}$, $\beta=0.025$, $\gamma=1.5$
   Partial Sum of the Tensor Nuclear Norm [26]	PSTNN	Sliding step: 40, $\lambda=\frac{0.6}{\sqrt{max(n1,n2)}},pathchsize:30\times 30$
   Multiple Subspace Learning and Spatial-temporal Patch-Tensor Model [27]	MSLSTIPT	$L=6$,$P=0.8$,$\lambda=\frac{1}{\sqrt{n_{3}\times max(n_{1},n_{2})}}$,$patchsize:30\times 30$
   Nonconvex tensor fibered rank approximation [28]	NTFRA	$Slidingstep:40$,$patchsize:40\times 40$,$\lambda=\frac{1}{\sqrt{n_{3}\times max(n_{1},n_{2})}}$,$\beta=0.01$
   Non-Convex Tensor Low-Rank Approximation [21]	ASSTV-NTLA	$L=3$, $H=6$, $\lambda_{tv}=0.5$, $\lambda_{s}=\frac{H}{\sqrt{max(M,N)\times L}}$, $\lambda_{3}=100$
   Tensor $L_{2,1}$ Norm Minimization via Tensor QR Decomposition	ASSTV-TLNMTQR	$r=180$, $L=3$, $H=6$, $\lambda_{tv}=0.5$, $\lambda_{s}=\frac{H}{\sqrt{max(M,N)\times L}}$, $\lambda_{3}=100$

   The solution of (35) is as follows:

   $${{\cal N}^{k+1}}=\frac{{{\mu^{k}}\left({{\cal K}-{{\cal B}^{k+1}}-{{\cal T}^{k+1}}}\right)+y_{1}^{k}}}{{{\mu^{k}}+2{\lambda_{3}}}}$$

   (36)

6. 6.

   Updating multipliers y1; y2; y3; y4; y5 with other variables being fixed:

   $$\left\{\begin{array}[]{l}y_{1}^{k+1}=y_{1}^{k}+{\mu^{k}}\left({{\cal K}-{{\cal B}^{k+1}}-{{\cal T}^{k+1}}-{{\cal N}^{k+1}}}\right)\\ y_{2}^{k+1}=y_{2}^{k}+{\mu^{k}}\left({{{\cal Z}^{k+1}}-{{\cal B}^{k+1}}}\right)\\ y_{3}^{k+1}=y_{3}^{k}+{\mu^{k}}\left({V_{1}^{k+1}-{{\cal K}_{h}}{{\cal B}^{k+1}}}\right)\\ y_{4}^{k+1}=y_{4}^{k}+{\mu^{k}}\left({V_{2}^{k+1}-{{\cal K}_{v}}{{\cal B}^{k+1}}}\right)\\ y_{5}^{k+1}=y_{5}^{k}+{\mu^{k}}\left({V_{3}^{k+1}-{{\cal K}_{z}}{{\cal B}^{k+1}}}\right)\end{array}\right.$$

   (37)

7. 7.

   Updating ${\mu^{k+1}}$ by ${\mu^{k+1}}=\min(\rho{\mu^{k}},{\mu_{\max}})$.

   0:  infrared image sequence ${k_{1}},{k_{2}},\cdots,{k_{P}}\in\mathbb{R}{{}^{{n_{1}}\times{n_{2}}}}$, number of frames $L$, parameters ${\lambda_{s}},\quad{\lambda_{tv}},\quad{\lambda_{3}},\quad\mu\textgreater 0$;

   1:  Initialize: Transform the image sequence into the original tensor ${\cal K}$, ${\cal B}^{0}$=${\cal T}^{0}$=${\cal N}^{0}$=${\cal V}_{i}^{0}$=0, $i=1,2,3$, ${y}_{i}^{0}$=0, ${\mu}_{0}=0.005$, ${\mu}_{max}=1e7$, $k=0$, $i=1,\dots,5$, $\rho=1,5$, $\xi=1e-6$.

   2:  while Not converged do

   3:     Update ${{\cal Z}^{k+1}}$ by 27

   4:     Update ${{\cal B}^{k+1}}$ by 28

   5:     Update ${{\cal T}^{k+1}}$ by 31

   6:     Update $V_{1}^{k+1}$,$V_{2}^{k+1}$,$V_{3}^{k+1}$ by 33

   7:     Update ${{\cal N}^{k+1}}$ by 35

   8:     Update multipliers $y_{i}^{k+1}$, $i=1,2,\cdots,5$ by 37

   9:     Updating ${\mu^{k+1}}$ by ${\mu^{k+1}}=\min(\rho{\mu^{k}},{\mu_{\max}})$

   10:     Check the convergence conditions $\frac{{\left\|{{\cal K}-{{\cal B}^{k+1}}-{{\cal T}^{k+1}}-{\cal N}+\frac{{y_{1}^{k}}}{{{\mu^{k}}}}}\right\|_{F}^{2}}}{{\left\|{\cal K}\right\|_{F}^{2}}}\leq\xi$

   11:     Update $k=k+1$

   12:  end while

   12:  ${{\cal B}^{k+1}}$,${{\cal T}^{k+1}}$,${{\cal N}^{k+1}}$

This section aims to demonstrate the robustness of our algorithm through a series of experiments related to infrared small target detection. We employ 3D Receiver Operating Characteristic Curve (ROC) [29] to assess the performance of our algorithm in separating tensor $\mathcal{T}$. Additionally, we have selected five comparative methods to highlight the superiority of our approach.

In our comparative experiments, we systematically evaluated the performance of our ASSTV-NTLA algorithm against five other algorithms, all of which were tested on the six sequences outlined in TABLE II. The specific parameter settings for each algorithm can be found in III.

Upon analyzing the experimental results depicted in Fig. 21 to Fig. 22, we observed that certain algorithms demonstrated detection rates either higher or nearly comparable to ASSTV-NTLA and ASSTV-TLNMTQR (ours) in specific sequences. However, a closer examination of TABLE IV reveals that these seemingly impressive detection rates are accompanied by high false alarm rates. This critical insight underscores the superior performance of ASSTV-NTLA and ASSTV-TLNMTQR (ours), as they consistently achieve the highest detection rates while maintaining lower false alarm rates. Furthermore, the comprehensive analysis presented in TABLE IV highlights an additional advantage of our ASSTV-TLNMTQR algorithm, which consistently performs at a speed approximately $25\%$ faster than ASSTV-NTLA across all six sequences. Notably, this accelerated speed is achieved without compromising accuracy, as evidenced by the mere $0.2\%$ difference in average accuracy compared to the ASSTV-NTLA method.