National Institute of Mental Health Intramural Research Program, NIH, Bethesda, MD 20892-1527, USA.
Med Phys. 2011 Oct;38(10):5582-9. doi: 10.1118/1.3633908.
To introduce a linear shift-invariant relationship between the partial derivatives of k space signals acquired using multichannel receive coils and to demonstrate that k space derivatives can be used for image unwrapping.
Fourier transform of k space derivatives contains information on the spatial origins of aliased pixels; therefore, images can be reconstructed by k space derivatives. Fully sampled phantom and brain images acquired at 3 T using a standard eight channel receive coil were used to validate the k space derivatives theorem by unwrapping aliased images.
Derivative encoding leads to new methods for parallel imaging reconstruction in both k space and image domains. Noise amplification in sensitivity encoding image reconstruction, which is considered to produce the optimal SNR, can be further reduced using k space derivative encoding without making any assumptions on the characteristics of the images to be reconstructed.
This work demonstrated that the partial derivative of the k space signal acquired from one coil with respect to one direction can be expressed as a sum of partial derivatives of signals from multiple coils with respect to the perpendicular k space direction(s). This relationship between the partial derivatives of k space signals is linear and shift-invariant in the Cartesian coordinate system.
介绍使用多通道接收线圈获得的 k 空间信号的偏导数之间的线性平移不变关系，并证明 k 空间导数可用于图像展开。
k 空间导数的傅里叶变换包含了混叠像素空间起源的信息；因此，可以通过 k 空间导数重建图像。使用标准的八通道接收线圈在 3T 上采集完全采样的幻影和大脑图像，通过展开混叠图像来验证 k 空间导数定理。
导数编码为 k 空间和图像域中的并行成像重建提供了新方法。灵敏度编码图像重建中的噪声放大被认为可以产生最佳信噪比，通过使用 k 空间导数编码，在不假设要重建的图像特征的情况下，可以进一步降低噪声放大。
这项工作表明，从一个线圈相对于一个方向获得的 k 空间信号的偏导数可以表示为相对于垂直 k 空间方向(s)的多个线圈信号的偏导数的和。在笛卡尔坐标系中，k 空间信号的偏导数之间存在线性平移不变关系。
