# CS 4501: Computer VisionHomework Assignment #3

## Tom Crossland (tcc4dr)

### Colorspaces and false-color visualizations

To create a false color representation of a number between 0 and 1, I wrote a matlab function called mapToColor. mapToColor has one parameter, p. p is a positive real number between 0 and 1. The function outputs a numerical triple [r g b]. These three numbers represent the color the input is being mapped to. r represents the red intensity of the color, g the green intensity, and b the blue intensity. These are positive real numbers between 0 and 1, increasing in color intensity with value. To perform this calculation, I first mapped the input p to a color in the hsv colorspace. hsv is made up of three numbers representing the hue, saturation, and value of the color. Each of these parameters are real numbers between 0 and 1. To get a unique color hue for a unique input, I set the hue value to the input p times .75. I kept the saturation and value levels at constant values of 1. This is turn produces bright distinct colors that could easily be indentified as distinct. For the hue value in the hsv colorspace, 0 corresponds to red. As the number increases, so does the color hue until it reaches one and wraps around to one again. Therefore, given constant value and saturation, two colors with hues 0 and 1 will look very similar to one another. To avoid this, I scaled p down by 3/4 for the hue value being mapped to. Therefore, all hues were in the range of red to purple instead of red to red. Because of the way I mapped the values to colors, lower inputs near zero mapped to a color close to red. In turn, inputs closer to one mapped to a color similar to purple. As the input value increases, the mapped color's hue increases, from red, following the colors of the rainbow, until purple. I then converted the hsv color triple to its respective rgb triple using the provided Matlab conversion function. Given a value between 0 and 1, I was able to produce a stable and unique color that mapped to it in both the hsv and the rgb color space.

To test my function, I wrote a test function which would perform false color imaging on different input pictures. From the input image, I was able to create an intensity or grayscale image that was representitive of it. Therefore, each pixel in the image was now an intensity value between 0 and 1. For each pixel, I called my function to map its intensity value to a specific rgb color. I then set the pixels' rgb values to the rgb triple outputted by the color mapping function. As a result, given an image as input, I was able to create a new image with color corresponding to the input image's pixel intensity. Below are five examples of false colored images produced using my color mapping function. Again, red and orange values correspond to pixels with low intensity, yellow and green correspond to medium intensity, and blue and purple correspond to high intensity.

### Optical flow

To detect optical flow, I wrote a matlab function called opticalFlow.m. This function takes 10 parameters: dir, fStart, fNum, sigma, nSize, spSize, tau, magTau, viewOpt, and record.

• dir is the name of the subfolder in which the frames of the video being used to detect optical flow in are stored. For simplicity's sake, it is assumed that this subfolder has a subfolder named 'output' to which frames with motion vectors overlayed from the video are saved to. In addition, it can be assumed that each directory contains sequentially numbered pictures starting at 1. These represent the frames of the video being looked at. Each file name is 8 digits long and is a '.jpg' image for ease of access.
• fStart is the frame number you want to start detecting optical flow at.
• fNum is the amount of frames you want to detect optical flow. fStart and fNum are thus both positive integers and fStart < fNum.
• Sigma is the width of the Gaussian blur being used. It is assumed that sigma would be a number greater than or equal to zero.
• nSize is the size of the neighborhood being considered when detecting optical flow. nSize is a positive integer. At a pixel at location (i,j), the resulting neighborhood is a square stretching from points (i-nSize,j-nSize) to (i+nSize,j+nSize). In turn, without running into a border, each neighborhood contains (2*nSize+1)^2 pixels.
• spSize represents the spacing for determining the optical flow in the frames. spSize is a positive integer. At every spSize-th pixel, a motion vector is computed or attempted to be computed.
• tau represents the threshold for determining if there exists a motion vector at a particular point. If the pixel's corresponding covariance matrix has a determinant less than tau, no solution exists.
• viewOpt determines how you want to visualize the arrows of the motion vectors in each frame. viewOpt has three possible values: 1, 2, and 3.
• When viewOpt = 1, the motion vectors are all colored green, showing the general optical flow.
• When viewOpt = 2, the motion vectors are colored based on error in the optical flow calculation.
• When viewOpt = 3, the motion vectors are colored based on stability in the optical flow calculation.
• Finally, record determines whether or not you want to record a video illustrating the optical flow on the frames. When record = 1, a video is record and saved. Else, it is not.

When determining optical flow amongst a set of frames from a video, I made some assumptions. First, I assumed that the frames would always be read in order. Therefore, for the i-th frame, the next frame being read and compared would be frame i+1. Because we assume that intensity and lighting are constant for this optical flow algorithm, frames being read in should be converted to intensity or grayscale images due to the ease of use and comparison of pixels represented by one value between 0 and 1 than by 3 values between 0 and 255. Then, the program goes into a loop processing frame by frame. Inside the loop, the program sets the current frame being processed to the former next frame. It then updates the next frame to process by reading in the next frame in sequence. To find the temporal gradiant of the frame, I simply subtracted the current frame's intensity values from the next frame's intensity values. This is simple finite differencing. The rate of change of the intensity at one pixel as a function of time is simply the difference between its value at time t+1 (the next frame) and its value at time t (the current frame). This definition won't let you compute dI/dt at the last frame. Therefore, when processing frames, we must ignore the last frame. Next the program computes the spatial gradient of each video frame by convolving it with partial derivatives of a Gaussian in the x- and y-dimensions.

At each pixel being looked at, I created an nSize neighborhood as stated before. If the neighborhood exceeded the edges of the frame, the neighborhood was cropped so it could fit without access errors. For each pixel in the neighborhood found, I made a resulting A and b matrix. The A matrix was an n by 2 matrix where n was the number of pixels in the given neighborhood. The first column of A was the spatial gradiant values of each pixel in the x direction. The second column of A was the spatial gradiant of each pixel in the y direction. b was an n by 1 matrix consisting of the temporal gradiant of each respective pixel in the neighborhood. Assuming there is no change in intensity between frames, Av + b = 0 where v represents the motion vector of the particular point being looked at. Using linear algebra, one can solve the equation for v given A and b. However, the system does not have a solution when the covariance matrix (A'A) is singular. Therefore, if the determinant of the covariance matrix is less than tau, the program assumes there is no solution for the motion vector and it skips over that point while looking for optical flow. Else, the program computes the motion vector.

At this time, the program also computes the relative error and stability of each point used to determine optical flow if there exists a motion vector at a given point. For relative error, I calculated the 'real' solutions for the change in intensity given A, v, and b. Thus, Av-b = [x y] where x and y equal the change in intensity in the x and y directions. I multiplied this resulting matrix by its transpose to get all positive numbers and summed up all of the elements in it. This gave a relative number representing how much error was there in calculating the optical flow at a particular pixel. Once all error values were determined, I divided all of them by the highest error value obtained. This normalized all of the answers, putting them between 0 and 1. Relative stability is proportional to the second largest eigenvalue of the covariance matrix of the point being looked at when determining optical flow. I thus found and stored this eigenvalue at every necessary point. This gave a relative number representing how stable it was calculating the optical flow at a particular pixel. Once all stability values were determined, I divided all of them by the highest stability value obtained. This normalized all of the answers, putting them between 0 and 1.

To avoid cluttering and unreadable results, I decided to only draw arrows representing optical flow whose magnitudes were greater than magTau. The program drew each noteworthy arrow after they were all normalized and scaled to certain lengths due to the ease of reading results. The colors of the arrows represented what the user wanted to visualize. When viewOpt = 1, the motion vectors are all colored green, showing the general optical flow. When viewOpt = 2, the motion vectors are colored based on error in the optical flow calculation. When viewOpt = 3, the motion vectors are colored based on stability in the optical flow calculation. For the latter two options, the pixel's respective values for error and stability were mapped to a color using my matlab function created earlier in the assignment. This process repeats for each frame until the second to last specified frame. Below are examples of optical flow across the video frames. For this and all following videos, I used a sigma blur value of 1, neighborhood size of 20, spacing size of 20, tau threshold for the determinant of 0.01, magTau threshold for the magnitude of the motion vectors of 1 and record on.

### hand.mp4

The video of the hand has a static background of a bookcase. Within a couple of frames, a hand comes down from the top right of the frame and moves slowly down. As seen in my videos, I was able to detect the optical flow across this video. The motion vectors were all near the hand and pointed in the direction of motion. This strongly indicates that the hand was moving in the video with nothing else. In addition, the program picked up on the shadow of the hand, indicating its movement. There were some anomalies where arrows appeared where there was no movement. This could have been because of a slight light intensity change or noise in the scene from frame to frame. This was greatly expected since there was little intensity change between frames and the movement in the video trying to be detected was a simple translation. There were some interesting finds related to which vectors showed up near the hand. There seemed to be a lack of vectors on the forehand of the subject's hand throughout the video. This could be because of lack of variation in that patch of skin. Looking at the forearm, there are little to no unique marks the program could use to determine translated motion. However, the program could do this on other parts of the arm with somewhat unique markings such as lines on the fingers and palm of the hand. This increase in texture seems to help identify motion.

In addition, one can also look at the error visualization of the optical flow. Arrows on the hand and arm tended to be red. This corresponds with a low error value. Arrows close to but not near the arm and those on the fingertips tended to be of a "higher" associated color. This indicates that there was more error in determining the motion vectors in places where it was harder to find unique places to track motion. A similar phenomenon occurred for the stability visulation, indicating a similar correlation.

### book.mp4

The video of the book has a static background of a bookcase. Within a couple of frames, a hand comes down from the top right of the frame and moves slowly down holding a book with a cactus on it. As seen in my videos, I was able to detect the optical flow across this video. The motion vectors were all near the hand and book and pointed in the direction of motion. This strongly indicates that the hand was moving with the book in the video with nothing else. In addition, the program picked up on the shadow of the hand and book, indicating its movement. Compared to the hand video, the book video did much better at displaying noteworthy motion vectors. The book moving in the video had many unique points on it - the picture of the cactus and text provided many unique points for the program to adequately detect translated motion. In my visualization, one can see how in almost every frame, the book is almost entirely covered in the correct motion vectors with no "bald" patches.

### flowers.mp4

The video of the flowers shows a camera itself moving around a scene. However, the program interprets this as if the camera is standing still and the scene moving around. Therefore, the program produced a vector field with vectors pointing opposite of the camera's movement. In almost every frame, the vector field is displayed across the whole image. This indicates that the whole scene was moving. However, when there was a patch of pixels with little variation, no arrows would be displayed. This is again due to the program not having enough unique points to adequately detect translated motion.

### traffic.mp4

The video of the traffic shows a camera attached to a moving car and observing what it sees. However, the program interprets this as if the camera is standing still and the scene is moving around. Therefore, the program produced a vector field with vectors pointing opposite of the car's movement. In almost every frame, the vector field is displayed across the foreground. This indicates that the foreground was moving. However due to perspective, smaller items farther away did not display too many arrows. This is because the deeper one goes in a field, the smaller the apparent change in movement is. Therefore, something moving at the same velocity closer to the camera will be more strongly detected than something further away moving at the same rate. However, when there was a patch of pixels with little variation, no arrows would be displayed. Some examples of this included patches of the sky and road. This is again due to the program not having enough unique points to adequately detect translated motion.