## 1 Introduction  
When imaging an object through a scattering medium, the transmitted light is consisted by the ballistic light and the scattered light [1]. The ballistic light that carries the object information migrates through the scattering medium without deviating from the forward direction. The scattered light that has been scattered randomly in all directions loses the object information and undermines the imaging quality [2]. Most methods only use the ballistic light for imaging, and the scattered light is suppressed by employing a short time gate based on nonlinear phenomena [3; 4; 5; 6]. However, these methods are only effective for the weak scattering medium. When imaging through a highly scattering medium, almost all the light has been scattered randomly. In order to solve this problem, some methods that use the scattered light for imaging have recently been proposed. These include computational ghost imaging [7; 8], wavefront shaping [9; 10], speckle correlation [11; 12; 13], and optical transmission matrix (TM) [14; 15; 16; 17]. Generally speaking, imaging 3D objects that are hidden behind scattering media is a challenging work. On one hand, compared with the traditional one-way imaging through a scattering medium where the light wave has only been distorted once, imaging an object that hidden behind scattering media is more difficult since the light wave has gone through a round-trip distortion: On the way in for the illumination and on the way out for the detection. In such a case, the information of the object seemingly cannot be recovered. On the other hand, two-dimensional (2D) imaging only needs to recover the intensity (amplitude) features of the object. Three-dimensional (3D) imaging is much more difficult, as both the intensity (amplitude) and the morphological (phase) features of the object should be recovered. Although a lot of methods have ever been proposed, they are not effective for imaging 3D objects hidden behind highly scattering media. For example, the computational ghost imaging method cannot achieve round-trip imaging through scattering media because the information of the illumination wave is lost via the on-the-way-in distortion [18]. The wavefront shaping method only can image the 2D intensity object as the phase information of the object cannot be recovered, and due to the scanning process, its imaging speed is also restricted [19]. Similarly, the speckle correlation method relies on the inherent correlations in scattered speckle patterns, which makes it only applied to thin opaque layers, as the field-of-view is inversely proportional to the scattering medium thickness [12]. Different from the above methods, the optical transmission matrix (TM) can inherently characterize the scattering medium by giving the relationship between the input optical field and the output optical field [14]. Theoretically, it has the ability to recover the information of the object without loss and achieve the 3D complex amplitude imaging. Recently, this method has shown potential in focusing [15], delivering imaging [16], controlling transmitted energy [20], multispectral control [21], and acoustically modulated light [22]. However, generally speaking, they all belong to the cases working in the transmission imaging mode where the light is only distorted once. More recently, we also show that a 2D intensity object, which is completely hidden behind a scattering medium, can be imaged directly from the distorted output optical field [23]. Nevertheless, to the best of our knowledge, using the TM method to recover the complex amplitude information of the object that is hidden behind highly scattering media has not yet been reported. In this paper, we show in principle that based on the round-trip TM, it is possible to recover the complex amplitude information of the object that is hidden behind scattering media. The imaging method and experiment setup are introduced, and a simple virtual complex amplitude object has also been constructed to verify the effectiveness of our method.