I am wondering what information does an Hessian Matrix of an image provides? Does it provide the information of the stable points? What is Hessian matrix used for?
Hessian matrix describes the 2nd order local image intensity variations around the selected voxel. For the obtained Hessian matrix, eigenvector decomposition extracts an orthonormal coordinate system that is aligned with the second order structure of the image. Having the eigenvalues and knowing the
(assumed) model of the structure to be detected and the resulting theoretical behavior of the eigenvalues, the decision can be made if the analyzed voxel belongs to the structure being searched.
The figure below illustrates the correspondence between eigenvalues of the hessian operation on the image and the local features (corner, edge, or flat region).
The Hessian operator is also widely used in 3D images, and it can reflect more local features:
It is widely used in vessel detection in medical images. For more details, please see M.Rudzki et al's Vessel Detection Method Based on Eigenvalues of the Hessian Matrix and its Applicability to Airway Tree Segmentation
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Currently, I am working at a short project about stereo-vision.
I'm trying to create depth maps of a scenery. For this, I use my phone from to view points and use the following code/workflow provided by Matlab : https://nl.mathworks.com/help/vision/ug/uncalibrated-stereo-image-rectification.html
Following this code I am able to create nice disparity maps, but I want to now the depths (as in meters). For this, I need the baseline, focal length and disparity, as shown here: https://www.researchgate.net/figure/Relationship-between-the-baseline-b-disparity-d-focal-length-f-and-depth-z_fig1_2313285
The focal length and base-line are known, but not the baseline. I determined the estimate of the Fundamental Matrix. Is there a way to get from the Fundamental Matrix to the baseline, or by making some assumptions to get to the Essential Matrix, and from there to the baseline.
I would be thankful for any hint in the right direction!
"The focal length and base-line are known, but not the baseline."
I guess you mean the disparity map is known.
Without a known or estimated calibration matrix, you cannot determine the essential matrix.
(Compare Multi View Geometry of Hartley and Zisserman for details)
With respect to your available data, you cannot compute a metric reconstruction. From the fundamental matrix, you can only extract camera matrices in a canonical form that allow for a projective reconstruction and will not satisfy the true baseline of the setup. A projective reconstruction is a reconstruction that differs from the metric result by an unknown transformation.
Non-trivial techniques could allow to upgrade these reconstructions to an Euclidean reconstruction result. However, the success of these self-calibration techniques strongly depends of the quality of the data. Thus, using images of a calibrated camera is actually the best way to go.
my aim is to classify the data into two sections- upper and lower- finding the mid line of the peaks.
I would like to apply machine learning methods- i.e. Discriminant analysis.
Could you let me know how to do that in MATLAB?
It seems that what you are looking for is GMM (gaussian mixture model). With K=2 (number of mixtures) and dimension equal 1 this will be simple, fast method, which will give you a direct solution. Given components it is easy to analytically find a local minima (which is just a weighted average of means, with weights proportional to the std's).
This is a follow-up question on the one below:
Second moments question
MATLAB's regionprops function estimates an ellipse from a given set of 2d-points. This is done by using the image moments, they claim to use normalized second central moments, the formulas also follow what is suggested by the wikipedia link on image moments.
Effectively the covariance matrix of the region is calculated (in a slightly more efficient way) and then the square root of the eigenvalues of this matrix are calculated and put out as the major and minor axes - with one change: They are multiplied by a factor of 4.
Why?
Essentially, covariance estimation assumes a multivariate normal distribution. However, an arbitrary image region is most likely not normally distributed, I would rather expect a factor based on the assumption that data is uniformly distributed. So what is the justification for choosing 4?
Meanwhile I found the answer: Factor 4 yields correct results for regions with an elliptical shape. For e.g. rectangular or non-solid regions, the estimated axis lengths are incorrect, and the error varies nonlinearly with changes in the region.
I am using PCA for face recognition. I have obtained the eigenvectors / eigenfaces for each image, which is a colomn matrix. I want to know if selecting the first three eigenvectors , since their corresponding eigenvalues amount to 70% of total variance, will be sufficient for face recognition?
Firstly, lets be clear about a few things. The eigenvectors are computed from the covariance matrix formed from the entire dataset i.e., you reshape each grayscale image of a face into a single column and treat it as a point in R^d space, compute the covariance matrix from them and compute the eigenvectors of the covariance matrix. These eigenvectors become a new basis for your space of face images. You do not have eigenvectors for each image. Instead, you represent each face image in terms of the eigenvectors by projecting onto (a possibly subset) of them.
Limitations of eigenfaces
As to whether the representation of your face images under this new basis good enough for face recognition depends on many factors. But in general, the eigenfaces method does not perform well for real world unconstrained faces. It only works for faces which are pixel-wise aligned, facing frontal, and has fairly uniform illumination conditions across the images.
More is not necessarily better
While it is commonly believed (when using PCA) that retaining more variance is better than less, things are much more complicated than that because of two factors: 1) Noise in real world data and 2) dimensionality of data. Sometimes projecting to a lower dimension and losing variance can actually produce better results.
Conclusion
Hence, my answer is it is difficult to say whether retaining a certain amount of variance is enough beforehand. The number of dimensions (and hence the number of eigenvectors to keep and the associated variance retained) should be determined by cross-validation. But ultimately, as I have mentioned above, eigenfaces is not a good method for face recognition unless you have a "nice" dataset. You might be slightly better off using "Fisherfaces" i.e., LDA on the face images or combine these methods with Local Binary Pattern (LBP) as features (instead of raw face pixels). But seriously, face recognition is a difficult problem and in general the state-of-the-art has not reached a stage where it can be deployed in real world systems.
It's not impossible, but a little rare to me that only 3 eigenvalues can achieve 70% variance. How many training samples do you have (what is the total dimension)? Make sure you are reshape each image from the database into a vector, normalize the vector data then align them into a matrix. The eigenvalues/eigenvectors are obtained from the covariance of the matrix.
In theory, 70% variance should be enough to form a human-recognizable face with the corresponding eigenvectors. However, the optimal number of eigenvalues is better to get from cross-validation: you can try to increase 1 eigenvector each time, observe the face formation and the recognition accuracy. You can even plot the cross validation accuracy curve, there may be a sharp corner on the curve, then the corresponding eigenvector number is hopefully applied in your test.
I am trying to develop a system for image classification. I am using following the article:
INDEPENDENT COMPONENT ANALYSIS (ICA) FOR TEXTURE CLASSIFICATION by Dr. Dia Abu Al Nadi and Ayman M. Mansour
In a paragraph it says:
Given the above texture images, the Independent Components are learned by the method outlined above.
The (8 x 8) ICA basis function for the above textures are shown in Figure 2. respectively. The dimension is reduced by PCA, resulting in a total of 40 functions. Note that independent components from different windows size are different.
The "method outlined above" is FastICA, the textures are taken from Brodatz album , each texture image has 640x640 pixels. My question is:
What the authors means with "The dimension is reduced by PCA, resulting in a total of 40 functions.", and how can I get that functions using matlab?
PCA (Principal Component Analysis) is a method for finding an orthogonal basis (think of a coordinate system) for a high-dimensional (data) space. The "axes" of the PCA basis are sorted by variance, i.e. along the first PCA "axis" your data has the largest variance, along the second "axis" the second largest variance, etc.
This is exploited for dimension reduction: Say you have 1000 dimensional data. Then you do a PCA, transform your data into the PCA basis and throw away all but the first 20 dimensions (just an example). If your data follows a certain statistical distribution, then chances are that the 20 PCA dimensions describe your data almost as well as the 64 original dimensions did. There are methods for finding the number of dimensions to use, but that is beyond scope here.
Computationally, PCA amounts to finding the Eigen-decomposition of your data's covariance matrix, in Matlab: [V,D] = eig(cov(MyData)).
Note that if you want to work with these concepts you should do some serious reading. A classic article on what you can do with PCA on image data is Turk and Pentland's Eigenfaces. It also gives some background in an understandable way.
PCA reduce the dimension of data,ICA extracts the components of the data of which dimension must <=
data dimension