I have a Radial Distribution Graph (having x-axis ie.,[r], y-axis ie., [g(r)] values).
One of the values in the graph has g(r) high value which is around 45 and the rest have around less than 5. Hence it makes the graphs unclear to differentiate.
Since the values are high in this case, I need to normalize it.
How can i normalize the values (the y-axis [g(r)])?
I have attached the graph for visualization.
Any suggestions?
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I have numerous return time series spanning over a couple of years. I want to see how stable these series are across time. So far I have winsorized and z-scored my data and created histograms and AVG vs. StdDev graphs. Using the histograms I can see how the distribution looks and check for positive or negative skew, with the Avg vs. StdDev chart I tried to get some kind of density measure within the data set (each data point represents a point in time), i.e a big blob means less stable than a dense one
I am looking for other ways to visualise my data. Any ideas welcome
Currently I am learning dense optical flow by myself. To understand it, I conduct one experiment. I produce one image using Matlab. One box with a given grays value is placed under one uniform background and the box is translated two pixels in x and y directions in another image. The two images are input into the implementation of the algorithm called TV-L1. The generated motion vector outer of the box is not zero. Is the reason that the gradient outer of the box is zero? Is the values filled in from the values with large gradient value?
In Horn and Schunck's paper, it reads
In parts of the image where the brightness gradient is zero, the velocity
estimates will simply be averages of the neighboring velocity estimates. There
is no local information to constrain the apparent velocity of motion of the
brightness pattern in these areas.
The progress of this filling-in phenomena is similar to the propagation effects
in the solution of the heat equation for a uniform flat plate, where the time rate of change of temperature is proportional to the Laplacian.
Is it not possible to obtain correct motion vectors for pixels with small gradients? Or the experiment is not practical. In practical applications, this doesn't happen.
Yes, in so called homogenous image regions with very small gradients no information where a motion can dervided from exists. That's why the motion from your rectangle is propagated outer the border. If you give your background a texture this effect will be less dominant. I know such problem when it comes to estimate the ego-motion of a car. Then the streat makes a lot of problems cause of here homogenoutiy.
Two pioneers in this field Lukas&Kanade (LK) and Horn&Schunch (HS) are developed methods for computing Optical Flow (OF). Both rely on brightness constancy assumption which feature location pixel values between two sequence frames not change. This constraint may be expressed as two equations: I(x+dx,y+dy,t+dt)=I(x,y,t) and ∂I/∂x dx+∂I/∂y dy+∂I/∂t dt=0 by using a Taylor series expansion I(x+dx,y+dy,t+dt) , we get (x+dx,y+dy,t+dt)=I(x,y,t)+∂I/∂x dx+∂I/∂y dy+∂I/∂t dt… letting ∂x/∂t=u and ∂y/∂t=v and combining these equations we get the OF constraint equation: ∂I/∂t=∂I/∂t u+∂I/∂t v . The OF equation has more than one solution, so the different techniques diverge here. LK equations are derived assuming that pixels in a neighborhood of each tracked feature move with the same velocity as the feature. In OpenCV, to catch large motions with a small window size (to keep the “same local velocity” assumption).
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Getting displacement from accelerometer data with Core Motion
Android accelerometer accuracy (Inertial navigation)
I am trying to use core motion user acceleration values, and double integrating them to derive distance covered. I move my iPhone linearly along its Y axis, against a 30 cm log ruler, on the table. First, I let the device be at rest for 10 seconds, and I calculate my offsets along the three axes, by averaging the respective user acceleration values.
The X, Y and Z offsets are subtracted from the acceleration values, when I try calculating the distance covered. After offset subtraction, these values are passed through a low pass filter and a median filter, separately of course. The filters are linear filters, and the cut-off frequency is specified by the number of neighbouring values whose mean is taken in low pass, and median in the median filter. I have experimented with varying values of this number from 1 to 100. In the end, these filtered values are double integrated using trapezoidal rule to get distances. But, the distance calculated is no where close to 30 cm. The closest value I got was some -22 cm(I am wondering why I am getting negative values even though I move the device in positive Y direction). I also came across this:
http://ajnaware.wordpress.com/2008/09/05/accelerating-iphones/
its an old post about the same thing, which says that the accelerometer readings returned appeared to come in quanta of about 0.18m/s^2 (ie. about 0.018g), resulting in a large cumulative error very quickly. Going by that, for this error to really not matter, one will have to accelerate the device by almost 1.8m/s^2, which is practically impossible for distance/length measurement purposes. for small movements, it does not look like there is a possibility of calculating distances by using an optimal filter and a higher order numerical integration method, without an impractical velocity/acceleration constraint like that. Is it possible?
How about using my acceleration vs timestamp data to interpolate a polynomial that grows over time, as I get more and more motion updates, which represents approximately an acceleration vs time curve. Double integration of ths polynomial would be a piece of cake. But, for small distances, the polynomial will have a big error component. Using a predictable known motion that my device will be subjected to, I wish to take a huge number of snapshots (calculated distance vs actual known distance) to calculate my error polynomial in a similar way, and then subtract it from my first polynomial. Can this work?
Although this does not fit StackOverflow, because it's not a question but a discussion, I'll try to sum up my thoughts about it.
As already said, the accelerometer is very inaccurate and you would need very good accuracy for this kind of task, especially if you are trying to measure such short distances. Plus, accelerometers differ from device to device, you will get different results for the same movements with different device. Plus a very huge random error.
My guess is, that you can get rid of a huge part of randomness/error by calibrating the device and making the "measurement move" a couple of times, like 10 times. After that you have enough data to get an average that might get close to the real value.
Calibration is a key part here, you have to think of a clever way to calibrate, like letting the user move the device over different distances in different speeds.
But all this is just theory. I would really like to see your results, but I doubt you get it working good enough even using the best possible filters/algorithms, since there is just too much noise.
I'm trying to read through PCA and saw that the objective was to maximize the variance. I don't quite understand why. Any explanation of other related topics would be helpful
Variance is a measure of the "variability" of the data you have. Potentially the number of components is infinite (actually, after numerization it is at most equal to the rank of the matrix, as #jazibjamil pointed out), so you want to "squeeze" the most information in each component of the finite set you build.
If, to exaggerate, you were to select a single principal component, you would want it to account for the most variability possible: hence the search for maximum variance, so that the one component collects the most "uniqueness" from the data set.
Note that PCA does not actually increase the variance of your data. Rather, it rotates the data set in such a way as to align the directions in which it is spread out the most with the principal axes. This enables you to remove those dimensions along which the data is almost flat. This decreases the dimensionality of the data while keeping the variance (or spread) among the points as close to the original as possible.
Maximizing the component vector variances is the same as maximizing the 'uniqueness' of those vectors. Thus you're vectors are as distant from each other as possible. That way if you only use the first N component vectors you're going to capture more space with highly varying vectors than with like vectors. Think about what Principal Component actually means.
Take for example a situation where you have 2 lines that are orthogonal in a 3D space. You can capture the environment much more completely with those orthogonal lines than 2 lines that are parallel (or nearly parallel). When applied to very high dimensional states using very few vectors, this becomes a much more important relationship among the vectors to maintain. In a linear algebra sense you want independent rows to be produced by PCA, otherwise some of those rows will be redundant.
See this PDF from Princeton's CS Department for a basic explanation.
max variance is basically setting these axis that occupy the maximum spread of the datapoints, why? because the direction of this axis is what really matters as it kinda explains correlations and later on we will compress/project the points along those axis to get rid of some dimensions
My lecturer has slides on edge histograms for image retrieval, whereby he states that one must first divide the image into 4x4 blocks, and then check for edges at the horizontal, vertical, +45°, and -45° orientations. He then states that this is then represented in a 14x1 histogram. I have no idea how he came about deciding that a 14x1 histogram must be created. Does anyone know how he came up with this value, or how to create an edge histogram?
Thanks.
The thing you are referring to is called the Histogram of Oriented Gradients (HoG). However, the math doesn't work out for your example. Normally you will choose spatial binning parameters (the 4x4 blocks). For each block, you'll compute the gradient magnitude at some number of different directions (in your case, just 2 directions). So, in each block you'll have N_{directions} measurements. Multiply this by the number of blocks (16 for you), and you see that you have 16*N_{directions} total measurements.
To form the histogram, you simply concatenate these measurements into one long vector. Any way to do the concatenation is fine as long as you keep track of the way you map the bin/direction combo into a slot in the 1-D histogram. This long histogram of concatenations is then most often used for machine learning tasks, like training a classifier to recognize some aspect of images based upon the way their gradients are oriented.
But in your case, the professor must be doing something special, because if you have 16 different image blocks (a 4x4 grid of image blocks), then you'd need to compute less than 1 measurement per block to end up with a total of 14 measurements in the overall histogram.
Alternatively, the professor might mean that you take the range of angles in between [-45,+45] and you divide that into 14 different values: -45, -45 + 90/14, -45 + 2*90/14, ... and so on.
If that is what the professor means, then in that case you get 14 orientation bins within a single block. Once everything is concatenated, you'd have one very long 14*16 = 224-component vector describing the whole image overall.
Incidentally, I have done a lot of testing with Python implementations of Histogram of Gradient, so you can see some of the work linked here or here. There is also some example code at that site, though a more well-supported version of HoG appears in scikits.image.