Python 2, Score: 1.205

I put together a quick Python solution some time ago but forgot to post it. Here it is. It finds the richest blocks in the image, then travels to each block, eating all the best blocks it comes to.

Results

bridge.jpg: 591.97866/515.41501 =                               1.14855 
BallsRender.png: 493.24711/387.80635 =                          1.27189 
Mandel_zoom_04_seehorse_tail.jpg: 792.23990/624.60579 =         1.26838 
Red_Vortex_Apophysis_Fractal_Flame.jpg: 368.26121/323.08463 =   1.13983 
The_Scream.jpg: 687.18565/555.05221 =                           1.23806
swirl.jpg: 762.89469/655.73767 =                                1.16341

AVERAGE                                                         1.205

Example Picture

Python 2.7 Code

from pygame.locals import *
import pygame, sys, random

fn = sys.argv[1]

screen = pygame.display.set_mode((1900,1000))
pic = pygame.image.load(fn)
pic.convert()
W,H = pic.get_size()
enough = W*H/3
screen.blit(pic, (0,0))

ORTH = [(-1,0), (1,0), (0,-1), (0,1)]
def orth(p):
    return [(p[0]+dx, p[1]+dy) for dx,dy in ORTH]

def look(p):
    x,y = p
    if 0 <= x < W and 0 <= y < H:
        return sum(pic.get_at(p))
    else:
        return -1

# index picture
locs = [(x,y) for x in range(W) for y in range(H)]
grid = dict( (p,sum(pic.get_at(p))) for p in locs )
rank = sorted( grid.values() )
median = rank[ len(rank)/2 ]
dark = dict( (k,v) for k,v in grid if v < median )
good = dict( (k,v) for k,v in grid if v > median )
pictotal = sum(rank)
picavg = 1.0 * pictotal/(W*H)
print('Indexed')

# compute zone values:
block = 16
xblocks, yblocks = (W-1)/block, (H-1)/block
zones = dict( ((zx,zy),0) for zx in range(xblocks) for zy in range(yblocks) )
for x,y in locs:
    div = (x/block, y/block)
    if div in zones:
        colsum = sum( pic.get_at((x,y)) )
        zones[div] += colsum

# choose best zones:
zonelist = sorted( (v,k) for k,v in zones.items() )
cut = int(xblocks * yblocks * 0.33)
bestzones = dict( (k,v) for v,k in zonelist[-cut:] )

# make segment paths:
segdrop = [(0,1)] * (block-1)
segpass = [(1,0)] * block
segloop = [(0,-1)] * (block-1) + [(1,0)] + [(0,1)] * (block-1)
segeat = ( segloop+[(1,0)] ) * (block/2)
segshort = [(1,0)] * 2 + ( segloop+[(1,0)] ) * (block/2 - 1)

def segtopath(path, seg, xdir):
    for dx,dy in seg:
        x,y = path[-1]
        newloc = (x+dx*xdir, y+dy)
        path.append(newloc)

# design good path:
xdir = 1
path = [(0,0)]
segtopath(path, segdrop, xdir)
shortzone = True
while True:
    x,y = path[-1]
    zone = (x/block, y/block)
    if zone in bestzones:
        if shortzone:
            seg = segshort
        else:
            seg = segeat
    else:
        seg = segpass
    segtopath(path, seg, xdir)
    shortzone = False
    # check end of x block run:
    x,y = path[-1]
    zone = (x/block, y/block)
    if not ( 0 <= x < xblocks*block ):
        del path[-1]
        segtopath(path, segdrop, xdir)
        shortzone = True
        xdir = xdir * -1
    if len(path) > enough:
        break
print('Path Found')

# show path on picture:
loc = path.pop(0)
eaten = 1
joetotal = grid[loc]
i = 0
while eaten <= enough:
    loc = path[i]
    i += 1
    pic.set_at(loc, (0,0,0))
    joetotal += grid[loc]
    eaten += 1
    if i % 1000 == 0:
        screen.blit(pic, (0,0))
        pygame.display.flip()
        for event in pygame.event.get():
            if event.type == QUIT: sys.exit(0)

# save picture and wait:
screen.blit(pic, (0,0))
pygame.display.flip()
pygame.image.save(pic, 'output/'+fn)
joeavg = 1.0 * joetotal/eaten
print '%s: %7.5f/%7.5f = %7.5f' % (fn, joeavg, picavg, joeavg/picavg)
while True:
    for event in pygame.event.get():
        if event.type == QUIT: sys.exit(0)