^{1}

^{*}

^{2}

^{3}

^{4}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

Traditionally, image registration of multi-modal and multi-temporal images is performed satisfactorily before land cover mapping. However, since multi-modal and multi-temporal images are likely to be obtained from different satellite platforms and/or acquired at different times, perfect alignment is very difficult to achieve. As a result, a proper land cover mapping algorithm must be able to correct registration errors as well as perform an accurate classification. In this paper, we propose a joint classification and registration technique based on a Markov random field (MRF) model to simultaneously align two or more images and obtain a land cover map (LCM) of the scene. The expectation maximization (EM) algorithm is employed to solve the joint image classification and registration problem by iteratively estimating the map parameters and approximate posterior probabilities. Then, the maximum

Remotely sensed images captured from satellites have been widely used for land cover mapping applications because of their capability to allow classification of different land cover types without having to physically assess the area of interest. In a situation where a single image does not provide sufficient classification performance, integrating multiple images of the same area is a common practice to increase the discrimination capability. Some applications, especially agricultural field mapping, particularly benefit from using multitemporal sequences of satellite images because vegetation appearance often changes according to the season. Moreover, multiple input images from different satellites can be used to further improve classification performance by providing better spectral separation characteristics that a single sensor alone cannot provide. A practical application is reported in [

The problems of multisource and multitemporal land cover mapping have been extensively studied in [

Mahapatra and Sun [

Another work by Chen

In this paper, we employ an approach similar to [

Based on our image model, the registration and classification process can be performed in the following fashion. First, we estimate the unknown map transformation parameters based on the maximum likelihood (ML) criteria, and then use these parameters to computer posterior probabilities for different arrangements of the land cover maps, where the MAP classifier selects the most likely LCM. However, in order to find the map parameters, the conditional probability of observed images given certain map parameters is needed. This conditional probability can only be obtained by summing the joint probabilities of observed images and LCM associated with the map parameters, over all possible LCMs. This is impossible to obtain in most practical scenarios. As a result, the expectation-maximization (EM) algorithm [

For a given iteration of the EM algorithm, our method computes the expected value of the logarithm of the probability of the observed images and land cover map given the map parameters, based on the

The remainder of this paper is organized as follows. The next section will define the problem and the model that we employed. In Section 3, we will derive the optimum land cover mapping and image registration process based on the model presented in Section 2. The optimization problem and its corresponding solution are presented in Section 4. Our experiments to evaluate our proposed approach are described in Section 5. Finally, Section 6 offers concluding remarks.

Let ^{𝒮}_{s}_{C}_{X}_{C ⊂ 𝒮} _{C}

For any two sites _{s}_{s}_{r}

Furthermore, we assume that there are _{n}_{n}^{𝒯n × Bn}; _{n}_{n}_{n}_{n}_{n}, v_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{1,n} and _{4,n} are scale parameters, _{2,n} and _{3,n} are skew parameters, and _{5,n} and _{6,n} are displacement parameters in the column- and row-direction, respectively. We refer to _{n}_{1,n}, _{2,n}, _{3,n}, _{4,n}, _{5,n}, _{6,n}] as the map parameter vector between coordinate systems _{n}

When all the map parameter vectors are given, one can remap all remotely sensed images to perfectly align with the LCM. Let us denote

As the remapped and resampled version of the _{1}, …, _{n}_{1}(_{1}), …, _{n}_{n}_{n}_{n,s}^{Bn} denotes the intensity vector of the remapped image _{n}

Furthermore, if we assume that the intensity vector at a pixel _{n}_{s}_{xs,n} and covariance matrix ∑_{xs,n},

By using the chain rule, the posterior probability of the LCM given the observed multispectral images and the map parameters can be written as
_{X∈Λ𝒮} ^{−E(X|Y, M)} is a normalizing constant and independent of the choice of _{s}^{4096} possible configurations for binary LCM of size 64 × 64.) As a result, we propose to use the mean field theorem [_{C}_{xr} [_{{s,r}} (_{r}_{xr} [_{{s,r}}(_{s}_{r}_{r}_{s∈𝒮} p_{s}

The standard approaches to multi-temporal and/or multi-modal image classification entail two steps. First, images from different sources and/or times are registered to produce a set of images in a common coordinate system. Then, a land cover map is derived from this set of registered images. In this work, even though we propose an algorithm to simultaneously register and classify images, we still treat image registration and classification as two separate problems in order to follow standard approaches. As a result, we propose different optimization criteria for image registration and land cover mapping. However, we will show in Section IV that both image registration and land cover mapping can be combined into a single algorithm so that the registration and land cover mapping can be performed simultaneously.

The maximum likelihood estimate (MLE) can be employed as the optimum map parameter estimator since the MLE is to known to be a consistent estimator [

In order to solve _{1}, …, _{N}_{1}, …, _{N}

Note here again that _{n}_{N}_{X∈Λ}^{𝒮}_{n}_{n}_{n}_{1} = [1,0,0,1,1,0] is also the solution of _{1}, _{1} =

Next, let us consider a small LCM of size 100 × 100 pixels. In this case, there are 2^{10,000} ≈ 2 × 10^{3,010} possible binary LCMs. Therefore, the direct calculation of _{1}, …, _{N}^{t}) ≥ (_{1}, …, _{N}^{t−1}), where ^{t−1} is the collection of estimated parameters at the (t−1)-th iteration. Here, and throughout the rest of the paper, we omit _{n} for the sake of brevity. In Section 4, we will discuss the details of the EM algorithm employed in this work and how it can be combined with the land cover mapping process. However, before going into the detail of the proposed algorithm, let us state the optimization criterion for the land cover mapping considered in this paper.

The classifier based on the maximum

In general, Pr(_{C}(X). Hence, by substituting

Since the optimizing function in ^{MF}_{s}

Since the EM algorithm is employed in this article as the parameter estimator, we begin our discussion with the details of the EM algorithm. The EM algorithm [_{1}, …, _{N}_{1}, …, _{N}

In the M-step, the expected value given in

Clearly, the terms log|∑_{xs,n}|, log(2^{Bn}, ∑_{C⊂𝒮} _{C}_{X}

To find the solution of

By substituting

Hence, in the M-step, the new map parameters can be obtained by maximizing the approximation given

Since _{n,s}_{n} and the right-hand side of _{n,s}

Using the approximations given above, the modified EM algorithm is displayed in ^{t}) is approximated by recalculating _{s}_{s}^{t}_{s}_{s}^{t}_{obv}_{s}_{ng}_{s}_{NG}

Since _{s}_{s}^{t}_{s}_{s}^{t}

Initialize map parameters, ^{MF}_{s}^{0}) based on some prior knowledge.

Compute

Obtain

Compute _{s}_{s}^{t}

Find the new LCM that minimizes _{s}_{s}^{t}

Let

The critical challenge in the successful implementation of the joint image registration and land cover mapping algorithm proposed above is how to solve ^{MF}(M||M^{t−1}) due to its non-convexity. The PSO exploits the cooperative behavior of a group of animals such as birds and insects. In the PSO, an individual animal is called a particle while a group of animals is called a swarm. Initially, these particles are distributed throughout the search space, and move around the search space. Based on some social and cooperative criteria, these particles will eventually cluster in the regions where the global optima can be found.

In our work, for a given image Y_{n}, each particle represents a mapping parameter and we denote the i-th particle as _{n,i}_{i}_{i}_{1} and φ_{2} are acceleration constants, and u_{1} and u_{2} are uniform random numbers between zero and one. The velocity is usually kept within the range of [V_{min}, V_{max}] to ensure that
_{1} and φ_{2}, and the number of iterations. In this paper, we set the number of particles to 80 and the maximum number of iterations to 200 as a suitable setup for our experiment. We acknowledge that different setups of these parameters may result in different convergence rates. However, the investigation of the optimum parameter selection of the PSO in term of convergence rate is out of the scope of this paper. We refer to the report studied by [

In this section, we provide the results of two experiments—based on the methodology derived in Section 4—to jointly register and classify a set of remotely sensed images. The first experiment was conducted over a simulated dataset in order for us to investigate many aspects of our proposed algorithm. Next, we examined the performance of our algorithm in an actual remote sensing image. For both examples, the goal was to examine the performance of the algorithm to different degrees of initial registration errors. If our algorithm performed perfectly, it would be able to align images together and produce an LCM from unregistered images as accurate as when images were registered.

In the first experiment, we examined the performance of the proposed algorithm in terms of classification performance and registration accuracy by attempting to produce a land cover map from a set of four simulated images. All the simulated images were of an equal size of 512 × 512 pixels (

Since our algorithm performed both image registration and land cover mapping simultaneously, the performance of our algorithm could be evaluated in terms of how much the resulting LCM deviated from the reference LCM, and the estimation error between our calculated map parameters and the actual parameters that registered the LCM to the simulated images. If our algorithm performed perfect registration and land cover mapping, the resulting percentages of mis-classified pixels would be zero, and the registration error between the images and LCM would also be zero. In this example, the correct mapping parameters for all observed images were identical and equal to _{Perfect} = [1,0,0,1,0,0] which correspond to unit scale, zero skew, and zero displacement. Next, since we wanted to examine the effects of the initial registration errors on the performance of our algorithm, we investigated different scenarios of initial registration errors by varying the initial mapping parameters between the observed images and LCM at different values of displacement, scale and skew parameters. In particular, we investigated three scenarios for only the displacement, only the scale and only the skew errors, respectively.

Before investigating the performance of our proposed algorithm, we examined the effect of registration errors on the performance of image classification. This value can be viewed as the worst case scenario where the LCM is derived directly from the set of mis-registered images. Here, we employed the maximum likelihood classifier (MLC) [

Next, the proposed algorithm was applied to the above datasets. The whole process was implemented using CUDA on NVIDIA Tesla M2090 with 1 GB memory. Here, we assigned
_{n} from two consecutive iterations where
_{i} from the nth at the tth iteration. In this example, the algorithm terminates when p_{changes} is less than p_{min} = 10^{−5}, and d_{movement,n} is less than 0.1 pixels for five consecutive iterations for ^{t} = ^{*}^{*} = _{Perfect}^{*} equal to the values given in

To ensure the statistical significance, we computed the pairwise t-statistics for unequal variance populations [

Since at

Another key performance metric in this example is the residual registration errors after processing. _{n}_{perfect}

Next, we examined the effects of image noise on the registration accuracy by varying the noise variance ^{2} from −30 dB to 0 dB, and the resulting averaged RMSEs for ^{2} of −30, −20 and −10 dB for both

For the performance comparison, we compared the registration accuracy of our proposed algorithm for various scenarios and

Next, we again performed the pairwise

A QuickBird dataset consisting of one multispectral image (MI) of size 150 × 300 pixels and one panchromatic image (PAN) of size 600 × 1,200 pixels was used in this experiment (^{2}, on 10 July 2008. Through visual interpretation, we classified the area into five classes, namely, water, shadow, vegetation, and impervious type 1 and impervious type 2. The ground truth image is shown in

In Experiment 2, we focused on the robustness of the proposed algorithm with respect to different degrees of the initial displacement, scale and rotation errors. In fact, there were six displacement errors in the _{changes}^{−5} and _{movement,MI}

Since the PAN had a higher resolution, we assumed that it was aligned with the LCM, and we only needed to find the map parameters of the MI. Here, the PAN has a higher resolution than the multispectral image by the factor of four, and both MI and PAN were obtained from the same satellite. The optimal map parameter vector relating the two images was therefore be equal to

To investigate the robustness of our algorithm to displacement, scale and rotation errors, we introduced the registration errors in displacement in the _{MI} (see ^{opt}^{t}^{opt}

The overall accuracies as a function of Δ

Similar to the previous example, we also compared the performance of our proposed algorithm (PA) with two extreme cases where images were perfectly registered (PR) and there was no registration error correction (NC). The results are provided in

Next, we also noticed that our proposed algorithm could sometimes achieve even higher accuracies than those of the perfect registration cases. This was due to the fact that our algorithm required more iterations than the scenarios where the image pair was perfectly registered since our algorithm terminated if both the estimated map parameters and the resulting LCM converged whereas, in the perfect registration case, the process terminated if only the resulting LCM converged. Hence, our algorithm might terminate at lower percentages of changes in the LCM, and lead to more accurate LCM which, in turn, resulted in higher precision.

Another key performance metric of our algorithm was the resulting registration errors.

For performance comparison, we applied the normalized cross correlation method [

Based on the experimental results shown in

In this paper, we propose a joint image registration and land cover mapping algorithm based on a Markov random field model. By combining image registration and classification into a single process, the classification performance of our proposed algorithm is not affected by the registration errors in image datasets, whereas the performance of traditional image classification algorithms can be significantly degraded due to registration errors.

In our work, the algorithm assumes that observed remote sensing images are derived from a hidden land cover map and captured with an unknown misalignment. Two adjacent pixels of the land cover map are more likely to belong to the same land cover class than different classes. By integrating this fact into the model, a large number of misclassified pixels, which often appear as isolated pixels, are removed from the resulting land cover map. Since the map parameter vector relating the different images is unknown, we employ the expectation-maximization procedure to simultaneously estimate the map parameters and use mean field theory to approximate the posterior probability.

We performed an experimental study using one simulated dataset, and one real remote sensing dataset of 2.4 m QUICKBIRD multispectral and 0.6 m QUICKBIRD panchromatic images. Our results show that, for the first data set, our algorithm can successfully classify image pairs and align them in different initial registration errors with proper selection of the Markov random field parameter. In fact, if the Markov random field parameter is chosen properly, our algorithm can classify mis-registered image pairs with a similar accuracy to the situation where images are perfectly aligned. For the real remote sensing dataset, we focused the investigation on the robustness of our algorithm to the initial alignment of image pairs. The study revealed that our algorithm is less sensitive to the initial alignment when the value of the Markov random field parameter is small since the expectation-maximization algorithm tends to converge faster.

One major limitation of our proposed algorithm is that the expectation-maximization algorithm employed in our algorithm tends to be trapped in local optima if the initial misalignment is large. Hence, in the future, we plan to investigate how to incorporate a different variation of the expectation-maximization algorithm that can escape from local optima in order to make our algorithm more robust. Another limitation of our algorithm is its sensitivity to the Markov random field parameter selection. To address this problem, we plan to investigate how to automatically tune the Markov random field parameter so that the joint image registration and classification can be performed without the initial Markov random field parameter section.

This work is supported in part by the Thailand Research Fund under Grant RSA5480031.

The authors declare no conflict of interest.

Clique types for (

Block diagram of the modified expectation maximization (EM) algorithm.

Noiseless Simulated Image in Example 1.

The ground data of Example 1.

An example of the noisy input image at σ = 1 in Example 1.

Examples of the maximum likelihood classifier (MLC)-based land cover maps (LCMs) for (

Examples of the resulting LCMs from our proposed algorithm (

The averaged number of iterations required before the termination criteria were satisfied for different scenarios in Example 1.

QUICKBIRD dataset of a part of Kasetsart University (

Ground truth image for Example 2 (green, blue, black, red and white colors for vegetation, water, shadow, impervious type 1 and impervious type 2, respectively).

LCMs for the perfect registration case for (

Overall accuracies for different values of

The effect of initial registration errors to the overall accuracies for (

The effect of the initial registration errors to the number of iterations (

The effect of the initial registration errors on the residual registration error of our proposed algorithm in Example 2.

Three scenarios for mapping parameter errors in Example 1.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | ||
---|---|---|---|---|---|---|---|

Scenario I: Displacement error ( |
1 | 1 | 0 | 0 | 1 | 0 | 0 |

2 | 1 | 0 | 0 | 1 | 0 | ||

3 | 1 | 0 | 0 | 1 | 0 | − | |

4 | 1 | 0 | 0 | 1 | − |
||

| |||||||

Scenario II: Scale error ( |
1 | 1 | 0 | 0 | 1 | 0 | 0 |

2 | 1+ |
0 | 0 | 1 | 0 | 0 | |

3 | 1 | 0 | 0 | 1+ |
0 | 0 | |

4 | 1− |
0 | 0 | 1− |
0 | 0 | |

| |||||||

Scenario III: Sheer error ( |
1 | 1 | 0 | 0 | 1 | 0 | 0 |

2 | 1 | 0 | 1 | 0 | 0 | ||

3 | 1 | 0 | 1 | 0 | 0 | ||

4 | 1 | − |
− |
1 | 0 | 0 |

Comparison of the averaged percentages of misclassified pixels (PMP) between two extreme cases and our proposed algorithm.

Scenario I with |
Scenario II with |
Scenario III with |
Scenario I with |
Scenario II with |
Scenario III with | ||
---|---|---|---|---|---|---|---|

0.0 | 25.65% | 28.66% | 26.87% | 27.05% | 28.65% | 26.07% | 27.12% |

0.25 | 0.43% | 4.81% | 5.96% | 6.45% | 0.45% | 0.43% | 0.43% |

0.5 | 0.039% | 4.24% | 5.65% | 6.21% | 0.039% | 0.041% | 0.043% |

0.75 | 0.021% | 4.19% | 5.56% | 6.13% | 0.024% | 0.032% | 0.026% |

The

Scenario I with |
Scenario II with |
Scenario III with |
Scenario I with |
Scenario II with |
Scenario III with | ||
---|---|---|---|---|---|---|---|

0.0 | 1 | 1.5 × 10^{−22} |
1.6 × 10^{−14} |
4.0 × 10^{−18} |
1.9 × 10^{−23} |
4.0 × 10^{−15} |
3.9 × 10^{−15} |

0.25 | 1 | 2.0 × 10^{−17} |
3.5 × 10^{−19} |
3.6 × 10^{−18} |
0.457 | 0.717 | 0.500 |

0.5 | 1 | 1.5 × 10^{−15} |
2.8 × 10^{−17} |
1.8 × 10^{−16} |
0.712 | 0.167 | 0.401 |

0.75 | 1 | 1.5 × 10^{−14} |
1.4 × 10^{−15} |
6.2 × 10^{−17} |
0.060 | 0.033 | 0.079 |

The averaged percentages of mis-classified pixels as a function of the initial registration error for all Scenarios.

PMP | PMP | PMP | |||

0 | 0.019% | −0.05 | 0.035% | −0.05 | 0.036% |

4 | 0.032% | −0.03 | 0.035% | −0.03 | 0.029% |

8 | 0.029% | −0.01 | 0.022% | −0.01 | 0.043% |

12 | 0.026% | 0.01 | 0.030% | 0.01 | 0.040% |

0.03 | 0.024% | 0.03 | 0.036% | ||

0.05 | 0.032% | 0.05 | 0.026% |

The residual registration errors of our proposed algorithm for various scenarios and values of

I ( |
Image 2 | Mean | 12 | 11.99 | 0.111 | 0.295 | 0.280 |

STD | - | 0.0015 | 0.259 | 0.139 | 0.100 | ||

Image 3 | Mean | 12 | 11.99 | 0.031 | 0.192 | 0.312 | |

STD | - | 0.0018 | 0.020 | 0.120 | 0.156 | ||

Image 4 | Mean | 16.97 | 16.96 | 0.213 | 0.338 | 0.212 | |

STD | - | 0.0017 | 0.566 | 0.088 | 0.136 | ||

| |||||||

II ( |
Image 2 | Mean | 14.06 | 13.56 | 0.028 | 0.281 | 0.327 |

STD | - | 0.072 | 0.010 | 0.130 | 0.113 | ||

Image 3 | Mean | 14.06 | 13.49 | 0.020 | 0.353 | 0.312 | |

STD | - | 0.032 | 0.080 | 0.102 | 0.106 | ||

Image 4 | Mean | 21.97 | 20.97 | 0.253 | 0.245 | 0.315 | |

STD | - | 0.095 | 0.636 | 0.120 | 0.082 | ||

| |||||||

III ( |
Image 2 | Mean | 14.76 | 14.71 | 0.025 | 0.295 | 0.296 |

STD | - | 0.204 | 0.020 | 0.149 | 0.098 | ||

Image 3 | Mean | 14.76 | 14.73 | 0.017 | 0.415 | 0.350 | |

STD | - | 0.182 | 0.006 | 0.090 | 0.136 | ||

Image 4 | Mean | 21.72 | 22.04 | 0.350 | 0.312 | 0.371 | |

STD | - | 0.0325 | 0.983 | 0.155 | 0.088 |

The residual registration errors for various noise variances and

| |||||||||
---|---|---|---|---|---|---|---|---|---|

Image 2 | Image 3 | Image 4 | Image 2 | Image 3 | Image 4 | Image 2 | Image 3 | Image 4 | |

−30 | 0.007 | 0.011 | 0.009 | 0.006 | 0.010 | 0.019 | 0.012 | 0.019 | 0.013 |

−20 | 0.010 | 0.012 | 0.009 | 0.023 | 0.016 | 0.012 | 0.017 | 0.016 | 0.011 |

−10 | 0.036 | 0.035 | 0.037 | 0.028 | 0.018 | 0.029 | 0.028 | 0.030 | 0.022 |

0 | 0.244 | 0.280 | 0.185 | 0.119 | 0.138 | 0.071 | 0.078 | 0.053 | 0.200 |

The residual registration errors for various noise variances and β = 0.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

−30 | 0.016 | 0.08 | 0.010 | 0.015 | 0.007 | 0.019 | 0.009 | 0.011 | 0.019 |

−20 | 0.017 | 0.012 | 0.014 | 0.015 | 0.018 | 0.015 | 0.010 | 0.015 | 0.017 |

−10 | 0.014 | 0.018 | 0.015 | 0.018 | 0.018 | 0.023 | 0.019 | 0.016 | 0.014 |

0 | 11.99 | 11.99 | 16.97 | 11.91 | 11.89 | 20.28 | 12.75 | 12.79 | 20.61 |

The residual registration errors using the minimum mean square error criteria for various noise variances.

| ||||||
---|---|---|---|---|---|---|

−30 | 0.008 | 0.0029 | 0.007 | 0.0041 | 0.010 | 0.0054 |

−20 | 0.422 | 0.0040 | 0.425 | 0.0033 | 0.423 | 0.0049 |

−10 | 0.663 | 0.0037 | 0.665 | 0.0014 | 0.664 | 0.0017 |

0 | 0.875 | 0.516 | 1.637 | 1.441 | 1.352 | 0.9744 |

The

Scenario I, |
Scenario II, |
Scenario III, | |||||||
---|---|---|---|---|---|---|---|---|---|

| |||||||||

Image 2 | Image 3 | Image 4 | Image 2 | Image 3 | Image 4 | Image 2 | Image 3 | Image 4 | |

−30 | 0.829 | 0.402 | 0.883 | 0.413 | 0.413 | 0.201 | 0.507 | 0.092 | 0.407 |

−20 | 1 × 10^{−18} |
4 × 10^{−14} |
2 × 10^{−21} |
1 × 10^{−13} |
1 × 10^{−13} |
3 × 10^{−15} |
2 × 10^{−13} |
2 × 10^{−13} |
5 × 10^{−17} |

−10 | 3 × 10^{−14} |
2 × 10^{−14} |
3 × 10^{−14} |
3 × 10^{−15} |
3 × 10^{−15} |
5 × 10^{−16} |
2 × 10^{−23} |
1 × 10^{−14} |
7 × 10^{−17} |

0 | 0.004 | 0.016 | 0.004 | 0.001 | 0.001 | 0.003 | 0.0010 | 0.007 | 0.004 |

The initial RMSE_{MI} in meters (pixels in LCM) for various cases in Example 2.

| |||||||
---|---|---|---|---|---|---|---|

Δ |
RMSE_{MI} |
Δ |
RMSE_{MI} |
Δ |
RMSE_{MI} |
Δ |
RMSE_{MI} |

−5 | 12 (20) | −5 | 12 (20) | −5% | 21.3 (36) | −3 | 11.12 (19) |

−3 | 7.2 (12) | −3 | 7.2 (12) | −2.5% | 10.7 (18) | −2 | 7.45 (12) |

−1 | 2.4 (4) | −1 | 2.4 (4) | 0% | 0.0 (0) | −1 | 3.72 (6.2) |

1 | 2.4 (4) | 1 | 2.4 (4) | 2.5% | 10.7 (18) | 1 | 3.72 (6.2) |

3 | 7.2 (12) | 3 | 7.2 (12) | 5% | 21.3 (36) | 2 | 7.45 (12) |

5 | 12 (20) | 5 | 12 (20) | 3 | 11.12 (19) |

Overall accuracies for different values of β in two extreme cases and our proposed algorithm for different initial displacement error in the

Δ |
Δ |
Δ |
Δ |
Δ |
Δ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||||

0.0 | 67.5 | 67.7 | 57.6 | 67.8 | 62.2 | 67.7 | 66.9 | 67.8 | 66.7 | 67.7 | 61.8 | 67.8 | 57.0 |

0.25 | 69.4 | 70.0 | 58.8 | 69.8 | 63.7 | 69.8 | 68.6 | 69.9 | 68.3 | 70.0 | 63.4 | 59.3 | 58.4 |

0.5 | 70.3 | 71.8 | 59.7 | 71.4 | 64.6 | 70.6 | 69.6 | 70.9 | 69.2 | 71.5 | 64.4 | 60.2 | 59.2 |

0.75 | 71.1 | 72.8 | 60.2 | 72.2 | 65.2 | 71.5 | 70.3 | 71.8 | 70.0 | 72.7 | 65.0 | 60.4 | 59.9 |

Overall accuracies for different values of β in two extreme cases and our proposed algorithm for different initial displacement error in the y-direction Δy where PA and NC denote the cases of the proposed algorithm and no registration error correction, respectively.

Δ |
Δ |
Δ |
Δ |
Δ |
Δ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||||

0.0 | 67.5 | 67.7 | 57.6 | 67.7 | 62.2 | 67.7 | 66.9 | 67.7 | 66.7 | 67.7 | 61.8 | 67.8 | 57.0 |

0.25 | 69.4 | 69.9 | 58.8 | 69.9 | 63.7 | 69.8 | 68.6 | 70.1 | 68.3 | 70.1 | 63.4 | 70.3 | 58.4 |

0.5 | 70.3 | 71.6 | 59.7 | 71.2 | 64.6 | 70.5 | 69.6 | 71.8 | 69.2 | 71.8 | 64.4 | 68.6 | 59.2 |

0.75 | 71.1 | 72.5 | 60.1 | 71.9 | 65.2 | 71.2 | 70.3 | 73.4 | 70.0 | 73.4 | 64.9 | 62.9 | 59.9 |

Overall accuracies for different values of β in two extreme cases and our proposed algorithm for different initial scale error Δs where PA and NC denote the cases of the proposed algorithm and no registration error correction, respectively.

Δ |
Δ |
Δ |
Δ |
Δ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||

0.0 | 67.5 | 67.8 | 52.7 | 67.7 | 61.0 | 67.7 | 67.5 | 67.8 | 64.9 | 67.8 | 57.8 |

0.25 | 69.4 | 69.6 | 53.4 | 69.5 | 62.4 | 70.0 | 69.4 | 70.3 | 66.1 | 70.2 | 58.9 |

0.5 | 70.3 | 71.1 | 54.2 | 70.6 | 63.3 | 71.0 | 70.3 | 71.6 | 67.0 | 72.1 | 59.7 |

0.75 | 71.1 | 72.1 | 54.7 | 71.5 | 64.2 | 71.1 | 71.1 | 72.7 | 67.6 | 73.4 | 60.1 |

Overall accuracies for different values of β in two extreme cases and our proposed algorithm for different rotation error Δθ where PA and NC denote the cases of the proposed algorithm and no registration error correction, respectively.

Δ |
Δ |
Δ |
Δ |
Δ |
Δ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||||

0.0 | 67.5 | 67.6 | 57.3 | 67.6 | 60.8 | 67.6 | 65.3 | 67.7 | 64.8 | 67.7 | 59.8 | 67.8 | 55.5 |

0.25 | 69.4 | 69.9 | 58.5 | 69.8 | 62.2 | 69.7 | 66.9 | 69.9 | 66.5 | 69.7 | 61.1 | 69.8 | 56.6 |

0.5 | 70.3 | 71.6 | 59.3 | 71.4 | 63.0 | 71.0 | 67.8 | 71.1 | 67.4 | 71.4 | 62.0 | 71.5 | 57.4 |

0.75 | 71.1 | 73.0 | 59.7 | 72.3 | 63.6 | 71.9 | 68.4 | 71.9 | 68.1 | 72.5 | 62.6 | 72.9 | 58.0 |