Enhancement of Sonar Detection in Karst Caves Through Advanced Target Location and Image Fusion Algorithms

(1) Underlying principles of karst cave detection via sonar

Sonar detection has perennially stood as a focal point of research. Its employment in karst cave exploration is recognized as a standard method. This technique is predominantly anchored in the identification of entities within cavities. Sound data are transmuted directly into electrical signals or alternative communication forms, aiming to furnish more lucid outcomes for operators. While this mode of detection has been shown to effectively map the cave's interior milieu and geological layout, it is not devoid of specific inaccuracies. A rigorous evaluation of its precision is mandated [13]. The intricate principles of karst cave detection via sonar are delineated in Figure 1.

As illustrated in Figure 1, sonar encompasses an array of acoustic sensors and mechanized instruments. Within this assembly, electrical signals, generated by the transmitter housed in the electronic cabinet, are relayed to the cavern post interfacing with the transducer. The strategic organization of these transducers constitutes the array, facilitating the mutual conversion of acoustic and electrical energies. When acoustic pulses navigate aquatic mediums, reflections arise upon target encounter. These reverberated sound waves are subsequently captured by the transducer, undergoing conversion into electrical impulses, which are then relayed to the electronic cabinet. Consequently, variations in echo waveforms and intensities across distinct frequency spectrums become visible on display panels.

In the context of karst cave detection using sonar, underground regions might encompass an amalgam of gaseous, liquid, and solid phases. Cave walls, perforated with orifices of varied dimensions and morphologies, exert differential impacts on acoustic signals. Hence, accurate interpretations necessitate the consideration of myriad external conditions and inherent challenges. The potential strengths and limitations of this technology in exploration pursuits will be analyzed. Furthermore, emphasis will be placed on elucidating the contributions of target localization algorithms and image fusion methodologies to the sonar-mediated identification of karst formations.

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Figure 1. The principle of sonar detection in caves

(2) Underpinnings of target localization techniques

Target localization serves as a cornerstone within radar systems. This process entails the derivation of the discrepancy between measured and actual observed values through the formulation of a mathematical model. Among the myriad steps involved, identifying the optimal solution stands out as the most intricate. With the burgeoning advancements in modern electronic information technology coupled with satellite navigation, technical prowess in this domain is observed to be increasingly refined. Its utility, especially in multi-sensor information fusion and real-time tracking, has been underscored by promising prospects [14]. The algorithmic principles guiding target localization are depicted in Figure 2.

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Figure 2. Algorithmic framework of target localization

Object localization is recognized as an adaptive learning algorithm predicated on fuzzy logic principles. Initially, upon input target perception, its location is promptly estimated by the sensor network. Subsequent outputs are deduced based on the prevailing state, thereby refining the prospective target domain. In the ensuing phase, visual characteristics are extracted from the procured image and supplementary convolutional layer features are garnered. A training set is constructed leveraging these features, where filtered pertinent data undergoes utilization for category detection and locational discernment. Ultimately, through a combination of image segmentation and feature extraction, classification outcomes are assessed and the real-time status of the target is ascertained, ensuring precision in target tracking and recognition. Owing to its efficacy in enhancing identification accuracy, the target localization algorithm has been found to be of significant merit in practical applications.

(3) Elucidation of the image fusion procedure

The progression of computer technology and the proliferation of internet usage have spurred heightened expectations concerning visual perception capabilities. Image fusion encompasses the amalgamation of data derived from disparate sources, typically gathered via a range of sensors. Once relayed for computational processing via networks, a coherent, precise, and intuitive aggregate assessment is rendered [15]. Generally initiated at the source, image processing assimilates salient environmental information, potentially mutable along its pathway, capitalizing on the interrelation among multi-source images to actualize target identification or detection. This technique is acknowledged for its superior real-time responsiveness and voluminous information capture. The intricate steps underpinning image fusion are delineated in Figure 3.

The fusion methodology can be segregated into four distinct stages. Initially, 'n' raw datasets or images are procured from external environments. Subsequent to this acquisition, unification and specialized treatment are conducted, aligning with data layer standards—a process colloquially termed as 'image registration'. Predominantly reliant on a reference image, the auxiliary images undergo intricate processing to synchronize temporally and spatially with the entirety of the image collection. This is followed by image preprocessing, a phase marked by novel representation of object interrelations within the backdrop and encapsulation of their intrinsic content. Concluding the sequence, the processed images are amalgamated to produce comprehensive visuals, which are subsequently presented to end-users or diverse application frameworks. Grounded in multi-scale image recognition paradigms, image fusion methodologies have emerged as avant-garde avenues for data acquisition. Their recurrent incorporation within an array of intelligent systems underscores their adaptability and resilience in multifaceted environments.

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Figure 3. Schematic representation of the image fusion procedure

2.1 Formulation of the target localization algorithm and image fusion algorithm

1) Target Localization Algorithm

Object localization has been established as a pivotal axis of research within deep learning algorithms. By fractionating intricate issues into several rudimentary sub-processes, this adaptive tracking technique, founded on the least squares principle, is manifested. It has been observed to surmount certain limitations inherent to conventional PID (Proportional Integral Derivative) control [16]. Furthermore, dynamic parameter adjustments, tailored to specific prerequisites, can be undertaken to achieve the anticipated outcome.

1. Elucidation of the Least Squares Matrix Solution

Let the matrix representation of the function f(n) be given as:

${{f}_{a}}(\text{n})=Na$                 (1)

Wherein N represents the matrix of dimension i×j, with i denoting the number of samples and j indicating the number of sample features. a is identified as the feature vector.

Subsequently, the formulation of the loss function is represented as:

$C(a)=\frac{1}{2}{{(Na-M)}^{q}}Na-M$                  (2)

On differentiating Eq. (2) concerning the eigenvector and equating to zero, the following is derived:

$\frac{\partial }{\partial a}C(a)={{N}^{q}}(Na-M)=0$                      (3)

Upon sequential arrangement, the expression becomes:

$NqNa=NqM$                     (4)

Multiplication on both sides by the inverse of (NqN) yields:

$a=(NqN)-1{{N}^{q}}M$                      (5)

From Eq. (5), the vector a can be computed. It has been discerned that sonar detection systems, leveraging the least square method, exhibit proficiency in discerning diverse hole types, thus bolstering classification accuracy.

2) Deep Neural Network

At the heart of deep learning lies object localization. It is posited that deep learning serves as the essential component of the deep neural network, recognized for its robust spatial reasoning capabilities and elevated prediction accuracy. Such attributes enable it to facilitate swift analyses of intricate challenges or unstructured data sets [17].

Within the perceptron of a neural network, the input layer's neurons are harnessed for their learning capacities during training sessions. Conversely, artificial neural network algorithms are employed by the output layer for data processing. A linear association between the output and the input is observed:

$s=\sum\limits_{a=1}^{i}{{{p}_{a}}{{q}_{a}}}+c$                     (6)

where, p stands as the coefficient delineating the linear relationship, while c ' acts as the bias coefficient.

Given the activation function λ(s) and deeming the output value of both the hidden and output layers as e, the subsequent output from the hidden layer is defined as:

$\begin{align}& {{e}_{1}}2=\lambda ({{s}_{1}}2) =\lambda ({{p}_{11}}2{{q}_{1}}+{{p}_{12}}2{{q}_{2}}+{{p}_{13}}2{{q}_{3}}+{{c}_{1}}2) \\\end{align}$                   (7)

Furthermore, the output layer's value is represented by:

$\begin{align}  & {{e}_{1}}3=\lambda ({{s}_{1}}3)=\lambda ({{p}_{11}}3{{q}_{1}}2+{{p}_{12}}3{{q}_{2}}2+{{p}_{13}}3{{q}_{3}}2+{{c}_{3}}3) \\\end{align}$                  (8)

Upon analysis of Eqs. (7) and (8), a pervasive pattern is discerned:

${{e}_{m}}n=\lambda ({{s}_{m}}n)=\lambda (\sum\limits_{k=1}{d}{{p}_{mk}}n{{e}_{k}}n-1+c_{m}^{n})$                  (9)

When expressed in matrix notation, the nth layer's output is articulated as:

$en=\lambda (sn)=\lambda (Anen-1+cn)$              (10)

where, A denotes a linear coefficient matrix.

Through this methodology, the accurate identification of surface features is efficiently achieved. For enhancement of target localization precision, diverse variants of the Kalman filter are routinely utilized.

3) Algorithm for image fusion

The algorithm for image fusion draws its foundation from multi-scale image recognition technology. In intricate environments, this approach is proficient in distinguishing between varying targets and background data. The transition from 2D to 3D processing is facilitated with heightened efficiency and speed. Such enhancement not only elevates monitoring accuracy but also fortifies adaptability and resilience in multifaceted scenes [18].

a. Local Standard Deviation

To some degree, the local standard deviation mirrors the relative stability of a region. This deviation is pivotal in gauging the trajectory of alterations in image contrast. Furthermore, the span of variation serves as a crucial yardstick in assessing the modulating patterns of gray values.

For an observational window with dimensions a×a, the equations for its standard deviation are articulated as:

$\partial =\sqrt{\frac{1}{{{a}^{2}}}de({{e}_{ij}}-\bar{e})}$                     (11)

$de=\sum\limits_{i=m-\frac{a-1}{2}}^{m+\frac{a-1}{2}}{{}}_{{}}^{{}}\sum\limits_{j=n-\frac{a-1}{2}}^{n+\frac{a-1}{2}}{{}}$                    (12)

where, e_ij is indicative of the pixel value within the given vicinity, juxtaposed against the mean value derived from all neighboring pixels.

b. Non-Negative Matrix Factorization

Non-negative Matrix FactorizationIn recent times, the introduction of non-negative matrix factorization has been noted as a contemporary method. This technique has not only showcased its aptitude to address a multitude of challenges that elude traditional algorithms but also boasts attributes such as simplicity, efficacy, and ease of programming. The utility of this method in computing decoupling and stabilizing nonlinear systems has been extensively recognized [19].

Given a matrix C, composed entirely of non-negative components, it can be represented by the multiplication of two distinct matrices as:

${{C}_{a\times b}}={{A}_{a\times r}}{{B}_{r\times b}}$                  (13)

where, matrix C exhibits dimensions of a×b, while matrix A has dimensions of a×r, and B holds the dimensions r×b. The factor r delineates the dimensionality pertinent to the non-negative matrix factorization.

The range of R is constrained by:

$r<\frac{ab}{a+b}$                   (14)

At the instance when r = 0, the feature base matrix, A, becomes synonymous with all the image attributes.

Eq. (14) can be depicted in the form of a vector product:

${{C}_{n}}=\sum\limits_{m=1}^{r}{{{A}_{m}}}{{B}_{mn}}$                (15)

where, C_n signifies the nth column of matrix C, while A_m characterizes the mth column of matrix A.

From the inference drawn from Eq. (15), each individual column of matrix C can be mirrored by the pertinent components found in matrices A and B. Consequently, matrix A is perceived as the foundational matrix, with matrix B taking on the role of the coefficient matrix. In scenarios where A embodies the intrinsic image characteristics, the following relationship emerges:

$C\approx AB$                   (16)

c. Wavelet Transform

The wavelet transform has been observed to effectively distil detailed information from images. Particularly in instances marked by substantial noise, wavelet decomposition proves superior in disentangling texture and colour attributes, consequently achieving an augmented compression ratio. Such capabilities underscore its pivotal role in image enhancement [20].

The continuous function characterizing the wavelet transform is delineated by:

$G(i,j)=\left\langle g \right.,\left. {{\lambda }_{i,j}} \right\rangle =\frac{1}{\sqrt{i}}\int{_{-\infty }^{\infty }}g(e)\lambda \left( \frac{e-j}{i} \right)ce$                   (17)

$g(e)=\frac{1}{{{D}_{\lambda }}}\int{_{-\infty }^{\infty }}G(i,j){{\lambda }_{i,j}}(e)\frac{cicj}{{{i}^{2}}}$                         (18)

In this context, $\lambda_{i, j}(e)$ epitomizes the shift and dilation of the foundational basis. The factor i, acting as the dilation factor, prescribes the support length of the wavelet function, ensuring i remains greater than 0. On the other hand, 'b' stands as the shift factor, concurrently serving as the wavelet's temporal parameter.

Upon setting constraints wherein _{$i_0$} exceeds 1 and $j_0$ is greater than 0, the expression for the discrete wavelet transform emerges as:

${{\lambda }_{x,y}}(e)={{i}_{0}}^{-\frac{x}{2}}\lambda ({{i}_{0}}^{-x}e-y{{j}_{0}})$                  (19)

where, both x and y are integral in nature.

In the realm of image processing, the unidimensional wavelet transform is extrapolated to a bidimensional transformation space:

$G(i,{{j}_{m}},{{j}_{n}})=\iint{g(m,n){{\lambda }_{i,{{j}_{m}},{{j}_{n}}}}}(m,n)cmcn$                 (20)

Within this equation, m and n denote the translation of the image across their specific dimensions.

Subsequently, the inverse operation associated with the two-dimensional wavelet transform is conceptualized as:

$\begin{align}  & \iint{g}(m,n)  =\frac{1}{{{D}_{\lambda }}}\iiint{G}\left( i,{{j}_{m}},{{j}_{n}} \right){{\lambda }_{i,{{j}_{m}},{{j}_{n}}}}(m,n)c{{j}_{m}}c{{j}_{n}}\frac{ci}{{{i}^{3}}} \\\end{align}$              (21)

From the accumulated data and the employed methods for sonar detection of karst caves, along with the specific findings of this study, several key observations and outcomes were deduced:

(1) A comprehensive review of the extant literature was conducted, intertwining it with both contemporary empirical findings and theoretical studies. Through this analysis, the present developmental trajectory of sonar systems was discerned. Its pivotal significance, as well as the inherent challenges in the domain of karst engineering exploration, was illuminated. Notably, the implications of the target positioning algorithm and image fusion algorithm in the realm of sonar detection of karst caves were accentuated.

(2) By pivoting on target positioning and image fusion, the feature vector was computed using the least square method, facilitating the differentiation of varied karst cave types. Viewing this through the lens of deep learning, a surge in positioning accuracy was perceived. The computation of the local standard deviation depicted the evolving trends in image contrast. Furthermore, wavelet transformation was employed to segregate features, which in turn, expedited the extraction of actionable information, thereby amplifying the image's overall efficacy.

(3) For a more empirical understanding, 20 adept karst cave detectors were enlisted for an investigative study, wherein comparative experiments were orchestrated. From this initiative, it was observed that sonar detection, post the incorporation of the target positioning and image fusion algorithms, often yielded more streamlined results. Such detections were marked by heightened accuracy and reliability, leading to images that were both sharper in clarity and more intuitive in interpretation.