A comparative analysis of intensity-based 2D–3D registration for intraoperative use in pedicle screw insertion surgeries

Abstract

Purpose

Although multiple algorithms have been reported that focus on improving the accuracy of 2D–3D registration techniques, there has been relatively little attention paid to quantifying their capture range. In this paper, we analyze the capture range for a number of variant formulations of the 2D–3D registration problem in the context of pedicle screw insertion surgery.

Methods

We tested twelve 2D–3D registration techniques for capture range under different clinically realistic conditions. A registration was considered as successful if its error was less than 2 mm and 2° in 95% of the cases. We assessed the sensitivity of capture range to a variety of clinically realistic parameters including: X-ray contrast, number and configuration of X-rays, presence or absence of implants in the image, inter-subject variability, intervertebral motion and single-level vs multi-level registration.

Results

Gradient correlation + Powell optimizer had the widest capture range and the least sensitivity to X-ray contrast. The combination of 4 AP + lateral X-rays had the widest capture range (725 mm²). The presence of implant projections significantly reduced the registration capture range (up to 84%). Different spine shapes resulted in minor variations in registration capture range (SD 78 mm²). Intervertebral angulations of less than 1.5° had modest effects on the capture range.

Conclusions

This paper assessed capture range of a number of variants of intensity-based 2D–3D registration algorithms in clinically realistic situations (for the use in pedicle screw insertion surgery). We conclude that a registration approach based on the gradient correlation similarity and the Powell’s optimization algorithm, using a minimum of two C-arm images, is likely sufficiently robust for the proposed application.

Appendices

Appendix 1: Details about the chosen similarity metrics

Gradient differences ($S_{\text{GD}}$): This metric estimates the similarity between horizontal and vertical gradients of the X-ray image ($I_{\text{F}}$) and the CT volume projected as a DRR ($I_{\text{M}}$):

$$ S_{\text{GD}} = \mathop \sum \limits_{i,j} \frac{{\sigma_{\text{v}}^{2} }}{{\sigma_{\text{v}} + \left( {M_{\text{v}} \left( {i,j} \right)} \right)^{2} }} + \mathop \sum \limits_{i,j} \frac{{\sigma_{\text{h}}^{2} }}{{\sigma_{\text{h}} + \left( {M_{\text{h}} \left( {i,j} \right)} \right)^{2} }} $$

where $\sigma_{\text{v}}^{2}$ and $\sigma_{\text{h}}^{2}$ are the variances of the X-ray images in the vertical and horizontal directions, $i,j$ are the pixel indices, and the terms $M_{\text{v}}$ and $M_{\text{h}}$ are defined as:

$$ M_{\text{v}} = \frac{{\partial I_{\text{F}} \left( {i,j} \right)}}{\partial i} - \frac{{\partial I_{\text{M}} \left( {i,j} \right)}}{\partial i} $$

$$ M_{\text{h}} = \frac{{\partial I_{\text{F}} \left( {i,j} \right)}}{\partial j} - \frac{{\partial I_{\text{M}} \left( {i,j} \right)}}{\partial j} $$

that define the differences in the vertical and horizontal gradients between images $I_{\text{F}}$ and $I_{\text{M}}$.

Gradient Correlation ($S_{\text{GC}}$): Vertical and horizontal gradients are used for calculation of this metric as:

$$ S_{\text{GC}} = \frac{{\mathop \sum \nolimits_{i,j} \left( {\partial_{i} I_{\text{F}} - \overline{{\partial_{i} I_{\text{F}} }} } \right)\left( {\partial_{i} I_{\text{M}} - \overline{{\partial_{i} I_{\text{M}} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{i,j} \left( {\partial_{i} I_{\text{F}} - \overline{{\partial_{i} I_{\text{F}} }} } \right)^{2} } \sqrt {\mathop \sum \nolimits_{i,j} \left( {\partial_{i} I_{\text{M}} - \overline{{\partial_{i} I_{\text{M}} }} } \right)^{2} } }} + \frac{{\mathop \sum \nolimits_{i,j} \left( {\partial_{j} I_{\text{F}} - \overline{{\partial_{j} I_{\text{F}} }} } \right)\left( {\partial_{j} I_{\text{M}} - \overline{{\partial_{j} I_{\text{M}} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{i,j} \left( {\partial_{j} I_{\text{F}} - \overline{{\partial_{j} I_{\text{F}} }} } \right)^{2} } \sqrt {\mathop \sum \nolimits_{i,j} \left( {\partial_{j} I_{\text{M}} - \overline{{\partial_{j} I_{\text{M}} }} } \right)^{2} } }} $$

where $\partial_{i} I_{\text{F}} = \frac{{\partial I\left( {i,j} \right)}}{\partial i }$, $\partial_{j} I_{\text{F}} = \frac{{\partial I\left( {i,j} \right)}}{\partial j }$ and $\overline{{\partial_{i} I_{\text{F}} }}$ denotes the respective mean value.

Pattern Intensity ($S_{\text{PI}}$): This metric operates based on characterized structures found in the difference image $D$ obtained by subtracting $I_{\text{F}}$ from $I_{\text{M}}$ [26]. The lower the intensity of these structures, the higher the pattern intensity similarity metric, $S_{\text{PI}}$. $S_{\text{PI}}$ is calculated within a region with a radius $r$:

$$ S_{\text{PI}} = \mathop \sum \limits_{i,j} \mathop \sum \limits_{{d^{2} \le r^{2} }} \frac{{\sigma^{2} }}{{\sigma^{2} + \left( {D\left( {i,j} \right) - D\left( {v,w} \right)} \right)^{2} }} $$

The term $r$ defines the size of a moving kernel and $\sigma$ is the desired sensitivity of the metric to considering an intensity difference as a structure, additionally:

$$ \begin{aligned} D\left( {i,j} \right) &= I_{\text{F}} \left( {i,j} \right) - I_{\text{M}} \left( {i,j} \right) // d^{2} &= \left( {i - v} \right)^{2} + \left( {j - w} \right)^{2} \end{aligned}$$

The terms $r$ and $\sigma$ are tunable parameters that need to be assigned according to the application.

Appendix 2: Optimization hyperparameters

Optimization hyperparameters. NI: number of iterations, NLI: number of line iterations, MSL: maximum step length, MIL: minimum step length, SL: step length, RF: relaxation factor, ST: step tolerance, VL: value tolerance.

Optimizer
RSGD	FRPR	POWL	SPSA#
NI = 50	NI = 20	NI = 20	NI = 1000
MSL* = 1	NLI = 10	NLI = 10	*α = 0.1
MIL = 0.001	VL = 0.001	SL* = 1	$\gamma$ = 0.01
RF = 0.6	SL* = 4	ST = 0.001	A = 10
	ST* = 0.08	VL = 0.001

1. *These parameters were halved at the beginning of each pyramid level (3 levels in total)
2. ^#The optimization parameters for this algorithm were tuned to best fit our application