# Oligoclonal Band Straightening Based on Optimized Hierarchical Warping for Multiple Sclerosis Diagnosis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Clinical Context

#### 1.2. The Automatic Analysis of IEF Membranes

#### 1.3. Geometric Band Distortions in IEF Images

#### 1.4. Related Research on Band Straightening

## 2. Materials and Methods

#### 2.1. Description of the Datasets

#### 2.1.1. IEF Image Acquisition and Pre-Processing

#### 2.1.2. The Real IEF Dataset

#### 2.1.3. The Synthetic Dataset

#### 2.2. Background Removal

#### 2.3. Correlation-Based Image Warping

#### 2.3.1. The Energy Minimization Approach for Image Warping

- External energy

- b.
- Internal energy

- c.
- The deformation constraint

#### 2.3.2. The Transformation Hierarchy

#### 2.3.3. Coupling of a Hierarchy of Image Resolutions to a Hierarchy of Transformations

#### 2.3.4. The Band-Straightening Algorithm

- Downsample the original lane image ${I}_{0}$ with the corresponding scale factors (${f}_{{r}_{j}},{f}_{{c}_{j}})$ (Section 2.3.3)
- Apply the deformation $\Delta Y$ to obtain the current image: $I\left(x,y+\Delta Y\left(x,y\right)\right)={I}_{0}\left(x,y\right)$
- For each type of move ${M}_{g}\in {\mathcal{M}}_{j}$ (${\mathcal{M}}_{j}$ being a list of types of move specific for each hierarchical step $j$) corresponding to a grid size $\langle {r}_{g},{c}_{g}\rangle $:
- Choose a random permutation ${L}_{g}$ of the set of the ${r}_{g}\times {c}_{g}$ grid points
- For each grid point $k$ in ${L}_{g}$ with coordinates $({x}_{k},{y}_{k})$ and for each possible sign $s\in \left\{-1,1\right\}$ of the shift (−1 being down, and 1 being up):
- Move vertically $k$ with ${\delta}^{k}=s\times {\delta}_{j}$$$\Delta {Y}^{\prime}\left({x}_{k},{y}_{k}\right)=\Delta Y\left({x}_{k},{y}_{k}\right)+s\times {\delta}_{j}$$
- Adjust the position of grid points on the same row as $k$, to comply with the band-straightening algorithm’s constraint $\int}\Delta {Y}^{\prime}\left(x,y\right)dx=0$ (Section 2.3.2)
- Linearly interpolate between grid points to obtain the warped lane image ${I}^{\prime}$
- If $E\left(\Delta {Y}^{\prime};{I}^{\prime}\right)<E(\Delta Y;I$):
- (a)
- Put the 4-neighbor grid points of $k$ at the end of ${L}_{g}$ (except when $k$ is located at border)
- (b)
- Move $k$ with a smaller shift (equal to $0.5\times s\times {\delta}_{j}^{y}$) and a larger shift (equal to $1.5\times s\times {\delta}_{j}^{y}$) and repeat steps 2 and 3 to obtain $\left({I}_{0.5}^{\prime},\Delta {Y}_{0.5}^{\prime}\right)$ and $\left({I}_{1.5}^{\prime},\Delta {Y}_{1.5}^{\prime}\right)$
- (c)
- Set $\left(I,\Delta Y\right)=\underset{\left(I,\Delta Y\right)\in \left\{\begin{array}{c}\left({I}_{0.5}^{\prime},\Delta {Y}_{0.5}^{\prime}\right)\\ \left({I}^{\prime},\Delta {Y}^{\prime}\right)\\ \left({I}_{1.5}^{\prime},\Delta {Y}_{1.5}^{\prime}\right)\end{array}\right\}}{\mathrm{argmin}}E\left(\Delta Y;I\right)$

#### 2.3.5. The Algorithm’s Settings

#### 2.3.6. Optimization of the Band-Straightening Algorithm

#### 2.4. Evaluation Methods

#### 2.4.1. Real IEF Dataset

- (i)
- SD $>$ 2 pixels correspond to a negligibly to-strongly deformed band.
- (ii)
- SD $>$ 3 pixels correspond to a weakly to-strongly deformed band.
- (iii)
- SD $>4\mathrm{pixels}$ corresponds to a moderately-to-strongly deformed band.
- (iv)
- SD $>$ 5 pixels correspond to a strongly deformed band.

#### 2.4.2. Synthetic Dataset

## 3. Results and Discussion

#### 3.1. Illustrative Results

#### 3.1.1. Results with the Real IEF Dataset

#### 3.1.2. Results with the Synthetic Dataset

#### 3.2. Statistical Evaluation of the Algorithm’s Performance with Real and Synthetic Data

#### 3.3. OCBSA-2021’s Contribution to OCB Detection

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of a CSF IEF membrane with different types of band deformation: the left and right contours of each lane are delineated (blue lines) using our automatic lane segmentation method described in [11].

**Figure 2.**Illustration of the importance of band straightening for faint, non-horizontal bands in a CSF lane: (

**Aa**): a lane with deformed bands; (

**Ba**): the same lane with straight bands; (

**Ab**,

**Bb**): a background-subtracted grayscale image (contrast ×12, for clarity); (

**Ac**,

**Bc**): three column profiles with and without band deformations, band peaks and signal valleys are aligned in (

**Bc**); (

**Ad**,

**Bd**): a 1D intensity profile. The band peak amplitudes in (

**Bd**) are higher than those in (

**Ad**). Non-horizontal, low-intensity bands 1, 2 and 5 are not detectable on (

**Ad**) but are detectable on (

**Bd**).

**Figure 3.**Illustration of the rolling ball approach for background and vertical irregularities subtraction: the ellipsoid falls into the valleys to be removed and rolls over the valleys to be preserved.

**Figure 4.**Illustration of the band-straightening method steps with a CSF lane: (

**a**) lane segmentation, (

**b**) lane straightening, (

**c**) grayscale conversion and removal of unambiguous artifacts (the black zones on the lane), (

**d**) Background removal (contrast ×12 for clarity), (

**e**–

**i**) are examples of intermediate steps chosen to illustrate the iterative process of band straightening with progressively finer moves (the deformed grid is superimposed on the image in red). Each blue arrow indicates the direction of the shift applied to the chosen grid point during the last iteration. (

**j**) The final result.

**Figure 5.**Results of band straightening on a CSF lane: (

**a**–

**c**): original colors; (

**d**–

**f**): grayscale converted, background subtracted, non-ambiguous artifact in black and contrast ×12 for clarity; (

**a**,

**d**): original deformation; (

**b**,

**e**): the output of OCBSA-2016; (

**c**,

**f**): the output of OCBSA-2021. Bands are shown as red dashed lines, and the corresponding SD are displayed on their right.

**Figure 6.**A comparison of band straightening with OCBSA-2016 (

**b**,

**e**,

**h**) vs. OCBSA-2021 (

**c**,

**f**,

**i**) on a control lane (

**a**–

**c**), a serum lane (

**d**–

**f**), and a tear lane (

**g**–

**i**).

**Figure 7.**Illustration of a band-straightening result for the CSF membrane in Figure 1.

**Figure 8.**Illustration of band straightening on two synthetic profiles: (

**a**,

**e**): non-deformed synthetic lanes; (

**b**,

**f**): digitally added non-uniform band deformations; (

**c**,

**g**): results of band straightening with OCBSA-2016; (

**d**,

**h**): results of band straightening with OCBSA-2021.

**Figure 9.**Band-straightening results for the real IEF training (

**A**) and test (

**B**) datasets: the number of bands with different deformation thresholds (SD > 2 pixels, 3 pixels, 4 pixels, 5 pixels) before straightening, after OCBSA-2016 and after OCBSA-2021 are displayed for the whole dataset and for each sample type (control, tears, CSF, and serum). The ${\rho}_{t}=\frac{nu{m}_{befor{e}_{t}}}{nu{m}_{afte{r}_{t}}}$ ratios with the different deformation thresholds are shown for the whole dataset (orange for OCBSA-2016, grey for OCBSA-2021).

**Table 1.**The average SD of the deformation introduced to the lanes without bands after processing with OCBSA-2016 or OCBSA-2021, for the training and test datasets.

Band-Straightening Algorithm | $\mathbf{Average}\mathbf{SD}\left(\Delta \mathit{Y}\right)$ | |
---|---|---|

Training—All(22 lanes) | OCBSA-2016 | 1.214 |

OCBSA-2021 | 0.976 | |

Test—All(18 lanes) | OCBSA-2016 | 1.416 |

OCBSA-2021 | 1.043 |

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**MDPI and ACS Style**

Haddad, F.; Boudet, S.; Peyrodie, L.; Vandenbroucke, N.; Poupart, J.; Hautecoeur, P.; Chieux, V.; Forzy, G.
Oligoclonal Band Straightening Based on Optimized Hierarchical Warping for Multiple Sclerosis Diagnosis. *Sensors* **2022**, *22*, 724.
https://doi.org/10.3390/s22030724

**AMA Style**

Haddad F, Boudet S, Peyrodie L, Vandenbroucke N, Poupart J, Hautecoeur P, Chieux V, Forzy G.
Oligoclonal Band Straightening Based on Optimized Hierarchical Warping for Multiple Sclerosis Diagnosis. *Sensors*. 2022; 22(3):724.
https://doi.org/10.3390/s22030724

**Chicago/Turabian Style**

Haddad, Farah, Samuel Boudet, Laurent Peyrodie, Nicolas Vandenbroucke, Julien Poupart, Patrick Hautecoeur, Vincent Chieux, and Gérard Forzy.
2022. "Oligoclonal Band Straightening Based on Optimized Hierarchical Warping for Multiple Sclerosis Diagnosis" *Sensors* 22, no. 3: 724.
https://doi.org/10.3390/s22030724