International Journal of Biomedical Imaging

Volume 2018, Article ID 7803067, 12 pages

https://doi.org/10.1155/2018/7803067

## Respiratory Motion Correction for Compressively Sampled Free Breathing Cardiac MRI Using Smooth -Norm Approximation

^{1}Electrical Engineering Department, International Islamic University, Islamabad, Islamabad, Pakistan^{2}Electrical Engineering Section, UniKL BMI, Gombak, Selangor, Malaysia^{3}Electrical Engineering Department, Air University Islamabad, Islamabad, Pakistan

Correspondence should be addressed to Muhammad Bilal; kp.ude.uii@lalib.m

Received 20 September 2017; Revised 19 November 2017; Accepted 21 December 2017; Published 23 January 2018

Academic Editor: Anne Clough

Copyright © 2018 Muhammad Bilal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Transformed domain sparsity of Magnetic Resonance Imaging (MRI) has recently been used to reduce the acquisition time in conjunction with compressed sensing (CS) theory. Respiratory motion during MR scan results in strong blurring and ghosting artifacts in recovered MR images. To improve the quality of the recovered images, motion needs to be estimated and corrected. In this article, a two-step approach is proposed for the recovery of cardiac MR images in the presence of free breathing motion. In the first step, compressively sampled MR images are recovered by solving an optimization problem using gradient descent algorithm. The -norm based regularizer, used in optimization problem, is approximated by a hyperbolic tangent function. In the second step, a block matching algorithm, known as Adaptive Rood Pattern Search (ARPS), is exploited to estimate and correct respiratory motion among the recovered images. The framework is tested for free breathing simulated and* in vivo* 2D cardiac cine MRI data. Simulation results show improved structural similarity index (SSIM), peak signal-to-noise ratio (PSNR), and mean square error (MSE) with different acceleration factors for the proposed method. Experimental results also provide a comparison between* k-t* FOCUSS with MEMC and the proposed method.

#### 1. Introduction

Compressed sensing (CS) has been successfully implemented to reduce the scan time of MR images [1–3]. Application of CS to transformed domain MR images includes brain [1], cardiac [4], and pediatric MR imaging [5]. According to CS approach, a sparse or compressible image can be recovered, by solving norm mixed optimization problem, from randomly undersampled data using a nonlinear recovery technique [2, 6]. Since the -norm is not differentiable everywhere, an approximation to the -norm is used in the reconstruction algorithm. Different approaches [1, 7] have been used to approximate -norm using some differentiable functions. Reference [7] uses hyperbolic tangent based function as an approximation of -norm to solve the CS recovery problem for static MR images.

Different types of motion during the data acquisition process cause artifacts like ghosting and blurring in the recovered cardiac MR images. In the presence of respiratory motion, high quality MR images can be produced by combining* k*-space profiles of the same cardiac phases in ECG-gated MR acquisition [8]. Addition of the same cardiac phases at different respiratory motions or states produces inconsistencies in* k*-space which results in motion artifacts in the combined reconstructed images. Sparsity, a necessary condition for CS, is also violated by this* k*-space profile combination [9]. Hence, to avoid periodic breath holds during the acquisition process and to take advantage of sparse signal recovery from undersampled data using CS methods, motion artifacts may need to be corrected. The combined approach of CS and motion correction has been implemented in [8–11]. Otazo et al. proposed 1D translational respiratory motion correction. Usman et al. introduced a reconstruction scheme for dynamic cardiac MRI by incorporating general motion framework directly into CS reconstruction. Their method uses data binning and intensity based nonrigid registration algorithm for estimating respiratory motions. Reference [11] proposed a CS based motion correction in the free breathing environment with multiple constraints. This method uses Demon based registration to estimate the motion between reference and other respiratory states.

Interframe motion estimation and compensation for time varying features of images has been used in video compression standards [12, 13]. These standards are based on different block matching algorithms [14] for motion estimation and compensation. Similar to video compression, dynamic MR images can be predicted by exploiting temporal redundancies between the images. Asif et al. proposed an algorithm, MASTeR [15], for the breath held condition, based on interframe motion to recover different cardiac MR images. MASTeR uses motion adaptive transform that models temporal sparsity using interframe motion estimation.* k-t* FOCUSS [16] also uses interframe motion estimation and compensation with a fixed reference frame during the image recovery process for breath held cardiac cine MRI.

In this article, a novel framework is presented for the recovery of highly undersampled free breathing cardiac MR images. Similar cardiac phases at different respiratory states are grouped like the frames of a video sequence. The Adaptive Rood Pattern Search (ARPS) technique, based on the interframe motion, is used to estimate and correct respiratory states among the grouped images. A two-step approach is adopted for the reconstruction of dynamic MR images. In the first step, free breathing cardiac phases without motion estimation are recovered from undersampled* k*-space data. Next, interframe motion between the reconstructed cardiac phases is calculated using ARPS to improve the image estimates iteratively. An approximation of the -norm penalty is used in the gradient descent algorithm to recover dynamic MR images. The adjustable parameters of the -norm approximation provide an extra benefit, as it can be adjusted to reflect the changing statistics of dynamic MR images. After the application of the proposed method, the combinations of similar cardiac phases at different respiratory states are clear and accurate as compared to the combined cardiac phase without motion estimation and correction.

The rest of the paper is organized as follows. Section 2 discusses the preliminaries for interframe motion estimation in dynamic MRI and CS. Section 3 describes the methodology of the proposed algorithm. Section 4 gives the details of algorithm, Section 5 presents the simulation parameters and results followed by Section 6 that discusses the merits and demerits of the scheme, and Section 7 concludes the work.

#### 2. Free Breathing Imaging Model and CS

Free breathing downsampled* k*-spaced data corrupted by motion states for cardiac phase is mathematically given as where is a two-dimensional complex MR image vector of length representing a cardiac phase at respiratory state , is a Fourier operator that transforms an image to* k*-space, is a random variable-density undersampling mask, different for all respiratory states and is a sensing matrix, and is a combined* k*-space measurement vector of length for* n*th cardiac phase acquired for all respiratory positions. A single cardiac phase* n* at respiratory state* d* in a specific heart cycle can be given asThe reduction or acceleration factor for MR images is given by *.* By increasing , the system in (1) becomes highly underdetermined. Compressed sensing solves such underdetermined system of equations effectively to recover MR images. The recovery of a sparse signal using CS can be achieved by solving the following convex optimization problem:where is the Lagrangian that provides a balance between sparsity and data consistency. The -norm keeps the solution consistent with the data and the -norm given by encourages sparsity in solution [2].

#### 3. Methods

##### 3.1. Smooth -Norm Approximation

CS algorithms recover sparse signals or images by solving the -norm regularized optimization problem such as that given in (3). In this article, we use gradient descent algorithm for solving the optimization problem with wavelet based penalty term. Nondifferentiability of the -norm at origin excludes the usage of mostly optimization approaches for the solution. Reference [1] uses an approximation of with the complex conjugate and a positive smoothing factor. Reference [7] proposes a new smooth function for approximating the -norm, used in this paper, which is given below

This function better approximates the absolute value and provides extra flexibility of adjusting the slope at the origin with the proper selection of and makes it more suitable for dynamic images. Mostly MR images are sparse in transformed domain so the modified version of cost function given in (3) for transformed MR images iswhere is a wavelet operator that transforms the image to a sparse domain.

In this article, we propose an iterative algorithm that uses the following approximation for the -norm penalty:where . The update equation for the algorithm, derived using the steepest descent method for a sparse vector , iswhere * is *positive valued step size and is the gradient operator that differentiates the cost function at th iteration. During each iteration, shrinkage given in (8) is applied in the wavelet domain after (7) to reconstruct the MR images.where is a thresholding parameter. By incorporating the approximation , the cost function can be written asThe gradient of the cost function is easy to compute:with

##### 3.2. Respiratory Motion Based Dynamical System

Two main problems with free breathing cardiac MRI are as follows.

(1) Blurring artifacts are generated by the combination of* k*-space samples for the same cardiac phases at different respiratory states.

(2) The combination of* k*-space data in free breathing decreases the sparsity level.

In this article, we use interframe motion to estimate the respiratory states between the same cardiac phases. Video standards MPEG and H.264 [12, 13] have successfully exploited interframe motion for compression. In the dynamic MRI images, pixels are not significantly displaced in the neighboring frames. Pixel locations can be predicted using interframe motion estimation. Temporal redundancy among the frames is advantageous for the prediction of pixel locations. Let and be images having th cardiac phases at respiratory states* d* and , respectively. The pixel values of at location are closest to the pixel values at in . The displacement of all pixels in from to in is represented by motion vectors According to [2], cardiac phase at* d*th respiratory state can be generated from the cardiac phase at th respiratory state by the following equation:where is a backward transformation that uses information about the physical changes between two datasets of the same cardiac phases. The motion compensated residual is computed by taking the difference between predicted and compensated image. Using the transformation , a motion dependent linear system can be written by combining (1) and (12) as follows:To recover the cardiac phases , we solve (13a) and (13b) by exploiting sparse structure in and . The process of complete high resolution image generation is shown in Figure 1. The data is acquired in segmented fashion because MRI is a slow imaging modality. During the data scanning process, a limited* k*-space sample is recorded at each heart phase in all cardiac cycles. To simulate this condition, each cardiac phase at different respiratory states is multiplied with different sampling matrix .