## 1 Introduction  
Thermal barrier coatings (TBCs), due to their excellent high temperature resistance, low thermal conductivity, corrosion protection, and anti-wear properties, are considered crucial protective layers. They have been extensively applied in the aerospace industry for hot-section components such as aero-engine blades [1, 2]. Blades are exposed to extreme service environments, such as the impact of high-temperature combustion gases, oxidizer erosion,and thermal stresses. These factors lead to increased blade surface temperature and material thermal expansion, resulting in issues such as thermal fatigue, oxidation, and mechanical failure [3]. The application of TBCs has effectively mitigated these detrimental effects on the blades, enhancing their heat resistance, corrosion resistance, and mechanical strength [4]. They have played a crucial role in reducing surface temperatures, improving component durability, and extending service life. With the rapid development of the aerospace industry and increasing demands for engine performance, the quality and performance of TBCs have become a focal point of attention [5]. TBCs have typically been applied with a thickness ranging from 100 to 600 \(\upmu\)m and consisting of three layers: a topcoat (TC) comprising an yttria-stabilized zirconia (YSZ) ceramic material, a band coat (BC) made of MCrAlY (where M represents Ni, Co, or other materials) as the metallic bond layer, and a high-temperature nickel-based alloy substrate. The topcoat possesses a low thermal conductivity and high thermal resistance, effectively reducing heat conduction and mitigating the surface temperature of the blades. The bond coat acts as an intermediary layer, providing adhesion and overall stability between the TBCs and the blade substrate [6]. However, as thermal barrier coatings are exposed to harsh operating conditions for prolonged periods, they might experience various forms of failure due to factors such as thermal fatigue, thermal stress mismatch, coating oxidation and corrosion, and particle impact [7; 8; 9]. As the TBCs are subjected to service conditions, the formation of thermal growth oxides (TGOs) occurs between the top ceramic layer and the bond coat [10]. Nevertheless, the presence of pore structures in TBCs is closely associated with various factors. The formation of these pores could be attributed to processes such as high-temperature spraying and sintering, which could result in particle accumulation and melting. Under thermal cycling conditions, TBCs experience complex effects such as temperature gradients, thermal stresses, and mechanical loads, leading to the formation of microcracks and delamination within the coating and subsequently generating pores. The porosity of TBCs significantly influences their performance. TBCs have typical porosity values which range from 3 to 20 vol%. Increasing the porosity of the coating is advantageous as it reduces its thermal conductivity, thereby enhancing its thermal insulation capabilities. The existence of air within the pores leads to a reduced effective thermal conductivity, acting as a thermal barrier and impeding the heat transfer through the coating. Nevertheless, excessive porosity could negatively impact the mechanical integrity and durability of the coating, rendering it susceptible to cracking and delamination [11; 12]. Therefore, the accurate evaluation of porosity in TBCs is of crucial importance to ensure coating quality, performance, and reliability. Accurate determination of the porosity assessment in TBCs has emerged as a key issue in research [13; 14; 15; 16; 17]. Various methods have been proposed to assess the porosity of these coatings, leading to significant research progress. Surface or cross-sectional observations and analyses of the coatings can be performed using a scanning electron microscope (SEM) [18] or transmission electron microscope (TEM) [19].