## 1 Introduction  
Crack distribution in rock fracture is crucial for finding and understanding the damage zone, which can be used to indicate the extent and the potential direction of crack propagation [1, 2]. Therefore, the distribution characterization of the cracks plays a fundamental role in the damage assessment of rock compression failure. The distribution of the cracks in rocks has been realized from different spatial dimensions, such as the 2D crack space assessment of cracks on the rock surface, including digital image correlation technology [3], infrared thermal imaging technology [4], optical observation technology [5], etc. There are methods for 3D crack space assessment of cracks inside the rock, including computer tomography [6], acoustic emission (AE) [7], etc. They demonstrated the power of acquiring crack distribution from light waves, electromagnetic waves, and sound waves. It is known that the original openings inside the rock are dislocated under loading, and the released energy during rupture propagates through the rock in the form of elastic waves, monitored as AE [8]. It has been widely used in rock mechanics and engineering for damage assessment and hazard warning [9, 10, 11]. AE monitoring was performed on rock failure experiments with different loading methods and paths, and the microcrack distribution results successfully indicated the real rupture location and the coincident damage shape [12; 13; 14; 15; 16; 17; 18]. As a result, the microcracks were found to be more prevalent in severely damaged zones in the laboratory and the field [9]. Therefore, for a better visualization of the damage zone directly by the microcrack, the density estimation method was adopted on the microcrack distribution results to find severely damaged areas [19; 20]. Furthermore, the micro-seismic events detected in earthquakes were displayed within a 2D map from different cross-sectional views [21]. The mainshocks could be found by the density contour [22]. The results from the high-density area demonstrate that microcracks have a potential elliptical distribution pattern [20]. Moreover, ellipsoidal-like results were observed in a granite boulder by a 3D kernel density estimation [23]. These studies provide strong evidence for the hypothesis theory that the single-cracked space is of ellipsoidal distribution, and demonstrate the feasibility of single-cracked space in characterizing rock failure studies. In addition, it is worth noting that crack recognition of multiple cracks was performed in the triaxial compression experiments, and the results of crack space characterized by ellipsoids were consistent with macrocrack rupture surfaces, and the distribution pattern has been proven [24; 25]. However, further investigation of the relationship between crack space and mechanics, and the optimization on the crack space size needs to be implemented for a more accurate result. The investigations of the spatial distribution of cracks have focused on hydraulic fracturing. In particular, the stimulated reservoir volume (SRV) was proposed to estimate the storage of oil and gas [26; 27; 28]. Convex optimization algorithms such as 3D convex hull and minimum volume enclosing ellipsoid (MVEE) were successfully adopted to evaluate the SRV from the perspective of microcracks [29; 30; 31]. However, the mechanism and mechanical behavior are different between hydraulic fracturing and compression failure in rocks. First, the fracturing fluid acts as a stressor in the process of hydraulic fracturing, accompanied by physical and chemical reactions. The fluid flows in via the hydraulic crack and forms a tensile stress zone near the non-closed fracture zone, resulting in the continuous expansion of the fracture and the eventual formation of the stimulated reservoir volume [32]. In contrast to hydraulic fracturing, the role of fluids is missing in regular rock compression failure. In the second, the mechanisms of rock compression failure include not only tensile failure, but also a part of shear failure [33; 34; 35; 36].