Sunday, May 6, 2018

Affine Transformation

I've been working on a personal project for data presentation and exploration. It is very much a work in progress but as part of the development I've come across some interesting little technical problems to solve. I hope to eventually have a longer series of posts along the lines of 'building your own data tools;' however, until I find the time to give that the attention it needs, I plan on posting some smaller entries centered around these technical problems.

One of the early challenges I came across when plotting data was mapping the input values to the fixed amount of space I had on screen. For example, if I generate a window that is 600px wide I can only support a range of input values less than or equal to 600 (using a naive approach of each value in the range maps to a pixel on screen). Supporting an input range greater than 600 values requires some level of mapping between the input values and the available pixels on the screen. There are simple conversions for specific input ranges (e.g. when the range is close to a multiple of the number of pixels) but a more general solution is necessary to handle arbitrary input data sets.

The way I ended up solving this was to use an affine transformation to map the input domain to a fixed pixel range. This is how scales work in D3.js if you are familiar with that JavaScript library. For example, if my input values come from [-100, 100] but I need my output values to fall within [28,63] then applying this transformation would look something like the image below:

Notice that the minimum value in the input interval (-100) maps directly to the minimum value in the output interval (28). Values along the input interval are mapped to the output interval preserving the relative position within each. The equation to do this is:

\[ range_{x} = (domain_{x} - domain_{a}) * ((range_{b} - domain_{a}) / (domain_{b} - domain_{a})) + range_{a} \]

Where domain and range are defined by the intervals \([domain_{a}, domain_{b}]\) and \([range_{a},range_{b}]\), respectively. Neither the domain nor the range is required to have a 0 as one of the endpoints and the intervals can be either increasing or decreasing. This generality is helpful in a variety of contexts. For example:

Setting Borders
The viewport used to plot data may not always use all available pixels of the window; multi-plot windows or viewports that have a border are two examples. In the case where there is a border around the plot (allowing for labels and tick marks) that area should not be considered as part of the range of the viewport. Instead, a window with 600px width that uses 30px on either side for border should use a range of [30,570] - the equation above makes this easy.

Inverted Axis
In most window coordinate systems the origin is at the top left of the screen. That means increasing y-axis values (from the window perspective) are typically decreasing data set y-axis values. Using a negative range makes this translation simple. Consider a 600px window height: mapping a domain to the range [600,0] automatically handles the inverted axis problem.

Another way that this simplifies things is when there is a need to 'flip' the layout of a graph. For example, in the two images below the data is the same; the only difference is the range. Having the ability to easily manipulate target ranges for the input data allows me to cycle through multiple views of the same data without much work on the backend.


Friday, May 12, 2017

Visualizing Network Traversal

As part of an experiment I was doing related to network traversal I started to trying to visualize coverage of a network by a single visitor traveling through the nodes. The idea was to set a gradient between two colors and as a node was visited more times it would slowly take on a change in color over the gradient. If the goal was to visit each node N times, then the visualization would indicate current progress as a 'heat map' of the network according to the number of visits to each node.

The outputs were interesting in that controlling the traversal policy would change the distribution of color over the image. This is perhaps obvious given the experimental construct but still provided an alternate mechanism for 'measuring' the accuracy of the traversal. For example, if the goal is to cover the network as evenly as possible during the traversal, the kurtosis of a distribution then becomes a measure of how close one is to this goal.

To demonstration this I set up an NxN grid network and ran three example traversals over that grid. I captured the 'heat map' when the median of all the visit values for the nodes was at half of the target visits. The results are below.

Random Choice Walk:
At any node, a visitor will choose to move up, down, left, or right by random selection.

Backpressure Walk:
Uses a notion of backpressure to control the traversal. At each node, a visitor checks for the the least visited neighboring node and travels there.

Not really a traversal, I just did it to see what it would look like. At each node, a visitor chooses a random node within the grid and jumps there directly.

Sunday, October 2, 2016

Water Bleed Image Effect

Recently, I had the occasion to play with some features of the amazingly powerful ImageMagick image manipulation tool. After playing with several of the command-line tools and manipulations I decided to look into the API provided through the MagickWand C interface to ImageMagick. To explore some of the features available at this layer I'm attempting to develop a bleed effect on an image; similar to if you had dipped a picture in water and then hung it up to dry. I'm hoping that my free time permits and I can turn this into a small series of posts detailing how I refine this tool (if I can eventually call it that).

Perhaps in a later post I'll get into what the code looks like and a dialog of what each part does but for now I'll just put up some of the early results and learning points that I've encountered. Ultimately, I'll get the code up on github and follow along with that here.

I'm using two images for this exercise: one that contains a rainbow of colors to test and another to represent the effect on a more likely picture.

Initially, I looked at just 'dragging' each color down the image while increasing the normalization scale along the vertical to give a wash out effect further down the image. This represents how the water would continue to wash color away as it 'pulled' higher colors down the hanging image. This worked well for the rainbow version of the image:

That result has a good 'bleed' in that stronger colors last longer down the vertical. However, with the colors in the house image the washout was too aggressive and wiped out much of the original content.

If you ignore the washout, though, it is clear that the draw of colors down the image is visible (especially around the windows) meaning that reducing the washout might still produce a more realistic effect if enough of the original image could be preserved.

With that in mind, I added the ability to preserve a percentage of the original image while applying the effect. This allows a user to tune the washout per image to get just the right look. For example, too much reuse (20%) on the house and the washout effect has little impact:

But taking that down to 7.5% gets closer to what I am looking for:

There is still plenty for me to do such as handling the edges and water line better and further manipulating the washed image. For example, adding additional transforms to the result to get more seep/bleed on the washed colors. The image below shows a simple kernel convolution to add blur to the pencils, for example.

Friday, May 6, 2016

Derive formula to convert Fahrenheit to Celsius

I had been revisiting linear regression the other day and as part of that review I challenged myself to use regression to derive a well known formula without actually looking it up (the formula, that is).

The first example that came to my mind was the formula for converting temperature in Fahrenheit to Celsius. I wanted to see if I could derive that formula using two sample data sets and a simple linear regression. If the data was accurate enough, I should be able to derive the exact equation for converting between the two formats. In essence, I wanted to be able to come to the following:

C = (F - 32) * 5/9

Since I didn't have a data set with both types of observations available I was faced with a little 'chicken or the egg' situation. Seeing how this is just a fun little exercise I generated my own data introducing some artificial error to stand in for true observations.

After the 'observations' were available the regression was as simple as loading the data into R and running lm. I ran through the entire manual procedure of how this works in a previous post so wont repeat it here. The result of calling lm is a list and one of the elements of that list is the coefficients - these represent the intercept and slope of:

y = mx + b

Since the Celsius observations are the response in my formula and the Fahrenheit observations are the predictors the I can create a similar equation where y represents the Celsius values and x represents the Fahrenheit values. Given that, I get the following (after plugging in the slope and intercept):

C = 0.555547 * F - 17.772318

Expanding the original equation for converting between Fahrenheit and Celsius yields:

C = (F * 5/9) - (32 * 5/9)
C = F * 0.555556 - 17.777778

So, given observations in both Celsius and Fahrenheit (for the same events, of course) it is possible to derive an equation to convert between the two using linear regression.

My observations are very highly correlated. Obviously, as this correlation falls the accuracy of the resulting equation will suffer. Fortunately there are tools to measure the correlation which helps quantify this accuracy.

You can find the code for this exercise on github.

Monday, December 1, 2014


I have written before about some of the differences between Ruby and Python and my quirks generally tend to the Ruby approach. I think readability is another dimension between the two languages that highlights this for me - especially as it applies to understanding new code.

I prefer to read code from left-to-right (LTR), top-to-bottom. This is natural for me as it models how I read other text. Code that processes right-to-left (RTL) and, in the severe case, bottom-to-top challenges my ability to easily understand intent. Method chaining highlights this quite nicely. For example, to transform the numbers of a list stored as a string in Python one might write:
','.join(map(lambda x : str(int(x) ** 2), "1,2,3,4".split(',')))
If I am reading that for the first time I need to mentally maintain a stack of operations (join, map, lambda) until I've parsed most of the statement and arrived at the object being operated on: "1,2,3,4". This is due to the RTL application of the code. I've then got to backtrack over each operation on my mental stack to understand what the type and/or result of the overall statement will be. This is complicated by the fact that Python allows for some LTR ("1,2,3,4".split(',')) mixed with the RTL.

For first-time readers of the language this process is even more difficult if the behavior of join or map are not yet well understood.

Ruby makes this significantly easier.
"1,2,3,4".split(',').map { |x| x.to_i ** 2 }.join(',')
When I read code similar to that I can store the type and result as I am parsing the statement. The initial object is immediately available and I can read the expression LTR as: split a string, apply a function to each element of the resulting array, and join that final array with the comma character. The fact that Ruby supports method chaining (on the built-in types) makes for much more readable code.

I've singled out Python above but that was only for the sake of the example. As far as RTL languages go I think Python is middle of the road. Haskell, for example, has a much nicer syntax to deal with function composition (a similar, but not identical situation). On the other end of the spectrum is Lisp which is basically a bottom-to-top, RTL language.

I can (and have) used these languages and many more; RTL vs. LTR in no way prevents one from being proficient over time. Certainly, most RTL code can be written in a way that it flows mostly LTR, top-to-bottom. Even when it isn't, well-written code can read by anyone with enough practice. For newcomers looking to read a new language however, there is less difficulty when the process more closely models how they read in general.

Tuesday, November 25, 2014

Order of Events

inet_ntoa uses static storage for the result it returns. The GNU implementation of inet_ntoa uses the following internally:

static __thread char buffer[18];

This makes the function thread-safe but this safety does not remove the need to worry about use within a single thread. Consider the following snippet of code:

#include <arpa/inet.h>
#include <stdio.h>
int main () {
    struct in_addr a = { .s_addr = 1234567 }, b = { .s_addr = 7654321 };
    return printf ("%s : %s\n", inet_ntoa (a), inet_ntoa (b));

Since inet_ntoa is used twice within the argument list the result is dependent on the order of evaluation of the arguments to printf. Regardless of which call gets evaluated first the output will always print the same IP address twice. On my system, the result is: :

This is a result of two things: arguments are evaluated before their results are used by printf; and inet_ntoa overwrites static storage on each invocation. Looking at the instructions for this C code makes this clear:

        .string "%s : %s\n"
        .globl  main
        .type   main, @function
        pushl   %ebp
        movl    %esp, %ebp
        pushl   %ebx
        andl    $-16, %esp
        subl    $32, %esp
        movl    $1234567, 24(%esp)
        movl    $7654321, 28(%esp)
        movl    28(%esp), %eax
        movl    %eax, (%esp)
        call    inet_ntoa
        movl    %eax, %ebx        ; pointer stored to static memory
        movl    24(%esp), %eax
        movl    %eax, (%esp)
        call    inet_ntoa
        movl    $.LC0, %edx
        movl    %ebx, 8(%esp)     ; arg2 to printf; pointer from above
        movl    %eax, 4(%esp)     ; arg1 to printf; new pointer, same static memory
        movl    %edx, (%esp)      ; arg0 (format string)
        call    printf
         movl    -4(%ebp), %ebx

The correct way to call inet_ntoa consecutively is to save each result to a local variable.

#include <arpa/inet.h>
#include <string.h>
#include <stdio.h>
int main () {
    struct in_addr a = { .s_addr = 1234567 }, b = { .s_addr = 7654321 };
    char ipa[18] = { 0 }, ipb[18] = { 0 };
    strcpy (ipa, inet_ntoa (a));
    strcpy (ipb, inet_ntoa (b));
    return printf ("%s : %s\n", ipa, ipb);

Sunday, October 5, 2014

Automating keystrokes via evdev

In a previous post I talked about how to capture keys out from under the X11 windowing system by reading from /dev/input/eventX. These character devices can also be useful to generate input simulating keyboard activity.

I circled back to this topic after having to automate user keyboard activity. I've accomplished similar tasks in the past with a tool named xdotool - unfortunately, in this case I did not have the luxury of being able to install software. The remainder of this post highlights the differences between consuming and producing events. (By the way, If you have the need to automate X actions I highly suggest looking at what xdotool can do for you.)

Consuming events is the easier of the two tasks: you simply read open the device and read events into the following structure:

/* See: /usr/include/linux/input.h */
struct input_event {
    struct timeval time;
    __u16 type;
    __u16 code;
    __s32 value;

Filter input with type == 1 and read the code to get the key and value to get the event (eg. press, release).

To produce a compliant event the process is a little more complicated since the input needs to be synchronized. For each event there are three distinct sets of data that are required: setup (EV_MSC); the event (EV_KEY); and event synchronize (EV_SYN). In addition to that, certain events are captured over time so this is a stateful process. An example of this is pressing Ctrl-L; the control key is held down while another key is pressed and then released.

The easiest way I found to initially grok the protocol is to capture all events while there is keyboard activity and see what the output looks like. Obviously, to produce fully compliant input you should consult API documentation or source code.

An example of automatically entering a URL in the Chrome browser (Ctrl-L [URL]) would require the following inputs (the type, code, and value members of struct input_event). The input goes to the focused window (the standard behavior for X) so you need to place focus on the Chrome window for the following example.

4,  4, 29  # Setup  
1, 29,  1  # Press Ctrl key
0,  0,  0  # Sync
4,  4, 29  # Setup  
1, 29,  2  # Ctrl (value == 2 -> autorepeat)
0,  0,  0  # Sync
4,  4, 38
1, 38,  1  # Press 'L' key
0,  0,  0
4,  4, 38
1, 38,  0  # Release 'L' key
0,  0,  0
4,  4, 29
1, 29,  0  # Release Ctrl key
0,  0,  0
# and so on for the URL string 

4,  4, 28
1, 28,  1  # Press Enter key
0,  0,  0
4,  4, 28
1, 28,  0  # Release Enter key
0,  0,  0