Keywords: neural net, image processing, phase shift holography, noise reduction
The mathematik of phaseshift can be visualized, using the IDS. The combination of the three intensities Ik at one image position is a vector in the IDS.
Fig.1: Composition of the Input--Data--Space (IDS). The holograms 1,2 and 3 constitute a set of phase shifted holograms of a MgO-crystal. The set of intensities I(1),I(2),I(3) measured in the holograms for each object point is interpreted as vector I in the IDS in the lower right diagram. Calculation shows that ideally all points lie on the surface of a paraboloid.
We rotate the IDS, to align the symmetry axis of the paraboloid with the z-axis of our coordinate system. The surface of the paraboloid represents the Gaussian plane. Different parameters in a phase-shift hologram lead to different shapes of the Gaussian plane shown in figure 2.
Fig.2a: contrast V=0.1 Fig.2b: contrast V=0.3 Fig.2c: contrast V=1.0
The intensity vector I in real holograms is always noise corrupted, this means it doesn´t point exact to the Gaussian plane. Best results are obtained, if the reconstruction process finds the minimum distance to the Gauss plane and if the surface is shallow in the IDS.
Fig.3: Top : Neural network before learning process. Bottom : Neural network after learning process for holograms with fringe contrast V=0.3. For readout of the neuron values, the neuron with the minimum distance to the intensity values at the position in the hologram is selected: its value represents the reconstructed image wave.
First Result :
We feeded our neural network with the three holograms in figure 1. The resulting amplitude and phase calculated from the complex result is shown in figure 4. Fig.4: Amplitude and phase of the hologram.
Where is the information in a hologram?
Every pixel i,j of the CCD contains information of the intensity of the image wave collected by this pixel. fig.5 This intensity I(x,y) results from the intensity-equation: (1)
The resulting intensity I in a pixel i,j is the integral over the pixelarea: (2)
a,b,u,v as in figure 7 for pixel 1 We are usually interested in amplitude and phase of the image wave (we suppress Iinel). This information is hidden, so we have to reconstruct these values. To solve an equation with two variables we need at least two equations. For reasons of symmetry, it is more convienient to use four equations. Therefore we look at a small spot of the CCD. figure 6
If amplitude and phase are constant over the area of pixel I, II, III and IV only the phase of the reference wave between the pixels is different -> we can apply the phase-shift calculation solving four equations. The overdetermination gives us a chance to reduce the noise.
We see, best results are obtained if we use space frequency q=0, but there is another condition: To use the equation system, the equations have to be linear independent. The phase shift between the four pixels has to be optimized. In the picture of the IDS, figure 2, the shape of the Gauss plane should be shallow resulting in a large volume surrounded by the Gauss plane. Figure 8 shows this volume depending on fringe direction and space frequency. fig. 8 This condition leads to space frequency q=1, combining these two demands lead to an optimum for the retrieved information if there is noise. fig. 9 The optimum fringe space frequency is q=0.8 / pixelwidth and the orientation is optimum at an rotation angle of =60° between fringe and CCD-camera. Reconstructing conventional holograms with phase shift calculation In this example we have reconstructed a hologram from using the intensity of nine pixels. The calculation is based on minimizing of the distance with ; the sufix ´in´ is for measured intensity while ´sim´ represents the calculated value using the intensity equation (2). The resulting amplitude and phase are shown in the following two images. fig 10
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