What is wrong with my cython implementation of erosion operation of mathematical morphology - numpy

Question

I have produced a naive implementation of "erosion". The performance is not relevant since I just trying to understand the algorithm. However, the output of my implementation does not match the one I get from scipy.ndimage. What is wrong with my implementation ?
Here is my implementation with a small test case:
import numpy as np
from PIL import Image
# a small image to play with a cross structuring element
imgmat = np.array([
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,1,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0],
])
imgmat2 = np.where(imgmat == 0, 0, 255).astype(np.uint8)
imarr = Image.fromarray(imgmat2).resize((100, 200))
imarr = np.array(imgrrr)
imarr = np.where(imarr == 0, 0, 1)
se_mat3 = np.array([
[0,1,0],
[1,1,1],
[0,1,0]
])
se_mat31 = np.where(se_mat3 == 1, 0, 1)
The imarr is .
My implementation of erosion:
%%cython -a
import numpy as np
cimport numpy as cnp
cdef erosionC(cnp.ndarray[cnp.int_t, ndim=2] img,
cnp.ndarray[cnp.int_t, ndim=2] B, cnp.ndarray[cnp.int_t, ndim=2] X):
"""
X: image coordinates
struct_element_mat: black and white image, black region is considered as the shape
of structuring element
This operation checks whether (B *includes* X) = $B \subset X$
as per defined in
Serra (Jean), « Introduction to mathematical morphology »,
Computer Vision, Graphics, and Image Processing,
vol. 35, nᵒ 3 (septembre 1986).
URL : https://linkinghub.elsevier.com/retrieve/pii/0734189X86900022..
doi: 10.1016/0734-189X(86)90002-2
Consulted le 6 août 2020, p. 283‑305.
"""
cdef cnp.ndarray[cnp.int_t, ndim=1] a, x, bx
cdef cnp.ndarray[cnp.int_t, ndim=2] Bx, B_frame, Xcp, b
cdef bint check
a = B[0] # get an anchor point from the structuring element coordinates
B_frame = B - a # express the se element coordinates in with respect to anchor point
Xcp = X.copy()
b = img.copy()
for x in X: # X contains the foreground coordinates in the image
Bx = B_frame + x # translate relative coordinates with respect to foreground coordinates considering it as the anchor point
check = True # this is erosion so if any of the se coordinates is not in foreground coordinates we consider it a miss
for bx in Bx: # Bx contains all the translated coordinates of se
if bx not in Xcp:
check = False
if check:
b[x[0], x[1]] = 1 # if there is a hit
else:
b[x[0], x[1]] = 0 # if there is no hit
return b
def erosion(img: np.ndarray, struct_el_mat: np.ndarray, foregroundValue = 0):
B = np.argwhere(struct_el_mat == 0)
X = np.argwhere(img == foregroundValue)
nimg = erosionC(img, B, X)
return np.where(nimg == 1, 255, 0)
The calling code for both is:
from scipy import ndimage as nd
err = nd.binary_erosion(imarr, se_mat3)
imerrCustom = erosion(imarr, se_mat31, foregroundValue=1)
err produces
imerrCustom produces

Answer 1

In the end, I am still not sure about it, but after having read several papers more, I assume that my interpretation of X as foreground coordinates was an error. It should have probably been the entire image that is being iterated.
As I have stated I am not sure if this interpretation is correct as well. But I made a new implementation which iterates over the image, and it gives a more plausible result. I am sharing it in here, hoping that it might help someone:
%%cython -a
import numpy as np
cimport numpy as cnp
cdef dilation_c(cnp.ndarray[cnp.uint8_t, ndim=2] X,
cnp.ndarray[cnp.uint8_t, ndim=2] SE):
"""
X: boolean image
SE: structuring element matrix
origin: coordinate of the origin of the structuring element
This operation checks whether (B *hits* X) = $B \cap X \not = \emptyset$
as per defined in
Serra (Jean), « Introduction to mathematical morphology »,
Computer Vision, Graphics, and Image Processing,
vol. 35, nᵒ 3 (septembre 1986).
URL : https://linkinghub.elsevier.com/retrieve/pii/0734189X86900022..
doi: 10.1016/0734-189X(86)90002-2
Consulted le 6 août 2020, p. 283‑305.
The algorithm adapts DILDIRECT of
Najman (Laurent) et Talbot (Hugues),
Mathematical morphology: from theory to applications,
2013. ISBN : 9781118600788, p. 329
to the formula given in
Jähne (Bernd),
Digital image processing,
6th rev. and ext. ed, Berlin ; New York,
2005. TA1637 .J34 2005.
ISBN : 978-3-540-24035-8.
"""
cdef cnp.ndarray[cnp.uint8_t, ndim=2] O
cdef list elst
cdef int r, c, X_rows, X_cols, SE_rows, SE_cols, se_r, se_c
cdef cnp.ndarray[cnp.int_t, ndim=1] bp
cdef list conds
cdef bint check, b, p, cond
O = np.zeros_like(X)
X_rows, X_cols = X.shape[:2]
SE_rows, SE_cols = SE.shape[:2]
# a boolean convolution
for r in range(0, X_rows-SE_rows):
for c in range(0, X_cols - SE_cols):
conds = []
for se_r in range(SE_rows):
for se_c in range(SE_cols):
b = <bint>SE[se_r, se_c]
p = <bint>X[se_r+r, se_c+c]
conds.append(b and p)
O[r,c] = <cnp.uint8_t>any(conds)
return O
def dilation_erosion(
img: np.ndarray,
struct_el_mat: np.ndarray,
foregroundValue: int = 1,
isErosion: bool = False):
"""
img: image matrix
struct_el: NxN mesh grid of the structuring element whose center is SE's origin
structuring element is encoded as 1
foregroundValue: value to be considered as foreground in the image
"""
B = struct_el_mat.astype(np.uint8)
if isErosion:
X = np.where(img == foregroundValue, 0, 1).astype(np.uint8)
else:
X = np.where(img == foregroundValue, 1, 0).astype(np.uint8)
nimg = dilation_c(X, B)
foreground, background = (255, 0) if foregroundValue == 1 else (0, 1)
if isErosion:
return np.where(nimg == 1, background, foreground).astype(np.uint8)
else:
return np.where(nimg == 1, foreground, background).astype(np.uint8)
# return nimg