The proposed method uses nonlinear conjugate gradients with alternating minimization to jointly estimate nonrigid motion and perform image reconstruction with data acquired using a non-Cartesian spiral pulse sequence. This method iteratively solves a nonconvex optimization problem for the reconstructed image and non-rigid motion. Different from previous approaches[1][2], this approach performs image reconstruction and motion estimation at the same time without using extra navigators. Thus it is faster because of the spiral pulse sequence and more motion robust. This research enables fast pediatric imaging without using sedation, facilitating more widespread use of MRI in pediatric imaging as an alternative to CT imaging.The proposed joint motion estimation and reconstruction is compared to the true motion and image from separate reconstructions.Motion during an MR scan, such as respiratory or patient motion, will cause different groups of k-space data to possess different motion, resulting in blurred images. In[1], image reconstruction is done by forming a data fitting term plus a regularization term. Based on that, a new data fitting term is formed by inserting motion deformation into the forward model and adding extra regularization for smoothness of the deformation, as shown below $$\underset{I_{c}, w_{c}}{\operatorname{argmin}} \parallel m_{c}-FT(w_{c})I_{c}\parallel_2^2 + \beta\parallel\triangledown_{x,y}w_{c} \parallel_2 $$ $$$I_{c}$$$ is the image to be reconstructed, $$$T(w_{c})$$$ is the deformation field using cubic B-spline interpolation[3], $$$F$$$ is the non-uniform fast Fourier transform, and $$$m_{c}$$$ are the measured k-space data. The l2-norm on the gradient of the deformation[4] ensures spatial smoothness of the estimated non-rigid motion. The conjugate gradient(CG) method is used to solve this nonlinear optimization problem. First, image $$$I_{c}$$$ is fixed and $$$T(w_{c})$$$ is found by minimizing the objective function for a certain number of iterations. Then,the deformation is updated and fixed, while $$$I_{c}$$$ is turned into the new variable and is solved using CG. By repeating these two steps iteratively, both $$$I_{c}$$$ and $$$T(w_{c})$$$ are updated during each iteration until convergence or reaching certain number of steps.K-space data of a T1-weighted spoiled gradient echo image using a constant density spiral pulse sequence from[5] is reconstructed to form a brain image without motion as the ground truth. The image reconstruction toolbox used can be found in[6][7] Figure 1 shows the reconstructed brain as well as six leaves out pf 120 of the spiral sampling trajectory. Four different deformations are added to the true image, and from each deformed image a group of k-space data is generated using the NUFFT operator from[6]. To simulate the blurry image from reconstruction without motion correction, 1/4 of the k-space data from each of the deformed image are merged together to form a full k-space. The reconstructed image from the merged k-space data can be seen in Figure 2. We then estimate the true image using the k-space data from the deformed images. The iteration steps for reconstructing the images and the motion are set to 50 each, and the whole alternating minimization is run for five iterations.Figure 3 compares estimated true image and the ground truth without motion, which generates a error value of 1.57% after total 500 iterations. Figure 4 shows the estimated deformation field and the true deformation field. The above results show that the joint motion estimation and image reconstruction method is very effective for estimating just a single image. Figure 5 plots the root mean squared error(RMSE)of the data fitting term for the first 100 iterations. The result shows that the convergence rate is very fast at the first 20 iterations.