An L1-norm phase constraint for half-Fourier compressed sensing in 3D MR imaging

Abstract

Objective

In most half-Fourier imaging methods, explicit phase replacement is used. In combination with parallel imaging, or compressed sensing, half-Fourier reconstruction is usually performed in a separate step. The purpose of this paper is to report that integration of half-Fourier reconstruction into iterative reconstruction minimizes reconstruction errors.

Materials and methods

The L1-norm phase constraint for half-Fourier imaging proposed in this work is compared with the L2-norm variant of the same algorithm, with several typical half-Fourier reconstruction methods. Half-Fourier imaging with the proposed phase constraint can be seamlessly combined with parallel imaging and compressed sensing to achieve high acceleration factors.

Results

In simulations and in in-vivo experiments half-Fourier imaging with the proposed L1-norm phase constraint enables superior performance both reconstruction of image details and with regard to robustness against phase estimation errors.

Conclusion

The performance and feasibility of half-Fourier imaging with the proposed L1-norm phase constraint is reported. Its seamless combination with parallel imaging and compressed sensing enables use of greater acceleration in 3D MR imaging.

Appendices

Appendix 1

Proof of the Convexity of L1PC: set $z = \alpha x + \beta y, \;0 \le \alpha \le 1.0, \;0 \le \beta \le 1.0, \;\alpha + \beta = 1.0$, W is an arbitrary constant matrix.

$$\begin{aligned} f_{1} \left( z \right) & = \left\| { \left( {z - \left| z \right|} \right) \circ W} \right\|_{1} \\ & = \mathop \sum \limits_{i} \left| {\left( {z_{i} - \left| {z_{i} } \right|} \right)} \right| \cdot W_{i} \\ \end{aligned}$$

Note that the real part of $z_{i} - \left| {z_{i} } \right|$ is a negative value, so

$$\begin{aligned} f_{1} \left( z \right) & \le \mathop \sum \limits_{i} \left| {\left( {z_{i} - \left( {\alpha \left| {x_{i} } \right| + \beta \left| {y_{i} } \right|} \right)} \right) \cdot W_{i} } \right| \\ & = \mathop \sum \limits_{i} \left| {\left( {\alpha \left( {x_{i} - \left| {x_{i} } \right|} \right) + \beta \left( {y_{i} - \left| {y_{i} } \right|} \right)} \right)} \right| \cdot W_{i} \\ & \le \alpha \mathop \sum \limits_{i} \left| {x_{i} - \left| {x_{i} } \right|} \right|W_{i} + \beta \mathop \sum \limits_{i} \left| {y_{i} - \left| {y_{i} } \right|} \right|W_{i} \\ & = \alpha f_{1} \left( x \right) + \beta f_{2} \left( y \right) \\ \end{aligned}$$

Therefore, $\left\| {\left( {x - \left| x \right|} \right) \circ W} \right\|_{1}$ is convex.

Proof of the Convexity of L2PC: set $z = \alpha x + \beta y,\; 0 \le \alpha \le 1.0, \;0 \le \beta \le 1.0, \;\alpha + \beta = 1.0$, W is an arbitrary constant matrix.

$$\begin{aligned} f_{2} \left( z \right) & = \parallel\!\left( {z - \left| z \right|} \right) \circ W\!\parallel_{2} \\ & = \sqrt {\mathop \sum \limits_{i} \left( {\left| {\alpha x_{i} + \beta y_{i} - \left| {\alpha x_{i} + \beta y_{i} } \right|} \right| \cdot W_{i} } \right)^{2} } \\ & \le \sqrt {\mathop \sum \limits_{i} \left( {\left| {\alpha \left( {x_{i} - \left| {x_{i} } \right|} \right) + \beta \left( {y_{i} - \left| {y_{i} } \right|} \right)} \right| \cdot W_{i} } \right)^{2} } \\ & \le \sqrt {\mathop \sum \limits_{i} \left( {\left( {\alpha \left| {x_{i} - \left| {x_{i} } \right|} \right| + \beta \left| {y_{i} - \left| {y_{i} } \right|} \right|} \right) \cdot W_{i} } \right)^{2} } \\ & = \sqrt {\alpha^{2} \mathop \sum \limits_{i} \left( {\left| {x_{i} - \left| {x_{i} } \right|} \right| \cdot W_{i} } \right)^{2} + \beta^{2} \mathop \sum \limits_{i} \left( {\left| {y_{i} - \left| {y_{i} } \right|} \right| \cdot W_{i} } \right)^{2} + 2\alpha \beta \mathop \sum \limits_{i} \left( {\left| {x_{i} - \left| {x_{i} } \right|} \right| \cdot W_{i} \cdot \left| {y_{i} - \left| {y_{i} } \right|} \right| \cdot W_{i} } \right) } \\ \end{aligned}$$

It has been proved that $\left| { < U,V > } \right| \le \left| U \right|\left| V \right|, \in R^{n}$. We have

$$\begin{aligned} f_{2} \left( z \right) & \le \sqrt {\alpha^{2} \left( {f_{2} \left( x \right)} \right)^{2} + \beta^{2} \left( {f_{2} \left( y \right)} \right)^{2} + 2\alpha \beta f_{2} \left( x \right)f_{2} \left( y \right)} \\ & = \alpha f_{2} \left( x \right) + \beta f_{2} \left( y \right) \\ \end{aligned}$$

Therefore, $\left\| {\left( {x - \left| x \right|} \right) \circ W} \right\|_{2}$ is convex.

Appendix 2

For L1PC, given $f_{1} = \left\| {\left( {x - \left| x \right|} \right) \circ W} \right\|_{1} = \left\| g \right\|_{1}$, where x and W are $n \times 1$ vectors. W is constant during the iterative optimization procedure.

The gradient operator of $f_{1}$ is

$$\nabla f_{1} = \left( {\frac{{\partial f_{1} }}{\partial x}} \right)^{*} = \left( {\frac{{\partial f_{1} }}{\partial g}\frac{\partial g}{\partial x} + \frac{{\partial f_{1} }}{{\partial \overline{g} }}\frac{{\partial \overline{g} }}{\partial x}} \right)^{*} = \left( {AB + CD} \right)^{*}$$

Here $\left( X \right)^{*}$ represents the conjugate transpose of matrix X. $\overline{X}$ represents the complex conjugate of X.

$$A = \frac{{\partial f_{1} }}{\partial g} = \left[ {\frac{{\partial f_{1} }}{{\partial g_{1} }}, \ldots \frac{{\partial f_{1} }}{{\partial g_{n} }}} \right]$$

Here,

$$\frac{{\partial f_{1} }}{{\partial g_{k} }} = \frac{{\partial \left| {g_{k} } \right|}}{{\partial g_{k} }} = \frac{1}{2} \frac{{\overline{{g_{k} }} }}{{\left| {g_{k} } \right| + \varepsilon }}$$

Here, $\varepsilon$ is a small scalar value to avoid “division by zero”.

$$B = \frac{\partial g}{\partial x} = \frac{{\partial \left( {\left( {x - \left| x \right|} \right) \circ W} \right)}}{\partial x}$$

$${\text{Its}}\,{\text{entry}}\,B_{kl} = \frac{{\partial g_{k} }}{{\partial x_{l} }} = \left\{ {\begin{array}{*{20}l} {0 \quad \quad \quad \quad \quad \quad \quad \quad {\text{when}}\, k \ne l} \hfill \\ {\left( {1 - \frac{1}{2} \frac{{\overline{{x_{k} }} }}{{\left| {x_{k} } \right| + \varepsilon }}} \right) \cdot W_{k} \quad {\text{when}}\, k = l} \hfill \\ \end{array} } \right.$$

$$C = \frac{{\partial f_{1} }}{{\partial \overline{g} }} = \overline{{\left( {\frac{{\partial f_{1} }}{\partial g}} \right)}} = \overline{A}$$

and

$$D = \frac{{\partial \overline{g} }}{\partial x} = \frac{{\partial \left( {\left( {\overline{x} - \left| x \right|} \right) \circ W} \right)}}{\partial x}$$

$${\text{Its}}\,{\text{entry}}\,D_{kl} = \frac{{\partial \overline{{g_{k} }} }}{{\partial x_{l} }} = \left\{ {\begin{array}{*{20}l} {0\quad \quad \quad \quad \quad \quad \quad \,\,\, {\text{when}}\, k \ne l} \hfill \\ {\left( { - \frac{1}{2} \frac{{\overline{{x_{k} }} }}{{\left| {x_{k} } \right| + \varepsilon }}} \right) \cdot W_{k} \quad {\text{when}}\, k = l} \hfill \\ \end{array} } \right.$$

For L2PC, given $f_{2} = \left\| { \left( {x - \left| x \right|} \right) \circ W} \right\|_{2} = g_{2}$, where x and W are $n \times 1$ vectors. W is constant during the iterative optimization procedure.

The gradient operator of $f_{2}$ is

$$\nabla f_{2} = \left( {\frac{{\partial f_{2} }}{\partial x}} \right)^{*} = \left( {\frac{{\partial f_{2} }}{\partial g}\frac{\partial g}{\partial x} + \frac{{\partial f_{2} }}{{\partial \overline{g} }}\frac{{\partial \overline{g} }}{\partial x}} \right)^{*} = \left( {AB + CD} \right)^{*}$$

Here $\left( X \right)^{*}$ represents the conjugate transpose of matrix X. $\overline{X}$ represents the complex conjugate of $X$.

$$A = \frac{{\partial f_{2} }}{\partial g} = \left[ {\frac{{\partial f_{2} }}{{\partial g_{1} }}, \ldots \frac{{\partial f_{2} }}{{\partial g_{n} }}} \right]$$

Its entry

$$\frac{{\partial f_{2} }}{{\partial g_{k} }} = \frac{{\partial (g_{k} \overline{{g_{k} }} )}}{{\partial g_{k} }} = \frac{{\overline{{g_{k} }} }}{{2f_{2} }}$$

$$B = \frac{\partial g}{\partial x} = \frac{{\partial \left( {\left( {x - \left| x \right|} \right) \circ W} \right)}}{\partial x}$$

$${\text{Its}}\,{\text{entry}}\,B_{kl} = \frac{{\partial g_{k} }}{{\partial x_{l} }} = \left\{ {\begin{array}{*{20}l} {0 \quad \quad \quad \quad \quad \quad \quad \quad {\text{when}}\, k \ne l} \hfill \\ {\left( {1 - \frac{1}{2} \frac{{\overline{{x_{k} }} }}{{\left| {x_{k} } \right| + \varepsilon }}} \right) \cdot W_{k} \quad {\text{when}}\, k = l} \hfill \\ \end{array} } \right.$$

$$C = \frac{{\partial f_{1} }}{{\partial \overline{g} }} = \overline{{\left( {\frac{{\partial f_{1} }}{\partial g}} \right)}} = \overline{A}$$

and

$$D = \frac{{\partial \overline{g} }}{\partial x} = \frac{{\partial \left( {\left( {\overline{x} - \left| x \right|} \right) \circ W} \right)}}{\partial x}$$

$${\text{Its}}\,{\text{entry}}\,D_{kl} = \frac{{\partial \overline{{g_{k} }} }}{{\partial x_{l} }} = \left\{ {\begin{array}{*{20}l} {0\quad \quad \quad \quad \quad \quad \quad \,\,\, {\text{when}} \; k \ne l} \hfill \\ {\left( { - \frac{1}{2} \frac{{\overline{{x_{k} }} }}{{\left| {x_{k} } \right| + \varepsilon }}} \right) \cdot W_{k} \quad {\text{when}} \; k = l} \hfill \\ \end{array} } \right.$$