# Image Reconstruction and Evaluation: Applications on Micro-Surfaces and Lenna Image Representation

## Abstract

**:**

## 1. Introduction

^{−1}-Function interpolation method, kernel-based modeling method, and Thin Plate Spline (TPS) method [6,7,8]. In the kernel-based method, a penalized likelihood of the projection data is used to reconstruct images from the estimated coefficient model. However, the method is challenging because the inverse problem is ill-posed and the resultant image is usually very noisy [8]. TPS algorithm is used in the representation of 3D object from 2D images. TPS requires no assumption to be made with respect to the distribution or the location of the DEM. This method has become popular in the medical imaging because the reduction of the registration from global into local allows the analysis of local changes of image [9,10]. While many researches showed the applications of TPS in the visualization of macroscopic objects, little information is known about its ability to characterize or visualize micro and nano structures. Towards this goal, we aim to expand upon the application of TPS algorithm for imagining of MEMS structures.

## 2. Materials and Methods

#### 2.1. Modified TPS Algorithm

#### 2.2. Characterization of MEMS Micro-Surfaces by Using Modified TPS Algorithm

^{2}in the SI unit system. In addition, it is used as a qualitative scaling factor. The TPS function, z, in Equation (3) minimizes the nonnegative quantity of I

_{z}; therefore, a planar micro-surface will have zero bending energy at areas where curvatures have high energy. A reconstructed topography of the bending energy is used as a qualitative measure of the geometry of a micro-surface at a location. For example, it views the negative features on a MEMS micro-surface image such as a micro-machined surface [15].

#### 2.3. Image Processing Based on Modified TPS Algorithms

#### 2.3.1. Modified TPS Based Image-Reconstruction Method

- A spatial neighborhood: this technique reconstructs a region within an image by using the information that is available within it, and without the knowledge from other neighboring regions. The method can be applied either globally for the entire image, or locally for a selected region within the image.
- Spatio-to-temporal neighborhood: this technique reconstructs an image by sequentially updating the information from multiple neighborhoods. For example, an image could be reconstructed and warped from several regions.

#### 2.3.2. Modified TPS Based Denoising Method

- Construct bending energy image, which is a greyscale intensity image of the defected image, and it is computed by using the bending energy index—Equation (18)—in the global coordinate. The impulsive defects are located in the constructed bending energy image because they are artifacts that tend to have black or white level with strong edges at their spatial boundaries [22]; therefore, a white pixel in the bending energy image corresponds to a black pixel in the original image, and it refers to a local maximum. A black pixel indicates a local minimum, and it corresponds to a white pixel in the original image.
- Obtain the locations of the defects from step (1), and then remove them to create a defect-free DEM; i.e., the original pixel intensity values are free of defect.
- Calculate the modified TPS coefficients from the updated DEM—from step (2)—by using Equation (8). There are two ways to calculate the modified TPS coefficients:
- Global: select all the available information from the updated DEM.
- Local: select a neighborhood for each defect.

- Interpolate the intensity value of the removed data by using Equation (3).
- If required, reconstruct high-resolution image by using the method described in Section 2.3.1.

#### 2.3.3. LSE-TPS Based Warping Algorithm

#### 2.3.4. LSE-TPS Based Rapid Reconstruction Method

- Reconstruct each patch by using its own control points.
- Warp the patches at their spatial boundaries—for example, regions Ω and $\mathcal{M}$ in Figure 3a—by selecting a neighboring set of control points for each boundary. Then, reconstruct the boundaries by using adequate method from Section 2.3.1, or the LSE-TPS algorithm that we described in Section 2.3.3. Figure 3b illustrate warping of multiple patches into a single image.

## 3. Discussion

#### 3.1. Performance Studies of the Modified TPS and Windwoing Techniques

^{2}image from Figure 1a, then reduced the resolution of the extracted image by half. This is done by removing the intermediate pixels from the 100 × 100 Pixel

^{2}image. The results are shown in Figure 4.

^{2}image in Figure 4b. The pixel intensity is based on a grayscale image with a maximum intensity value of 256. The intermediate points, which bisect the mid distances between every two adjacent pixels in column 37 of Figure 4b, are interpolated using Equation (3).

^{2}shown in Figure 4b. The total error based on the modified TPS with ɛ = 10

^{−6}is calculated at column 37 and it is equal to 363.5187. It should be noted that the intermediate points suffer from small deviation from the true value as shown in Figure 5 and as noted by the non-zero values of accumulative error ${e}_{MTPS}$. The estimated value of a pixel at the DEM location is equal to the true value, i.e., coincident. This is shown in Figure 6, which plots three superimposed micro-surfaces in three dimension: (1) the triangulation of true pixel of image in Figure 4b; (2) the interpolation of the entire image whose pixels are evaluated at the location of each true pixel by using the Modified TPS method described in Section 2.3.1; and (3) the intermediate pixels which are interpolated along column 37.

^{2}is used to reconstruct 100 × 100 Pixel

^{2}image. The SNR values of the entire true images—50 × 50 Pixel

^{2}and 100 × 100 Pixel

^{2}—are 14.9 and 15.1, respectively. Then, we compare the values of the SNR which is obtained for each method to reconstruct 100 × 100 Pixel

^{2}image. Table 2 shows that the nearest method has the lowest SNR among other methods and it is sufficiently equal to the SNR for the true image of 50 × 50 Pixel

^{2}. This is due to the locality nature of the “nearest” method, which emphasizes the information available around the region of interpolated pixel. However, the modified TPS has the highest SNR value of 16.4 which could be explained by the global nature of the applied modified TPS method where all DEMs in the 50 × 50 Pixel

^{2}is weighted in the interpolation.

^{2}in Figure 4b to increase the image resolution by using modified TPS method, and with ɛ = 10

^{−6}. The interpolation is carried out to interpolate the intermediate points between pixels, and with step up resolutions as indicated in Table 3. It can be concluded that the value of the calculated SNR increases as the resolution of the reconstructed image increase, however, to understand how such change is related to the size of the reconstructed image relative to the true image, we define SNR error percentage by $\mathsf{\beta}=\left(SN{R}_{Re}-SN{R}_{Tr}\right)/SN{R}_{Tr}\times 100\%$. Where $SN{R}_{Re}$ is the signal-noise ratio of the reconstructed image, and $SN{R}_{Tr}$ is the signal-noise-ratio of the true image used in the reconstruction. The simulation results in Figure 7 shows exponential convergence of β. i.e., the signal-to-noise ratio of a reconstructed image has maximum limit where it reaches saturation at certain pixel resolution. In this case study, the pixel resolution that corresponds to the saturation limit could be obtained from the following suggested fit-model: ${\mathsf{\beta}}^{*}=\left({p}_{1}x+{p}_{2}\right)/\left(x+{q}_{1}\right)$, where $x$ is the pixel intensity expressed by $Log\left(Pixe{l}^{2}\right)$, and $\left\{{p}_{1},{p}_{2},{q}_{1}\right\}$ is a set of undetermined coefficients. The fit analysis is carried out by using MATLAB “fittool” GUI function. The coefficients $\left\{{p}_{1},{p}_{2},{q}_{1}\right\}$ are evaluated and found equal to $\left\{12.91,-101,-7.425\right\}$. The “goodness of fit” is evaluated by using the regression coefficient, R

^{2}, which is equal to 0.9999. The final value of the fit-model, ${\mathsf{\beta}}^{*}$, can be found by using “limit theory”, and it gives ${\mathsf{\beta}}^{*}\left(x\to \infty \right)=12.91\%$, i.e., $SN{R}_{Re}\left(Pixel\text{}Intensity\to \infty \right)=16.82$. The pixel intensity that corresponds to 1% within the range of the final value of $SN{R}_{Re}$ is ~381 × 381 Pixel

^{2}. This explains that the SNR value of the reconstructed image would not change significantly at high resolution.

^{3}+ O(n

^{2}) [24], where n is the total number of control points within an image. The Householder is a stable and efficient algorithm that is typically used for the inversion of large symmetrical matrices [25]; however, when an image is divided into L patches, the number of the computations becomes 3L/2(n/L)

^{3}+ O(n

^{2}); therefore, the number of the computation decreases when the number of patches increases. For example, suppose an image contains 1024 pixels. The number of computation required to inverse H matrix with L = 1 and L = 24 are 1.6 × 10

^{9}and 2.8 × 10

^{6}, respectively.

#### 3.2. Previous Studies

## 4. Application Results

#### 4.1. Characterization of Micro-Surfaces Topology

**ɛ**, in the TPS model makes it possible to obtain contentious derivatives of the model in the spatial domain. Therefore, in this illustrative example, we characterize the topology of a micro-surface that contains roughness and waviness components. The height variation of a MEMS micro-surface in Figure 8a is measured at different location $\left(x,y\right)$, and then it is registered in DEM. A 3D MEMS profiler, “Veeco” [37] is used to scan the micro-surface at spatial pixel resolution of 0.193 µm, and with a total number of control points of 479 × 639 Pixel

^{2}. Figure 8b shows a 3D visualization of the pad micro-surface plotted by using all the measured control points. The characterization of such data-rich model is computationally inefficient; therefore, a sufficiently small number of control points would be preferred.

^{2}is uniformly sampled from the DEM at a lateral pixel resolution of 0.965 µm. Figure 9a shows a linear triangulation of 100 control points covering an area of 92.5 µm × 123.3 μm. Equation (8) is applied to extract the TPS coefficients, and then the micro-surface is constructed at a smaller pixel size of 0.5 µm by using Equation (3). Figure 9b shows a continuous and smooth micro-surface that passes through all the 100 control points. The variation of the slopes in the x and y coordinates are obtained from Equations (9) and (10), and then they are plotted at 0.5 µm resolution, as shown in Figure 9c,d, respectively. Similarly, the curvature variation in the x and y coordinates are constructed by using Equations (11) and (12), with the results shown in Figure 9e,f, respectively. These later micro-surfaces measure the topography fluctuation; where a spiky point implies that there is a saddle point close to right angle. The strike angle in Equation (15) is a measure of the topographical contour slopes. Figure 9g shows a reconstruction of the strike micro-surface; i.e., non-uniform fluctuation of strike angles within ±90° range. The dips micro-surface in Figure 9h is constructed from Equation (17), and it shows positive inclination orthogonal to the strike line. The bending energy in Figure 9i is constructed from Equation (18), and it shows the bent variation in the micro-surface due to curvatures. The nodes, which are located on the infinite part of the extrapolated micro-surface, have negligible bending energies; therefore, the energy index can be used to detect the sudden changes within micro-surface such as holes, groves, and standing features in MEMS structures.

#### 4.2. Visual Enhancement of Grayscale Lenna Image

^{2}, and it is used to study the image processing techniques discussed in Section 4. A sub-image of 51 × 51 Pixel

^{2}is extracted from the Lena image as shown in Figure 10b. This later image will be used throughout this section.

^{2}as shown in Figure 11a. The down sampling scheme followed here is conducted by removing the intermediate pixels located between every two adjacent columns and rows of the image in Figure 10b. The control points, which are contained in this low resolution image, is used to extract the TPS coefficients by using Equation (8), then a higher resolution image of 260 × 260 Pixel

^{2}is reconstructed in the global coordinate by using Equation (3). The result is shown in Figure 11b.

^{2}as shown in Figure 13a. The defect is detected by using the bending energy algorithm. The control points, which are corresponding to the defect, are removed from the image, leading to a DEM defect-free. This new DEM is used in Equation (3) to reconstruct a higher resolution image of 51 × 51 Pixel

^{2}by using the global TPS algorithm, and the result is shown in Figure 13b.

^{2}image. Figure 14a is divided into 5 × 5 square windows. The modified TPS algorithm is then implemented at the local coordinates of each window with the goal to increase the overall image resolution. The overall image resolution has increased from 100 × 100 Pixel

^{2}to 250 × 250 Pixel

^{2}as shown in Figure 14b,c, respectively.

^{2}are 198, 977 and 37,500, respectively; however, the boundaries between the square windows are not smooth, and they become visually sharp when the resolution increases. This is observed in Figure 10b,c. To mitigate such shortcoming, the rest of this section studies the application of warping by using the procedures that we suggested in Section 2.3.3 and Section 2.3.4.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scaning electron microscopy (SEM) image of a 3D interconnected microelectricomechanical system (MEMS) structures: (

**a**) the low magnification image shows the entire structure; and (

**b**) the high magnification image shows the area of interest.

**Figure 4.**Gray scale images extracted from SEM picture in Figure 1b: (

**a**) Resolution is reduced down to 100 × 100 Pixel

^{2}; (

**b**) Resolution is reduced down to $50\times 50$ Pixel

^{2}.

**Figure 6.**Three dimensional micro-surface of the interpolated points superimposed on the true image surface.

**Figure 7.**Semi log plot of the SNR error percentage, β, and the size of the reconstructed image based on modified TPS method.

**Figure 8.**Gold coated electrical-pad of a MEMS silicon micro-surface: (

**a**) 50X optical image; (

**b**) Triangulation of a measured micro-surface area of 92.5 μm × 123.3 μm.

**Figure 9.**Isometric views of: (

**a**) Measured micro-surface; (

**b**) Reconstructed micro-surface; (

**c**) Interpolated slope micro-surface in x-direction; (

**d**) Interpolated slope micro-surface in y-direction; (

**e**) Interpolated curvature micro-surface in x-direction; (

**f**) Interpolated y-direction curvature; (

**g**) Interpolated micro-surface of strike angles; (

**h**) Interpolated micro-surface of dip angles; (

**i**) Interpolated bending energy micro-surface.

**Figure 10.**Grey intensity image: (

**a**) Lena Image; (

**b**) A sub-image zoomed in the eye area of 51 × 51 pixels

^{2}.

**Figure 11.**Reconstruction from few control points by using global TPS algorithm: (

**a**) Low resolution image containing 26 × 26 Pixels

^{2}; (

**b**) Reconstructed at a higher resolution of 260 × 260 Pixels

^{2}.

**Figure 12.**Detection of defects: (

**a**) Defected image; (

**b**) Defect detection by using the bending energy algorithm.

**Figure 13.**Reconstruction of a poor image: (

**a**) Defected and distorted image; (

**b**) Automatic denoising and reconstruction.

**Figure 14.**Reconstruction by using local TPS method: (

**a**) A 50 × 50 Pixel

^{2}image is divided into 25 square windows; (

**b**) Reconstructed image at resolution of 100 × 100 Pixel

^{2}; (

**c**) Reconstructed image at resolution of 250 × 250 Pixel

^{2}.

**Figure 16.**Warping by using local TPS algorithm: (

**a**) Torn image; (

**b**) Reconstruction by using available information in the left part; (

**c**) Reconstruction by using available information in the right part.

**Figure 17.**Application of LSE-TPS algorithm: (

**a**) Propagation from left to right; (

**b**) Propagation from right to left.

ɛ | 10^{−6} | 10^{−5} | 10^{−4} | 10^{−3} | 10^{−2} | 10^{−1} | 1 | 10 | 100 |
---|---|---|---|---|---|---|---|---|---|

${\mathit{e}}_{\mathit{M}\mathit{T}\mathit{P}\mathit{S}}$ | 363.5187 | 363.5192 | 363.5243 | 363.5745 | 364.0648 | 368.0492 | 381.5303 | 392.1309 | 77852.1 |

Method | $\mathit{S}\mathit{N}\mathit{R}$ ^{Ϙ} | ${\mathit{e}}_{\mathit{M}\mathit{T}\mathit{P}\mathit{S}}$ ^{†} |
---|---|---|

DEM: 100 × 100 Pixel^{2} | 15.1 | - |

DEM: 50 × 50 Pixel^{2} | 14.9 | - |

Interpolation, Modified TPS with ɛ = 10^{−6} ** | 16.4 | 363.57 |

Interpolation, linear method ** | 15.4 | 2767.75 |

Interpolation, cubic method ** | 15.7 | 2799.50 |

Interpolation, nearest method ** | 14.9 | 2825.00 |

Interpolation, spline method ** | 16.2 | 2829.80 |

^{2}by using DEM size of 50 × 50 Pixel

^{2};

^{†}Evaluated along column 37;

^{Ϙ}Evaluated for the entire reconstructed image with a resolution of 100 × 100 Pixel

^{2}.

Resolution Enhancement (Pixel^{2}) Interpolation by Modified TPS with ε = 10^{−6} ^{*} | SNR ^{ω} |
---|---|

50 × 50 Pixel^{2} | 14.90 |

$100\times 100$ Pixel^{2} | 16.39 |

$164\times 164$ Pixel^{2} | 16.56 |

$246\times 246$ Pixel^{2} | 16.60 |

$491\times 491$ Pixel^{2} | 16.67 |

^{2};

^{ω}Evaluated for the entire reconstructed image.

Method | Description |
---|---|

SIFT | The method uses corner detection algorithm, image matching, affine deformation, viewpoint change, noise, and illumination changes [26]. It shows stability in scale and rotation, robustness to localization error, and reduces the time and computational complexity of images [27]. |

Harris Corner Detector | The method is based on the local auto-correlation function of a signal particularly to detect gray scale image location that has large gradients in all directions at a predetermined scale. The method has good consistency on natural imagery [28]. |

SUSAN | The method is based on non-linear filtering, and it associates each pixel of the image a small area of similar brightness from neighboring pixels. It is used as edge finder [29]. |

SURF | The method is based on multiscale space theory and feature detection by using Hessian matrix, which has good performance and accuracy [26], but needs further improvement when the rotation is large [27]. |

This Study | Our method uses density information of the extracted features, which is incorporated to regularize the overall energy function to reduce the impact from outliers. It is based on corner detection using local or global properties, which is an effective tool to implement the registration of local image parts, especially when the key points are evenly distributed across the whole image. |

Method | Description |
---|---|

Spatial Filtering Method | Typically, smoothing filters are used for blurring and for noise reduction. Sharpening of the image is done by spatial differentiation which enhances edges and other discontinuities and de-emphasizes areas with slowly varying intensities [9]. Example of spatial methods include the Sobel Method [30], Prewitt Method [31], and Laplacian method [32]. |

Genetic Algorithm | The method is a stochastic optimization algorithm that is based on four steps: selection, crossover, mutation and fitness function [33]. A genetic algorithm needs less information to solve any problem as compared to conventional optimization methods. |

Neural Network Method | Restoration is carried out by parameter evaluation and image construction procedure using learning BP neural network to determine the best image transform function [34]. Neural Network is used to remove noise, motion blur, out-of-focus blur, and distortion caused by low-resolution images [35]. |

Fuzzy Method | Used to handle uncertainty to enhance contrast of the images and to remove unclearness in edges, boundaries, regions and features of image [36]. |

This study | We described modified TPS methods, which are based on the original pixel intensity values to evaluate model parameter based on logarithmic RBFs of bending surface of minimum deformation. The model can restore selected areas or entire image, and it is suitable to increase the image resolution, and restore lost information. We combined TPS and LSE in a windowing technique to improve the computation time of reconstruction. |

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Mayyas, M.
Image Reconstruction and Evaluation: Applications on Micro-Surfaces and Lenna Image Representation. *J. Imaging* **2016**, *2*, 27.
https://doi.org/10.3390/jimaging2030027

**AMA Style**

Mayyas M.
Image Reconstruction and Evaluation: Applications on Micro-Surfaces and Lenna Image Representation. *Journal of Imaging*. 2016; 2(3):27.
https://doi.org/10.3390/jimaging2030027

**Chicago/Turabian Style**

Mayyas, Mohammad.
2016. "Image Reconstruction and Evaluation: Applications on Micro-Surfaces and Lenna Image Representation" *Journal of Imaging* 2, no. 3: 27.
https://doi.org/10.3390/jimaging2030027