I'm trying to get a depth map with an uncalibrated method. I can obtain the fundamental matrix by finding correspondent points with SIFT and then using cv2.findFundamentalMat. I then use cv2.stereoRectifyUncalibrated to get the homography matrices for each image. Finally I use cv2.warpPerspective to rectify and compute the disparity, but this doesn't create a good depth map. The values are very high so I'm wondering if I have to use warpPerspective or if I have to calculate a rotation matrix from the homography matrices I got with stereoRectifyUncalibrated.

I'm not sure of the projective matrix with the case of homography matrix obtained with the stereoRectifyUncalibrated to rectify.

A part of the code:

#Obtainment of the correspondent point with SIFT
sift = cv2.SIFT()

###find the keypoints and descriptors with SIFT
kp1, des1 = sift.detectAndCompute(dst1,None)
kp2, des2 = sift.detectAndCompute(dst2,None)

###FLANN parameters
FLANN_INDEX_KDTREE = 0
index_params = dict(algorithm = FLANN_INDEX_KDTREE, trees = 5)
search_params = dict(checks=50)

flann = cv2.FlannBasedMatcher(index_params,search_params)
matches = flann.knnMatch(des1,des2,k=2)

good = []
pts1 = []
pts2 = []

###ratio test as per Lowe's paper
for i,(m,n) in enumerate(matches):
    if m.distance < 0.8*n.distance:
        good.append(m)
        pts2.append(kp2[m.trainIdx].pt)
        pts1.append(kp1[m.queryIdx].pt)
    
    
pts1 = np.array(pts1)
pts2 = np.array(pts2)

#Computation of the fundamental matrix
F,mask= cv2.findFundamentalMat(pts1,pts2,cv2.FM_LMEDS)


# Obtainment of the rectification matrix and use of the warpPerspective to transform them...
pts1 = pts1[:,:][mask.ravel()==1]
pts2 = pts2[:,:][mask.ravel()==1]

pts1 = np.int32(pts1)
pts2 = np.int32(pts2)

p1fNew = pts1.reshape((pts1.shape[0] * 2, 1))
p2fNew = pts2.reshape((pts2.shape[0] * 2, 1))
    
retBool ,rectmat1, rectmat2 = cv2.stereoRectifyUncalibrated(p1fNew,p2fNew,F,(2048,2048))

dst11 = cv2.warpPerspective(dst1,rectmat1,(2048,2048))
dst22 = cv2.warpPerspective(dst2,rectmat2,(2048,2048))

#calculation of the disparity
stereo = cv2.StereoBM(cv2.STEREO_BM_BASIC_PRESET,ndisparities=16*10, SADWindowSize=9)
disp = stereo.compute(dst22.astype(uint8), dst11.astype(uint8)).astype(np.float32)
plt.imshow(disp);plt.colorbar();plt.clim(0,400)#;plt.show()
plt.savefig("0gauche.png")

#plot depth by using disparity focal length `C1[0,0]` from stereo calibration and `T[0]` the distance between cameras

plt.imshow(C1[0,0]*T[0]/(disp),cmap='hot');plt.clim(-0,500);plt.colorbar();plt.show()

Here are the rectified pictures with the uncalibrated method (and warpPerspective):

Here are the rectified pictures with the calibrated method:

I don't know how the difference is so important between the two kind of pictures. And for the calibrated method, it doesn't seem aligned.

The disparity map using the uncalibrated method:

The depths are calculated with : C1[0,0]*T[0]/(disp) with T from the stereoCalibrate. The values are very high.

------------ EDIT LATER ------------

I tried to "mount" the reconstruction matrix ([Devernay97], [Garcia01]) with the homography matrix obtained with "stereoRectifyUncalibrated", but the result is still not good. Am I doing this correctly?

Y=np.arange(0,2048)
X=np.arange(0,2048)
(XX_field,YY_field)=np.meshgrid(X,Y)

#I mount the X, Y and disparity in a same 3D array 
stock = np.concatenate((np.expand_dims(XX_field,2),np.expand_dims(YY_field,2)),axis=2)
XY_disp = np.concatenate((stock,np.expand_dims(disp,2)),axis=2)

XY_disp_reshape = XY_disp.reshape(XY_disp.shape[0]*XY_disp.shape[1],3)

Ts = np.hstack((np.zeros((3,3)),T_0)) #i use only the translations obtained with the rectified calibration...Is it correct?


# I establish the projective matrix with the homography matrix
P11 = np.dot(rectmat1,C1)
P1 = np.vstack((np.hstack((P11,np.zeros((3,1)))),np.zeros((1,4))))
P1[3,3] = 1

# P1 = np.dot(C1,np.hstack((np.identity(3),np.zeros((3,1)))))

P22 = np.dot(np.dot(rectmat2,C2),Ts)
P2 = np.vstack((P22,np.zeros((1,4))))
P2[3,3] = 1

lambda_t = cv2.norm(P1[0,:].T)/cv2.norm(P2[0,:].T)


#I define the reconstruction matrix
Q = np.zeros((4,4))

Q[0,:] = P1[0,:].T
Q[1,:] = P1[1,:].T
Q[2,:] = lambda_t*P2[1,:].T - P1[1,:].T
Q[3,:] = P1[2,:].T

#I do the calculation to get my 3D coordinates
test = []
for i in range(0,XY_disp_reshape.shape[0]):
    a = np.dot(inv(Q),np.expand_dims(np.concatenate((XY_disp_reshape[i,:],np.ones((1))),axis=0),axis=1))
    test.append(a)

test = np.asarray(test)

XYZ = test[:,:,0].reshape(XY_disp.shape[0],XY_disp.shape[1],4)

TLDR; Use StereoSGBM (Semi Global Block Matching) and use some post filtering if you want it even smoother

OP didn't provide original images, so I'm using Tsukuba from the Middlebury data set.

Result with regular StereoBM

Result with StereoSGBM (tuned)

Best result I could find in literature

See the publication here for details.

Example of post filtering (see link below)

Theory/Other considerations from OP's question

The large black areas of your calibrated rectified images would lead me to believe that for those, calibration was not done very well. There's a variety of reasons that could be at play, maybe the physical setup, maybe lighting when you did calibration, etc., but there are plenty of camera calibration tutorials out there for that and my understanding is that you are asking for a way to get a better depth map from an uncalibrated setup (this isn't 100% clear, but the title seems to support this and I think that's what people will come here to try to find).

Your basic approach is correct, but the results can definitely be improved. This form of depth mapping is not among those that produce the highest quality maps (especially being uncalibrated). The biggest improvement will likely come from using a different stereo matching algorithm. The lighting may also be having a significant effect. The right image (at least to my naked eye) appears to be less well lit which could interfere with the reconstruction. You could first try brightening it to the same level as the other, or gather new images if that is possible. From here out, I'll assume you have no access to the original cameras, so I'll consider gathering new images, altering the setup, or performing calibration to be out of scope. (If you do have access to the setup and cameras, then I would suggest checking calibration and using a calibrated method as this will work better).

You used StereoBM for calculating your disparity (depth map) which does work, but StereoSGBM is much better suited for this application (it handles smoother edges better). You can see the difference below.

This article explains the differences in more depth:

Block matching focuses on high texture images (think a picture of a tree) and semi-global block matching will focus on sub pixel level matching and pictures with more smooth textures (think a picture of a hallway).

Without any explicit intrinsic camera parameters, specifics about the camera setup (like focal distance, distance between the cameras, distance to the subject, etc.), a known dimension in the image, or motion (to use structure from motion), you can only obtain 3D reconstruction up to a projective transform; you won't have a sense of scale or necessarily rotation either, but you can still generate a relative depth map. You will likely suffer from some barrel and other distortions which could be removed with proper camera calibration, but you can get reasonable results without it as long as the cameras aren’t terrible (lens system isn't too distorted) and are set up pretty close to canonical configuration (which basically means they are oriented such that their optical axes are as close to parallel as possible, and their fields of view overlap sufficiently). This doesn't however appear to be the OPs issue as he did manage to get alright rectified images with the uncalibrated method.

Basic Procedure

1. Find at least 5 well-matched points in both images you can use to calculate the Fundamental Matrix (you can use any detector and matcher you like, I kept FLANN but used ORB to do detection as SIFT isn't in the main version of OpenCV for 4.2.0)
2. Calculate the Fundamental Matrix, F, with findFundamentalMat
3. Undistort your images with stereoRectifyUncalibrated and warpPerspective
4. Calculate Disparity (Depth Map) with StereoSGBM

The results are much better:

Matches with ORB and FLANN

Undistorted images (left, then right)

Disparity

StereoBM

This result looks similar to the OPs problems (speckling, gaps, wrong depths in some areas).

StereoSGBM (tuned)

This result looks much better and uses roughly the same method as the OP, minus the final disparity calculation, making me think the OP would see similar improvements on his images, had they been provided.

Post filtering

There's a good article about this in the OpenCV docs. I'd recommend looking at it if you need really smooth maps.

The example photos above are frame 1 from the scene ambush_2 in the MPI Sintel Dataset.

Full code (Tested on OpenCV 4.2.0):

import cv2
import numpy as np
import matplotlib.pyplot as plt

imgL = cv2.imread("tsukuba_l.png", cv2.IMREAD_GRAYSCALE)  # left image
imgR = cv2.imread("tsukuba_r.png", cv2.IMREAD_GRAYSCALE)  # right image


def get_keypoints_and_descriptors(imgL, imgR):
    """Use ORB detector and FLANN matcher to get keypoints, descritpors,
    and corresponding matches that will be good for computing
    homography.
    """
    orb = cv2.ORB_create()
    kp1, des1 = orb.detectAndCompute(imgL, None)
    kp2, des2 = orb.detectAndCompute(imgR, None)

    ############## Using FLANN matcher ##############
    # Each keypoint of the first image is matched with a number of
    # keypoints from the second image. k=2 means keep the 2 best matches
    # for each keypoint (best matches = the ones with the smallest
    # distance measurement).
    FLANN_INDEX_LSH = 6
    index_params = dict(
        algorithm=FLANN_INDEX_LSH,
        table_number=6,  # 12
        key_size=12,  # 20
        multi_probe_level=1,
    )  # 2
    search_params = dict(checks=50)  # or pass empty dictionary
    flann = cv2.FlannBasedMatcher(index_params, search_params)
    flann_match_pairs = flann.knnMatch(des1, des2, k=2)
    return kp1, des1, kp2, des2, flann_match_pairs


def lowes_ratio_test(matches, ratio_threshold=0.6):
    """Filter matches using the Lowe's ratio test.

    The ratio test checks if matches are ambiguous and should be
    removed by checking that the two distances are sufficiently
    different. If they are not, then the match at that keypoint is
    ignored.

    https://stackoverflow.com/questions/51197091/how-does-the-lowes-ratio-test-work
    """
    filtered_matches = []
    for m, n in matches:
        if m.distance < ratio_threshold * n.distance:
            filtered_matches.append(m)
    return filtered_matches


def draw_matches(imgL, imgR, kp1, des1, kp2, des2, flann_match_pairs):
    """Draw the first 8 mathces between the left and right images."""
    # https://docs.opencv.org/4.2.0/d4/d5d/group__features2d__draw.html
    # https://docs.opencv.org/2.4/modules/features2d/doc/common_interfaces_of_descriptor_matchers.html
    img = cv2.drawMatches(
        imgL,
        kp1,
        imgR,
        kp2,
        flann_match_pairs[:8],
        None,
        flags=cv2.DrawMatchesFlags_NOT_DRAW_SINGLE_POINTS,
    )
    cv2.imshow("Matches", img)
    cv2.imwrite("ORB_FLANN_Matches.png", img)
    cv2.waitKey(0)


def compute_fundamental_matrix(matches, kp1, kp2, method=cv2.FM_RANSAC):
    """Use the set of good mathces to estimate the Fundamental Matrix.

    See  https://en.wikipedia.org/wiki/Eight-point_algorithm#The_normalized_eight-point_algorithm
    for more info.
    """
    pts1, pts2 = [], []
    fundamental_matrix, inliers = None, None
    for m in matches[:8]:
        pts1.append(kp1[m.queryIdx].pt)
        pts2.append(kp2[m.trainIdx].pt)
    if pts1 and pts2:
        # You can play with the Threshold and confidence values here
        # until you get something that gives you reasonable results. I
        # used the defaults
        fundamental_matrix, inliers = cv2.findFundamentalMat(
            np.float32(pts1),
            np.float32(pts2),
            method=method,
            # ransacReprojThreshold=3,
            # confidence=0.99,
        )
    return fundamental_matrix, inliers, pts1, pts2


############## Find good keypoints to use ##############
kp1, des1, kp2, des2, flann_match_pairs = get_keypoints_and_descriptors(imgL, imgR)
good_matches = lowes_ratio_test(flann_match_pairs, 0.2)
draw_matches(imgL, imgR, kp1, des1, kp2, des2, good_matches)


############## Compute Fundamental Matrix ##############
F, I, points1, points2 = compute_fundamental_matrix(good_matches, kp1, kp2)


############## Stereo rectify uncalibrated ##############
h1, w1 = imgL.shape
h2, w2 = imgR.shape
thresh = 0
_, H1, H2 = cv2.stereoRectifyUncalibrated(
    np.float32(points1), np.float32(points2), F, imgSize=(w1, h1), threshold=thresh,
)

############## Undistort (Rectify) ##############
imgL_undistorted = cv2.warpPerspective(imgL, H1, (w1, h1))
imgR_undistorted = cv2.warpPerspective(imgR, H2, (w2, h2))
cv2.imwrite("undistorted_L.png", imgL_undistorted)
cv2.imwrite("undistorted_R.png", imgR_undistorted)

############## Calculate Disparity (Depth Map) ##############

# Using StereoBM
stereo = cv2.StereoBM_create(numDisparities=16, blockSize=15)
disparity_BM = stereo.compute(imgL_undistorted, imgR_undistorted)
plt.imshow(disparity_BM, "gray")
plt.colorbar()
plt.show()

# Using StereoSGBM
# Set disparity parameters. Note: disparity range is tuned according to
#  specific parameters obtained through trial and error.
win_size = 2
min_disp = -4
max_disp = 9
num_disp = max_disp - min_disp  # Needs to be divisible by 16
stereo = cv2.StereoSGBM_create(
    minDisparity=min_disp,
    numDisparities=num_disp,
    blockSize=5,
    uniquenessRatio=5,
    speckleWindowSize=5,
    speckleRange=5,
    disp12MaxDiff=2,
    P1=8 * 3 * win_size ** 2,
    P2=32 * 3 * win_size ** 2,
)
disparity_SGBM = stereo.compute(imgL_undistorted, imgR_undistorted)
plt.imshow(disparity_SGBM, "gray")
plt.colorbar()
plt.show()