Remove white noise between white horizontal line

edit

There are 2 steps. In cv.GaussianBlur, You can change odd number btween 15 to 21, but not even numbers to suit our needed. Step 1:

#!/usr/bin/env python3
#OpenCV 4.0.1
#Date: 20th April, 2019

import cv2 as cv
import numpy as np

img = cv.imread('lines.jpg')
img = cv.cvtColor(img, cv.COLOR_BGR2GRAY)
gray = cv.GaussianBlur(img, (19, 19), 0)
img = cv.bitwise_not(gray)
adp_thresh = cv.adaptiveThreshold(img, 255, cv.ADAPTIVE_THRESH_MEAN_C,cv.THRESH_BINARY,15,-2)

horizontal_inv = cv.bitwise_not(adp_thresh.copy())
masked = cv.bitwise_and(img, img, mask=horizontal_inv)

inv_masked = cv.bitwise_not(masked)
cv.imshow('Masked', inv_masked)
cv.imwrite('masked.jpg', inv_masked)
cv.waitKey(0)
cv.destroyAllWindows()

Output:

Step 2:

#!/usr/bin/env python3
#OpenCV 4.0.1
#Date: 20th April, 2019

import cv2 as cv
import numpy as np

img = cv.imread('lines.jpg')
img=cv.cvtColor(img, cv.COLOR_BGR2GRAY)
gray = cv.GaussianBlur(img, (19, 19), 0)

img = cv.bitwise_not(gray)
adapt_thresh = cv.adaptiveThreshold(img, 255, cv.ADAPTIVE_THRESH_MEAN_C, cv.THRESH_BINARY, 15, -2)
cv.imshow('adapt_thresh', adp_thresh)
#cv.imwrite('adapt_thresh.jpg', adp_thresh=)
cv.waitKey(0)
cv.destroyAllWindows()

Output:

Well, that's an ugly image!

I suggest to clean up the original (RGB) image, it's much easier than the binary image.

Try also to improve the binarization (e.g. work on HSV or LAB color space, use several channels, combine the RGB channels) to get a cleaner result. If you post the original image, you could get better ideas.

If you must work with the binary image, a directional diffusion (Perona&Malik method, ~~not part of OpenCV~~ correction: see below) could be a solution.

[EDIT] I was wrong, Perona-Malik is part of the Extended image processing module: cv::ximgproc::anisotropicDiffusion