I am going through a paper in computer vision, and I came through this line :
the L values, or the luminance values, for these pixels are then linearly and horizontally interpolated between the pixels on the (one pixel wide) brightest column in region B, and the pixels in regions A and C.
What does linear and horizontal interpolation mean?
So I tried looking for linear interpolation, so does it mean that we average out the values of pixels which are linear to each other? As I can't see any proper definition.
Paper : http://140.118.9.222/publications/journal/haircut.pdf
Every programmer should know linear interpolation!!! Especially if you're entering the domain of image-processing.
Please read this and never ever forget about it.
https://en.wikipedia.org/wiki/Linear_interpolation
The paper describes pretty well what is going on. They synthesize skin texture by sampling the face and then interpolating between those samples. They sample 3 regions A, B and C.
They pick the brightest column of B, the left-most column of A and the right-most column of C.
Then for every row they linearly interpolate between the columns' pixels.
Related
I'd like to know if this is a known algorithm with a name.
I've never done any image processing, but I'm picturing an image as a 2-d matrix of 3-d vectors (ignore transparency).
The only input parameter is distance. Every pixel is tested against its neighbors. If they are closer than the parameter, they join a group and their values are averaged. As groups grow by gaining new pixels all pixels get the average value of the group.
For your typical selfie the result might resemble quantizing or posterizing, but unlike quantizing or posterizing, there is no fixed count of output colors. If absolutely no pixels are close enough to their neighbors, the result is a 1:1 mapping of every pixel to its own group.
Is there a name for this?
I have an Image I
I am trying to do Automatic Object Extraction using Quantum Mechanics
Each pixel in an image is considered as a potential field, V(x,y) and hence each wave (eigen) function represents a meaningful region.
2D Time-independent Sschrodinger's equation
Multiplying both sides by
We get,
Rewriting the Laplacian using Finite Difference approach
where Ni is the set of neighbours with index i, and |Ni| is the cardinality of, i.e. the number of elements in Ni
Combining the above two equations, we get:
where M is the number of elements in
Now,the left hand side of the equation is a measure of how similar the labels in a neighbourhood are, i.e. a measure of spatial coherence.
Now, for applying this to images, the potential V is given as the pixel intensities.
Here, V is the pixel intensities
The right hand side is a measure of how close the pixel values in a segment are to a constant value E.
Now, the wave functions can be numerically calculated by solving the eigenvectors of Hamiltonian operator in matrix form which is
for i = j
for
and elsewhere 0
Now, in this paper it is said that first we have to find the maximum and minimum eigenvalues and then calculate the eigenvectors with eigenvalues closest to a number of values regularly selected between the minimum and maximum eigenvalues. the number is 300.
I have calculated the 300 eigenvectors.
And then the absolute square of the eigenvectors are thresholded to obtain the segments.
Fine upto this part.
Now, how do I reconstruct the eigenvectors into a 2D image so as to get the potential segments in the image?
I am not able to under stand the formula ,
What is W (window) and intensity in the formula mean,
I found this formula in opencv doc
http://docs.opencv.org/trunk/doc/py_tutorials/py_feature2d/py_features_harris/py_features_harris.html
For a grayscale image, intensity levels (0-255) tells you how bright is the pixel..hope that you already know about it.
So, now the explanation of your formula is below:
Aim: We want to find those points which have maximum variation in terms of intensity level in all direction i.e. the points which are very unique in a given image.
I(x,y): This is the intensity value of the current pixel which you are processing at the moment.
I(x+u,y+v): This is the intensity of another pixel which lies at a distance of (u,v) from the current pixel (mentioned above) which is located at (x,y) with intensity I(x,y).
I(x+u,y+v) - I(x,y): This equation gives you the difference between the intensity levels of two pixels.
W(u,v): You don't compare the current pixel with any other pixel located at any random position. You prefer to compare the current pixel with its neighbors so you chose some value for "u" and "v" as you do in case of applying Gaussian mask/mean filter etc. So, basically w(u,v) represents the window in which you would like to compare the intensity of current pixel with its neighbors.
This link explains all your doubts.
For visualizing the algorithm, consider the window function as a BoxFilter, Ix as a Sobel derivative along x-axis and Iy as a Sobel derivative along y-axis.
http://docs.opencv.org/doc/tutorials/imgproc/imgtrans/sobel_derivatives/sobel_derivatives.html will be useful to understand the final equations in the above pdf.
Assuming that I have a grayscale (8-bit) image and assume that I have an integral image created from that same image.
Image resolution is 720x576. According to SURF algorithm, each octave is composed of 4 box filters, which are defined by the number of pixels on their side. The
first octave uses filters with 9x9, 15x15, 21x21 and 27x27 pixels. The
second octave uses filters with 15x15, 27x27, 39x39 and 51x51 pixels.The third octave uses filters with 27x27, 51x51, 75x75 and 99x99 pixels. If the image is sufficiently large and I guess 720x576 is big enough (right??!!), a fourth octave is added, 51x51, 99x99, 147x147 and 195x195. These
octaves partially overlap one another to improve the quality of the interpolated results.
// so, we have:
//
// 9x9 15x15 21x21 27x27
// 15x15 27x27 39x39 51x51
// 27x27 51x51 75x75 99x99
// 51x51 99x99 147x147 195x195
The questions are:What are the values in each of these filters? Should I hardcode these values, or should I calculate them? How exactly (numerically) to apply filters to the integral image?
Also, for calculating the Hessian determinant I found two approximations:
det(HessianApprox) = DxxDyy − (0.9Dxy)^2 anddet(HessianApprox) = DxxDyy − (0.81Dxy)^2Which one is correct?
(Dxx, Dyy, and Dxy are Gaussian second order derivatives).
I had to go back to the original paper to find the precise answers to your questions.
Some background first
SURF leverages a common Image Analysis approach for regions-of-interest detection that is called blob detection.
The typical approach for blob detection is a difference of Gaussians.
There are several reasons for this, the first one being to mimic what happens in the visual cortex of the human brains.
The drawback to difference of Gaussians (DoG) is the computation time that is too expensive to be applied to large image areas.
In order to bypass this issue, SURF takes a simple approach. A DoG is simply the computation of two Gaussian averages (or equivalently, apply a Gaussian blur) followed by taking their difference.
A quick-and-dirty approximation (not so dirty for small regions) is to approximate the Gaussian blur by a box blur.
A box blur is the average value of all the images values in a given rectangle. It can be computed efficiently via integral images.
Using integral images
Inside an integral image, each pixel value is the sum of all the pixels that were above it and on its left in the original image.
The top-left pixel value in the integral image is thus 0, and the bottom-rightmost pixel of the integral image has thus the sum of all the original pixels for value.
Then, you just need to remark that the box blur is equal to the sum of all the pixels inside a given rectangle (not originating in the top-lefmost pixel of the image) and apply the following simple geometric reasoning.
If you have a rectangle with corners ABCD (top left, top right, bottom left, bottom right), then the value of the box filter is given by:
boxFilter(ABCD) = A + D - B - C,
where A, B, C, D is a shortcut for IntegralImagePixelAt(A) (B, C, D respectively).
Integral images in SURF
SURF is not using box blurs of sizes 9x9, etc. directly.
What it uses instead is several orders of Gaussian derivatives, or Haar-like features.
Let's take an example. Suppose you are to compute the 9x9 filters output. This corresponds to a given sigma, hence a fixed scale/octave.
The sigma being fixed, you center your 9x9 window on the pixel of interest. Then, you compute the output of the 2nd order Gaussian derivative in each direction (horizontal, vertical, diagonal). The Fig. 1 in the paper gives you an illustration of the vertical and diagonal filters.
The Hessian determinant
There is a factor to take into account the scale differences. Let's believe the paper that the determinant is equal to:
Det = DxxDyy - (0.9 * Dxy)^2.
Finally, the determinant is given by: Det = DxxDyy - 0.81*Dxy^2.
Look at page 17 of this document
http://www.sci.utah.edu/~fletcher/CS7960/slides/Scott.pdf
If you made a code for normal Gaussian 2D convolution, just use the box filter as a Gaussian kernel and the input image will be the same original image not integral image. The results from this method will be same with the one you asked.
I'm validating an image segmentation algorithm applied to 2D images. The algorithm generates a contour segment, i.e. a set of connected pixels that form a freecurve in 2D space. The idea is to compare this set of pixels with a ground-truth, in my case another contour segment manually traced by an expert. An image showing what would be a segmentation result and the corresponding manual (ground-truth) segmentation is shown below:
I'm trying to think of an adequate comparison metric to validate the segmentation results. Ideally the best metric would be the point-to-point euclidean distance between corresponding pairs of pixels on each segment, however (as seen in previous figure) the segments don't have the same length (i.e. differ by the total number of pixels) so pixel-to-pixel comparisons have to be discarded.
Can you suggest me an adequate metric for validating my algorithm? Thanks for any suggestion!
For each pixel in the ground truth, take the distance to the nearest pixel in the segmentation result. Then take the sum of that for all ground truth pixels as the total error.
That's basically recall weighted by distance. If you start with the pixels in the result, it would resemble precision instead.
If the curves are closed, you can compute the area between the curves. If you can tell which pixels belong to a segment, that is as easy as computing XOR set of the 2 pixel sets.
Here is an example using that I've created using Matlab:
You could divide each line into n segments of equal length, then compute the euclidean distance between each segment and its pair on the other line.