Basically, you use the findContours function, in combination with many other functions OpenCV provides for especially this purpose.
Useful functions used (surprise, surprise, they all appear on the Structural Analysis and Shape Descriptors page in the OpenCV Docs):
example code (I have all the properties from Matlab's regionprops except WeightedCentroid and EulerNumber - you could work out EulerNumber by using cv2.RETR_TREE in findContours and looking at the resulting hierarchy, and I'm sure WeightedCentroid wouldn't be that hard either.
# grab contours
cs,_ = cv2.findContours( BW.astype('uint8'), mode=cv2.RETR_LIST,
method=cv2.CHAIN_APPROX_SIMPLE )
# set up the 'FilledImage' bit of regionprops.
filledI = np.zeros(BW.shape[0:2]).astype('uint8')
# set up the 'ConvexImage' bit of regionprops.
convexI = np.zeros(BW.shape[0:2]).astype('uint8')
# for each contour c in cs:
# will demonstrate with cs[0] but you could use a loop.
i=0
c = cs[i]
# calculate some things useful later:
m = cv2.moments(c)
# ** regionprops **
Area = m['m00']
Perimeter = cv2.arcLength(c,True)
# bounding box: x,y,width,height
BoundingBox = cv2.boundingRect(c)
# centroid = m10/m00, m01/m00 (x,y)
Centroid = ( m['m10']/m['m00'],m['m01']/m['m00'] )
# EquivDiameter: diameter of circle with same area as region
EquivDiameter = np.sqrt(4*Area/np.pi)
# Extent: ratio of area of region to area of bounding box
Extent = Area/(BoundingBox[2]*BoundingBox[3])
# FilledImage: draw the region on in white
cv2.drawContours( filledI, cs, i, color=255, thickness=-1 )
# calculate indices of that region..
regionMask = (filledI==255)
# FilledArea: number of pixels filled in FilledImage
FilledArea = np.sum(regionMask)
# PixelIdxList : indices of region.
# (np.array of xvals, np.array of yvals)
PixelIdxList = regionMask.nonzero()
# CONVEX HULL stuff
# convex hull vertices
ConvexHull = cv2.convexHull(c)
ConvexArea = cv2.contourArea(ConvexHull)
# Solidity := Area/ConvexArea
Solidity = Area/ConvexArea
# convexImage -- draw on convexI
cv2.drawContours( convexI, [ConvexHull], -1,
color=255, thickness=-1 )
# ELLIPSE - determine best-fitting ellipse.
centre,axes,angle = cv2.fitEllipse(c)
MAJ = np.argmax(axes) # this is MAJor axis, 1 or 0
MIN = 1-MAJ # 0 or 1, minor axis
# Note: axes length is 2*radius in that dimension
MajorAxisLength = axes[MAJ]
MinorAxisLength = axes[MIN]
Eccentricity = np.sqrt(1-(axes[MIN]/axes[MAJ])**2)
Orientation = angle
EllipseCentre = centre # x,y
# ** if an image is supplied with the BW:
# Max/Min Intensity (only meaningful for a one-channel img..)
MaxIntensity = np.max(img[regionMask])
MinIntensity = np.min(img[regionMask])
# Mean Intensity
MeanIntensity = np.mean(img[regionMask],axis=0)
# pixel values
PixelValues = img[regionMask]