Week 7 Homework -- Nov 29, 2011
					 see images: 07week_HW_Q*.jpg

HW7Q1 Perspective Projection

 Object X -- size of thing out there somewhere
 Range Z -- distance object to lens
 Focal Length f -- distance of lens to image
 Image x  -- size of image

 dropping minus sign, want all numbers to be positive in this example...

	X(meters)	Z(meters)	f(mm)		x(mm)
   x = X * f / Z
	 10			100			 10			???			xmm = 1mm
	 10,000mm	100,000mm	 10mm

   f = Z * x / X
	 20			400			???			  1			fmm = 20mm
	 20,000mm	400,000mm				  1mm
		!!!miscopied this into the answers as 30mm !!gdmit!!! so got it wrong

   Z = X * f / x
	  2			???			 40			  1			 Zm = 80m
	  2m				     .040m		  .001m

   X =  Z * x / f
	???			300			100			  1			 Xm = 3m
				300m		.1m			  .001m


HW7Q2 Linear or Not -- BTFOM

 a function is linear if the resulting pixel is
  a linear combination of input pixels --
  could be weighted by constants or summed...but squares, etc are non-linear

  							Linear	Non-linear
  gradient kernel -1,1			L
    -- any kernel convolution is linear --

  5x5 gaussian kernel		    L
    -- see kernel above, gaussian just has many different values --

  absolute value				L  <***wrong, it is NON
   (if pix is negative just
   map back to abs value)
    -- abs value in NON-linear, "kink" at 0 where direction changes

  gradient magnitude kernel				N
   sqr root of the sum of
   image gradients:
    sqrt( Ix^2 + Iy^2 )
	-- non-lin because of squares and roots,
	   ...but I still don't understand if/how this turns into
	      the "regular" 2d edge finder kernels... --

  average brightness			L
   of all greyscale
   image pixels
   -- same as gaussian with a single multiplier value in every location --


HW7Q3 Gradient Image
  calculate a gradient image

  given a 3x3 image with pixel values:
  		
		  2	  0	  2
		  4	100	102
		  2	  4	  2
  	
  and all values outside the image are 0

  want to obtain another 3x3 image by convolving image with the kernel:
  	-1,0,1  and taking the absolute value of each pixel (ignore minus sign)
  to get a new 3x3 matrix (for each value you get a linear combination...)
  ==
		  0	  0	  0
		100	 98	100
		  4	  0	  4
  		
GDM -- I guessed the right way to go about it...


HW7Q4 Stereo
		see image: 07week_HW_Q4.jpg

 for a well calibrated stereo camera using two pinhole cameras
  whose displacement is B, distance to image is f, distance to Object is Z
  (he said distance to image but drew distance to pinholes...)
  and the displacement of the two image x's "dx" in mm,
  f is in mm, B and Z are in meters

	dx(mm)	f(mm)	B(m)	Z(m)
   Z = f * B / dx
	4		 40		 0.1	???			  Zm = 1m
	.004m	 .04m	 0.1m

   B = dx * Z / f
	0.05	 50		 ???	100			  Bm = .1m
    0.00005m 0.05m			100m	

   f = dx * Z / B
	0.1		???		 0.2	 50			 fmm = 25mm
	0.1mm			 200mm	 50,000mm

   dx = f * B / Z
	???		200		 1		 50			dxmm = 4mm
			200mm	 1000mm	 50,000mm


HW7Q5 Correspondence in Stereo
 dynamic programming approach using scan lines

 left  scan line: R R B B B R R
 right scan line: R R R B B B R
 ...obviously there's a shift going on...

 where does the left line middle B pixel (4 from left) go on the right line?

 cost of bad match (of two colors that don't correspond) = 20
 cost of (an assumed) occlusion or dis-occlusion = 10

 find the optimal alignment and check the box for the correspondence


 then change the cost of occlusion to 100
 find the optimal alignment and check the box for the correspondence

 for occlusion = 10:      R R R B B B R
                                  ^

 for occlusion = 100:     R R R B B B R
                                ^

HW7Q6 Structure from Motion 
 not full grown which is hard to do on paper

	see image: 07week_HW_Q6.jpg
 have two cameras A,B (at 90deg to each other in drawing)
  and three features to see
 each camera sees different features on Left, Center, Right (of image)
 just call features 1,2,3 (don't know which is which)

 if Camera A sees 1,2,3 L,C,R what does camera B see?

 if Camera B sees 1,2,3 L,C,R what does camera A see?

 ARE THE IMAGES INVERTED??? no, not according to the example in:
	Video 18.2 Projection Question
		see image: 18class_SFM_Quiz.jpg

 for A123 -> B 3,1,2
 for B123 -> A 2,3,1