MRI with non-linear spatial encoding magnetic (SEM) fields has many advantages over conventional linear SEM MRI. To name a few faster gradient switching time without peripheral nerve stimulation (PNS) ¹, enhanced edge resolution ^2,3, higher image quality at high acceleration rates ^2,4 can be the examples. However, research on MRI with non-linear SEM has mostly focused on 2-dimensional image acquisitions. In this study, we introduce 3D O-Space, and focus on image reconstruction of 3D encoded data from single channel receiver coil. A 3D projection radial gradient echo sequence was modified to implement 3D O-Space sequence, Fig.1 (a), on a 3T Siemens Tim Trio system (Siemens Healthcare, Erlangen, Germany) with a birdcage Tx coil and an 8-channel receiver coil. The number of readout point was 128 and 128 x 32 (φ x Ɵ) different sets of linear gradient fields were used for full SEM. To generate Z2 SEM a 10 channel gradient insert from Resonance Research Inc. (Billerica, MA, USA) was used. The basic pulse sequence is shown below. To reconstruct the experimental data various types of Kaczmarz methods were examined. Standard Kaczmarz solves y = Ax by minimizing ||y – Ax||₂ iteratively and choosing i th row of A sequentially. Randomized Kaczmarz chooses i th row randomly and update ‘x’, and randomized Kaczmarz with a probability distribution selects i th row with the probability. The probability comes from norm of the i th row, but in our case the inverse of the k-vector at the i th row is used ⁵. To improve the image quality compressed sensing (CS) is applied after Kaczmarz. One of the biggest obstacle implementing CS in 3D non-linearly encoding system is that the encoding matrix size is extremely large. In our case it is 2 TB, = (128 x 128 x 32)² x 8/1024³. We address this issue by generating encoding matrix with an initial condition and a difference, which has drastically reduced the size of the encoding matrix by 64 times. This has huge benefit in speed compared to generating every line to save memory space. Even though 16 GB is a huge memory size but it is manageable and applicable to reconstructing highly accelerated MRI acquisitions.Reconstructed images from various methods, Regridding, standard Kaczmarz (Kaczmarz 1), randomized Kaczmarz (Kacmarz 2), and randomized with a probability distribution (Kaczmarz 3) are shown in Fig.2. In above experiment Z2 gradient was not applied to the sequence. Kaczmarz 1 has been shown to find a solution very efficiently in 2-dimensional cases ⁶, but its convergence speed is not fast enough in a 3-dimensional case. Similarly, randomizing iteration order does not improve the convergence. Additionally, it increases background noise level as shown in Fig.2 (c). However, Kaczmarz 3 proves its fast convergence speed compared to the other iterative methods and results in agreeable images compared to those from the regridding method. CS is applied to weakly accelerated image, R = 2, and successfully denoises in the phantom area. The less sharp image is due to single channel acquisition which would be improved with parallel receivers. Traditional Kaczmarz converges to a solution very slowly with 3D reconstruction and this severely limits the application of 3D O-Space. We have addressed this issue by applying a randomized Kaczmarz with a probability distribution which becomes a fast and precise iterative method for 3D data. Additionally, our optimized CS for 3D non-linearly encoded data is adaptable to most acquisition strategies and improves image quality for higher accelerations in data acquisition. Further tests with multi-channel receiver coils and contrast detail phantoms in presence of Z2 gradient are being investigated.