We Keep Coding

Sampling an arbitrary point within a DFT?

What I'm trying to do: I want to compress a 2D grey-scale map (2D array of float values between 0 and 1) into a DFT. I then want to be able to sample the value of points in continuous coordinates (i.e. arbitrary points in between the data points in the original 2D map).
What I've tried: So far I've looked at Exocortex and some similar libraries, but they seem to be missing functions for sampling a single point or performing lossy compression. Though the math is a bit above my level, I might be able to derive methods do do these things. Ideally someone can point me to a C# library that already has this functionality. I'm also concerned that libraries that use the row-column FFT algorithm don't produce sinusoid functions that can be easily sampled this way since they unwind the 2D array into a 1D array.
More detail on what I'm trying to do: The intended application for all this is an experiment in efficiently pre-computing, storing, and querying line of sight information. This is similar to the the way spherical harmonic light probes are used to approximate lighting on dynamic objects. A grid of visibility probes store compressed visibility data using a small number of float values each. From this grid, an observer position can calculate an interpolated probe, then use that probe to sample the estimated visibility of nearby positions. The results don't have to be perfectly accurate, this is intended as first pass that can cheaply identify objects that are almost certainly visible or obscured, and then maybe perform more expensive ray-casting on the few on-the-fence objects.

Determine if an image is blurred

I saw a lot a topic about this, I understood the theory but I'm not able to code this.
I have some pictures and I want to determine if they are blurred or not. I found a library (aforge.dll) and I used it to compte a FFT for an image.
As an example, there is two images i'm working on :
My code is in c# :
public Bitmap PerformFFT(Bitmap Picture)
{
//Loade Image
ComplexImage output = ComplexImage.FromBitmap(Picture);
// Perform FFT
output.ForwardFourierTransform();
// return image
return = output.ToBitmap();
}
How can I determine if the image is blurred ? I am not very comfortable with the theory, I need concret example. I saw this post, but I have no idea how to do that.
EDIT:
I'll clarify my question. When I have a 2D array of complex ComplexImage output (image FFT), what is the C# code (or pseudo code) I can use to determine if image is blurred ?

The concept of "blurred" is subjective. How much power at high frequencies indicates it's not blurry? Note that a blurry image of a complex scene has more power at high frequencies than a sharp image of a very simple scene. For example a sharp picture of a completely uniform scene has no high frequencies whatsoever. Thus it is impossible to define a unique blurriness measure.
What is possible is to compare two images of the same scene, and determine which one is more blurry (or identically, which one is sharper). This is what is used in automatic focussing. I don't know how exactly what process commercial cameras use, but in microscopy, images are taken at a series of focal depths, and compared.
One of the classical comparison methods doesn't involve Fourier transforms at all. One computes the local variance (for each pixel, take a small window around it and compute the variance for those values), and averages it across the image. The image with the highest variance has the best focus.
Comparing high vs low frequencies as in MBo's answer would be comparable to computing the Laplace filtered image, and averaging its absolute values (because it can return negative values). The Laplace filter is a high-pass filter, meaning that low frequencies are removed. Since the power in the high frequencies gives a relative measure of sharpness, this statistic does too (again relative, it is to be compared only to images of the same scene, taken under identical circumstances).

Blurred image has FFT result with smaller magnitude in high-frequency regions. Array elements with low indexes (near Result[0][0]) represent low-frequency region.
So divide resulting array by some criteria, sum magnitudes in both regions and compare them. For example, select a quarter of result array (of size M) with index<M/2 and indexy<M/2
For series of more and more blurred image (for the same initial image) you should see higher and higher ratio Sum(Low)/Sum(High)
Result is square array NxN. It has central symmetry (F(x,y)=F(-x,-y) because source is pure real), so it is enough to treat top half of array with y<N/2.
Low-frequency components are located near top-left and top-right corners of array (smallest values of y, smallest and highest values of x). So sum magnitudes of array elements in ranges
for y in range 0..N/2
for x in range 0..N
amp = magnitude(y,x)
if (y<N/4) and ((x<N/4)or (x>=3*N/4))
low = low + amp
else
high = high + amp
Note that your picture shows jumbled array pieces - this is standard practice to show zero component in the center.

Why don't some images show proper Fourier and Inverse spectrums?

I am trying to develop an application for image processing.
Here is my complete code in DotNetFiddle.
I have tested my application with different images from the Internet:
Cameraman is GIF.
Baboon is PNG.
Butterfly is PNG.
Pheasant is JPG.
Butterfly and Pheasant are re-sized to 300x300.
The following two images show correct Fourier and Inverse Fourier spectrum:
The following two images do not show the expected outcome:
What could be the reason?
Are there any problem with the later two images?
Do we need to use images of specific quality to test Image-processing applications?

The code you linked to is a radix-2 FFT implementation which would work for any image with sizes that are exact powers of 2.
Incidentally, the Cameraman image is 256 x 256 (powers of 2) and the Baboon image is 512 x 512 (again powers of 2). The other two images, being resized to 300 x 300 are not powers of 2. After resizing those images to an exact power of 2 (for example 256 or 512), the output of FrequencyPlot for the brightness component of the last two images should look somewhat like the following:
butterfly
pheasant
A common workaround for images of other sizes is to pad the image to sizes that are exact powers of 2. Otherwise, if you must process arbitrary sized images, you should consider other 2D discrete Fourier transform (DFT) algorithms or libraries which will often support sizes that are the product of small primes.
Note that for the purpose of validating your output, you also have option to use the direct DFT formula (though you should not expect the same performance).

I got not time to dig through your code. Like I said in my comments you should focus on the difference between those images.
There is no reason why you should not be able to calculate the FFT of one image and fail for another. Unless you have some problem in your code that can't handle some difference between those images. If you can display them you should be able to process them.
So the first thing that catches my eye is that both images you succeed with have even dimensions while the images your algorithm produces garbage for have at least one odd dimension. I won't look into it any further as from experience I'm pretty confident that this causes your issue.
So befor you do anything else:
Take one of those images that work fine, remove one line or row and see if you get a good result. Then fix your code.

How can I compare two images to detect duplicates and cropped duplicates?

How can I compare two images and determine if they are 100% similar, or only altered in color, or cropping?

Well, abstractly speaking, you need to define a similarity function, that compares two images. To determine if the images are "100% similar" (equal) you can do the following:
compare the sizes of the images
if the image sizes are the same simply subtract the pixels from each other
if ( sum( abs( pixel_1_i - pixel_2_j ) ) / num_pixels < threshold ) return true
For the case that images are differently colored, or cropped
apply an edge detector to both images
compute the cross-correlation (in the frequency domain, FFT)
find the highest peak
place the (smaller) edge map in the determined position
calculate the absolute error
if (error < threshold) return true
BTW: This approach will not work if your images are scaled or rotated.
Further Research:
cross-correlation: FFT (fast fourier transformation, link1, link2, FFT in C#), zero-padding (needed for the FFT if the input signals have different sizes)
edge detection: Sobel, Canny (these are very common image processing filters, they should be available in a C# library, just like the FFT)

The following is a fairly simplistic approach to the problem and won't work well with two different photographs of the same subject taken from slightly different angles, but would work if you had two copies of the same image that you wanted to verify.
The case of two identical images is straightforward - just loop through the pixel arrays subtracting on RGB value from the other. If the difference is less than a small tolerance then the pixel is identical. Thus as soon as you find a pixel difference greater than the tolerance you know that the images are different.
You could allow for a certain number or percentage of differences to allow for differences causes by compression artefacts.
To check for alterations in colour you could look at the HLS (Hue, Lightness and Saturation) values instead. If the pixels have the same L & S values but a different H value then it's just the colour that's different (I think).
Cropping is more difficult as you have to try to find the location of the smaller image in the larger one.

You can use object descriptors such as:
SIFT - http://en.wikipedia.org/wiki/Scale-invariant_feature_transform
SURF - http://en.wikipedia.org/wiki/SURF
Then compare images by using calculated descriptors. Those descriptors will enable you to deal with rotated, scaled and slightly changed images.
Also the descriptors consist of oriented gradients meaning that those descriptors are robust to illumination and color changes as well.
You can use Accord.NET (SURF implementation).

How to implement smoothing in frequency domain?

I want to do smoothing to an image in the frequency domain. when i use google to see any articles it gave some Matlab codes which i don't need. i could do FFT to an image but i don't know how to implement any smoothing techniques(ILPF, BLPF, IHPF, BHPF) in frequency domain. if you can provide any code samples for any of the above techniques WITHOUT using any image processing libraries it will be really helpful and C# is preferred.
Thanks,

Could you define what you mean by 'smoothing in the frequency domain'? You can generate a spectrum image using FFT and multiply the image by some function to attenuate particular frequencies, then convert the spectrum back to an image using the inverse-FFT. However, for this kind of filtering (multiplication by some scaling function in frequency), you can achieve the same result more quickly by convolving with the dual function in the spatial domain.
In any case, if you wish to implement this yourself, read up on FFT (the fast Fourier transform) and convolution. You might also check out a signal processing textbook, if you're interested, as the theory behind discrete filtering is fairly deep. The algorithms won't make a whole lot of sense without that theory, though you can certainly apply them without understanding them.
If you want to implement your own DSP algorithms, check out this book online. In particular, Ch 33 describes the math and algorithm behind Butterworth filter design. Ch 12 describes how to implement FFT.

There is a great series on Code Project by Christian Graus which you might find useful, especially part 2 which deals amongst others with smoothing filters:
Image Processing for Dummies with C# and GDI+ Part 1 - Per Pixel Filters
Image Processing for Dummies with C# and GDI+ Part 2 - Convolution Filters
Image Processing for Dummies with C# and GDI+ Part 3 - Edge Detection Filters
Image Processing for Dummies with C# and GDI+ Part 4 - Bilinear Filters and Resizing
Image Processing for Dummies with C# and GDI+ Part 5 - Displacement filters, including swirl
Image Processing for Dummies with C# and GDI+ Part 6 - The HSL color space

Keshan, it is simple. Imagine the FFT is another two pictures where low frequencies lie in the middle and high frequencies away from the middle. If the pixels are numbered from -w/2 to w/2 and -h/2 to h/2 you can simply measure the distance from the middle as a(x,y)=sqrt(x^2+y^2). Then take some arbitrary monotonic decreasing function like f(x)=1/(1+x) and multiply each point in the fft with f(a(x,y)). Then transform back using the FFT.
There are different choices for f(x) which will look different. For example a gaussian function or bessel or whatever. I did this for my undergrad and it was great fun. If you send me a mail I will send you my program :-).
One bit caveat is the ordering in output of the fft. The arrays it generates can be ordered in weird ways. It is important that you find out which array index corresponds to which x/y-position in the "analytical" fourier transform!

For all image/signal processing I recommend OpenCV.
This has a managed C# wrapper: Emgu.
http://www.emgu.com/wiki/index.php/Main_Page