1. Introduction

A brain tumor is a mass of abnormal cells involved in a chaotic process of somatic driver mutations [1,2], where they cause various symptoms and increase the risk of brain damage. In fact, the secondary tumor infiltrates neighboring healthy tissues and proliferates within the brain or its membranes, making it critical to determine its shape and volume to ensure effective management of patients in the early stages of cancer. Magnetic resonance imaging (MRI) is the most commonly used non-invasive imaging modality for brain tumor detection [3–5]. MRI uses radio waves and a strong magnetic field to create a series of cross-sectional images of the brain. In other words, the 3D anatomical details of a tumor are represented as a set of parallel 2D cross-sectional images. Representing 3D data as projected 2D slices results in a loss of information and can raise questions about tumor prognosis. In addition, 2D slices do not accurately represent the complexity of brain anatomy. Therefore, interpretation of 2D images requires specialized training. Therefore, volume reconstruction from sequential parallel 2D cross-sectional slices is a necessity for 3D tumor visualization, In 2013, the authors in [6] proposed an improved interpolation technique to estimate missing slices and the Marching Cubes (MC) algorithm to mesh the tumor. For surface rendering, they applied the Phong shading and lighting model to better compute the tumor volume.

The 3D reconstruction of tumors first requires an appropriate segmentation of the region of interest. This 3D reconstruction helps radiologists to better diagnose patients and subsequently remove the entire tumor when surgical intervention is considered. Techniques presented in [7,8] are based on preprocessing, image enhancement, and contouring prior to reconstruction. In 2012, authors in [9] used a technique based on phase-contrast projection tomography to calculate the 3D density distribution in bacterial cells. In addition, an approach proposed by [10] in 2019 provides a technique for segmenting brain tumors in the 3D volume using a 2D convolutional neural network for tumor prediction. Authors in [11,12] conducted a comparison between conventional machine learning based techniques and deep learning based techniques. The latter are further categorized into 2D CNN and 3D CNN techniques. However, the results of techniques based on deep convolutional neural networks out perform those of machine learning techniques. As for the authors in [13,14], they introduced a two-stage optimal mass transport technique (TSOMT), which involve stransforming an irregular 3D brain image into a cube with minimal deformation, for segmenting 3D medical images. Automatic segmentation of a brain tumor from two-dimensional slices (coronal, sagittal, and axial), facilitated by convolutional neural networks [15], significantly aids in delineating the region of interest in 3D.

Conventional holography was invented by [16] in 1948 during his research to improve the resolution of electron microscopes. This invention evolved in the 1960s with the advent of lasers, and holograms were recorded on plates or photosensitive films based primarily on silver ions that darkened under the influence of light. With the progress of high-resolution matrix detectors, digital holography was generalized in 1994 by the authors in [17], paving the way for numerous applications: holographic microscopy [18–20], quantitative phase imaging [21–23], color holography [24–26], metrology [27–29], holographic cameras [30], 3D displays [31–34], and head-up displays [35,36]. Authors in [37] were the first to use phase-shifting holography to eliminate unwanted diffraction orders from the hologram. They introduced spatial phase shifting using a piezoelectric transducer with a mounted mirror [38], and slight frequency modifications of acousto-optic modulators (AOMs) [39–41]. The latter technique is closely related to heterodyne detection methods.

Optical Scanning Holography (OSH) is considered an intelligent application for processing the pupil interaction. The pupil interaction scheme was implemented using optical heterodyne scanning by Korpel and Poon in 1979, and the use of an interaction scheme in a scanning illumination mode was developed by Indebetouw and Poon in 1992. By modifying one of the twolenses relative to the other (specifically, one lensis an open mask and the other is a pinhole mask) and defocusing the optical system, in 1985 author in [42] developed an optical scanning system capable of holographically recording a scanned object. This technique led to the invention of optical scanning holography. OSH has various applications, including optical scanning microscopy, 3D shape recognition, 3D holographic TV, 3D optical remote sensing, and more.

Early work on preprocessing in the OSH system dates back to 1985. Later, it was shown that placing a Gaussian annular aperture instead of a flat lens is useful for recovering the edge of a cross-sectional image in a hologram [43,44]. In 2010, authors in [45] demonstrated that by choosing a pupil function such as the Laplacian of the Gaussian, the performance of the method is an efficient means to extract the edge of a 3D scanned object by the OSH system. The authors in [46] proposed a 1D image acquisition system for auto stereoscopic display consisting of a cylindrical lens, a focusing lens, and an imaging device. By scanning an object over a wide angle, the synthesized image can beviewed as a 3D stereoscopic image.

The aim of this paper is to propose and develop a fully automatic 3D segmentation of brain tumors from magnetic resonance images. The proposed model is based on the same principle as our previous articles [47,48] and specifically for the segmentation of high and low grade glioblastoma brain tumors. To achieve this goal, the following contributions are made:

• Proposing a fully automated 3D method for brain tumor segmentation from MRI image sequences.
• Improving the conventional optical scanning holography technique by exploiting the properties of the cylindrical lens to optimize the scanning process and shorten the holographic recording process.
• Transitioning active contour theory from semi-automatic to fully automatic status with reliable tumor detection.
• Perform tests to demonstrate the effectiveness of our method using the well-known BraTS 2019 and BraTS 2020 databases.

2. Materials And Methods

2.1. Data used

The database of brain tumor images used in this study was obtained from the Medical Image Computing and Computer Assisted Intervention (MICCAI) 2019 and MICCAI 2020 Multimodal Brain Tumor Segmentation Challenges, organized in collaboration with B. Menze, A. Jakab, S. Bauer, M. Reyes, M. Prastawa, and K. Van Leemput. It includes 40 glioma patients, including 20 high-grade (HG) and 20 low-grade (LG) cases, with four MRI sequences (T1, T1C, T2, and FLAIR) available for each patient. The challenge data base contains fully anonymized images collected from the ETH Zurich, the University of Debrecen, the University of Bern, and the University of Utah. All images are linearly co-registered and craniocaudally oriented. Institutional review board approval was not required as all human subject data are publicly available and de-identified [1,3].

2.2. Methodology

Figure 1 shows the optical scanning holography system used in our method. A laser beam of frequency ω isshifted in frequency to ψ and ψ+Δψ by Acousto-Optic Modulators (AOM) 1 and 2, respectively. The beams from the AOMs are then collimated by collimators BE1 and BE2. The out going beam from BE2 is considered as a plane wave of frequency ω+ψ+Δψ, which is projected onto the object by the x-y scanner. Our novel method involves the integration of a cylindrical lens L1 into the chosen imaging system, which provides a cylindrical wave at ω+Ω that is projected onto the object. A focusing lens is also used to capture a large number of elementary images containing extensive parallax data. These elemental images are transformed into a matrix of elemental images, where each captured elemental image corresponds to a vertical line in ray space [46]. Using this linear scanning technique, object images are captured in a single pass rather than point-by-point, and the shape of the surface is adjusted after each iteration, saving computational time. Accordingly, after appropriate sampling for viewing conditions, we achieved fully automatic segmentation through the improved algorithm and arrangement of color filters. This allowed the transformation of two-dimensional element images into Three-Dimensional (3D) images, as shown in Figures 5 and 6.

The x-y scanner is used to scan the 3D object uniformly, line by line. As a result, each scan line of the object corresponds to a line in the hologram at the same vertical position. Along each scan line, Photo-Detectors PD1 and PD2 are used to capture the optical signal scattered by the object and the heterodyne frequency information ∆Ω as a reference signal, respectively, and convert them into electrical signals for the lock-in amplifier. The in-phase and quadrature-phase outputs of the lock-in amplifier circuit produce a sine hologram, H_sin (x, y), and a cosine hologram, H_cos (x, y), to achieve a complete 2D scan of the object, as shown below:

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